Lattice Kinetic Methods & Thermal Rectification Phenomena

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Transport: Lattice KineticMethods & Thermal RectificationPhenomena

Doctoral Thesis

Author(s):Solórzano, Sergio

Publication date:2017

Permanent link:https://doi.org/10.3929/ethz-b-000228259

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Diss. ETH No. 24682

Transport: Lattice KineticMethods & Thermal

Rectification Phenomena

A thesis submitted to attain the degree of

Doctor of Sciences of ETH Zurich

(Dr. sc. ETH Zurich)

presented by

Sergio Daniel Solorzano Rocha

MSc. Physics, ETH Zurich

born 11.10.1989

citizen of Colombia

accepted on the recommendation of

Prof. Dr. Hans J. Herrmann, examiner

Prof. Dr. Sauro Succi, co-examiner

2017

Acknowledgments

Unas gracias infinitas a mi mama, Tomasa, mi papa, la pilla, y el nuevo

miembro de la familia Cutu la pantera negra, por el apoyo incondicional du-

rante este proyecto 24/7/365 en cualquier continente y huso horario. Merci

beaucoup ma petite amie Rocıo pour couter et me montrer que tout est juste

de petits morceaux de la mosaıque de la vie. To Prof. Dr. Hans Herrmann

deep gratitude for giving me the opportunity to work on very interesting

and challenging problems in a very rich environment. Under his supervision

I learnt not only about science, but also and perhaps more important how

to do science. To Dr. Miller Mendoza Deep and Wide gratitude for his vital

guidance and support throughout these years, his ideas and insights always

enriched my work. He patiently helped me through every project, always

having plenty of time for good news and just a little for bad news, but al-

ways having time to guide, discuss and teach. I am also grateful with Miller

for pushing me the right amount and teaching me that things can be done.

To Prof. Dr. Sauro Succi, for the interesting ideas and input regarding the

Wigner function as well as for his patience and thoughtful comments in, the

perhaps, most complex project of my Ph.D. To Prof. Dr. Nuno Araujo for

his patient guidance in the, rich and full of surprises, ratchet project. A

los amigos de la U: J, Henry-Kathe-Chiara, Dani, Alfred, Checho, Laura,

Dra. L. Martin, Nico, El paisa, Juan Pablo del Risco, Monica, Nataly, Juan

David, Rafa, por todas las pendejadas alegres, paseos a Mariquita, Holanda,

Alemania, tareas innecesarias y cosas serias. A mis amigos Luis German y

Angelica por nunca perder el contacto y siempre tener cosas para hablar.

To the set of Zurich friends: Sidharta-Sidharta, Ekin, Ryan, Deep, Nastya,

i

Tom, Hassam, Leila, Tim, Timo, Mathia, Jan, Raphaela, Gabriel, Nishant

Johu. Too many movies, dinners, conversations to remember. Thanks for

the great time together. To the group friends: Robin, Jens-Daniel, Ilario,

Farhang, Lucas, Ryuta, Oliver, Dominik, Trivik, Gautam, Kyriakos, Lau-

rens, Pavel, Kornel, Mirko, Jan, Rodrigo, Juliana. I learn from them, and

they made the time at the group a very enjoyable experience. Thanks for

the time together.

Contents

Zusammenfassung i

Abstract iii

Related publications v

1 Introduction 1

1.1 Overview and motivation . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Structure and organization . . . . . . . . . . . . . . . . . . . 7

2 General formulation of lattice kinetic methods 9

2.1 Lattice kinetic methods . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 From Boltzmann to Lattice Boltzmann . . . . . . . . 9

2.1.2 Velocity space discretization . . . . . . . . . . . . . . 11

2.1.3 Equilibrium and source term distributions . . . . . . 14

2.1.4 Algorithmic details . . . . . . . . . . . . . . . . . . . 14

3 Lattice kinetic approach to Density Functional Theory 17

3.1 Density functional theory and imaginary time propagation . 17

3.1.1 Density functional theory . . . . . . . . . . . . . . . 17

3.1.2 Imaginary time propagation . . . . . . . . . . . . . . 20

3.1.3 Kinetic approach to DFT . . . . . . . . . . . . . . . 21

3.2 Lattice Kinetic model for DFT . . . . . . . . . . . . . . . . . 26

3.2.1 Formal correction of discrete lattice effects . . . . . . 26

3.2.2 Equilibrium and source distributions construction . . 29

3.2.3 Semi implicit correction . . . . . . . . . . . . . . . . 30

3.3 Application of the improved lattice kinetic approach . . . . . 31

3.3.1 Model comparison . . . . . . . . . . . . . . . . . . . 31

3.3.2 Ethane molecule test . . . . . . . . . . . . . . . . . . 35

3.3.3 Pseudopotentials . . . . . . . . . . . . . . . . . . . . 35

3.3.4 Lattice performance . . . . . . . . . . . . . . . . . . 38

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Lattice Wigner Model 41

4.1 Wigner formalism . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Lattice Wigner model . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.1 Harmonic potential . . . . . . . . . . . . . . . . . . . 48

4.3.2 Anharmonic potential . . . . . . . . . . . . . . . . . 52

4.4 Computational cost . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Lattice Wigner application . . . . . . . . . . . . . . . . . . . 56

4.5.1 1D system . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5.2 2D system . . . . . . . . . . . . . . . . . . . . . . . . 62

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Thermal gas rectification using a sawtooth channel 67

5.1 Ratchet systems . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Canonical molecular dynamics . . . . . . . . . . . . . . . . . 69

5.2.1 Molecular dynamics . . . . . . . . . . . . . . . . . . . 69

5.2.2 Predictor corrector integration . . . . . . . . . . . . . 69

5.2.3 Canonical molecular dynamics . . . . . . . . . . . . . 71

5.3 Model and methods . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Conclusion and outlook 83

Appendix 85

A.1 Wigner function derivation . . . . . . . . . . . . . . . . . . . 85

A.2 Generalized Hermite polynomials . . . . . . . . . . . . . . . 89

A.3 Wigner Forcing term calculation . . . . . . . . . . . . . . . . 89

A.4 General Wigner function calculation . . . . . . . . . . . . . . 91

A.5 Multiple barriers effective transmission coefficient . . . . . . 93

A.6 Lattice specification . . . . . . . . . . . . . . . . . . . . . . . 93

References 97

Zusammenfassung

Diese Arbeit besteht aus zwei Teilen. Im Ersten verfolgen wir die Idee

die sogenannten Lattice-Kinetic Methoden, d.h. Erweiterungen der Lattice-

Boltzmann Methode, welche normalerweise fr Fluiddynamik verwendet wer-

den, auf andere Gebiete anzuwenden. Konkret wird eine neue halb-implizite

Korrekturmethode zweiter Ordnung fur das kinetische Kohn-Sham Modell

[1] entwickelt. Mit der neuen Methode werden die Austauschkorrelations-

energien von Atomen und Dimeren fur die ersten beiden Reihen des Peri-

odensystems berechnet und mit Erfahrungswerten verglichen. Dabei wur-

de eine ausgezeichnete Ubereinstimmung gefunden. Des Weiteren wird das

Ethanmolekul simuliert, wobei die Bindungslangen bestimmt und mit Stan-

dardmethoden vergleichen werden. Zuletzt wird gepruft inwiefern Pseudopo-

tentiale im Lattice-Kinetic Kohn-Sham Ansatz Anwendung finden konnen.

Wir verfolgen die Idee Lattice-Kinetic Methoden jenseits der Hydrodynamik

zu untersuchen weiter und entwickeln eine neue Methode um die Wigner-

Gleichung zu losen. Wir validieren den Ansatz fur die Falle eines harmo-

nischen und unharmonischen Potentials. Zudem untersuchen wir ein- und

zweidimensionale offene Quantensysteme mit Potentialbarrieren. Wir zeigen

auf, dass die Methode auch auf dreidimensionale Systeme erweitert werden

kann.

Im zweiten Teil untersuchen wir die Gleichrichtung eines zweidimensionalen

thermischen Gases in einem Kanal mit dissipativen Wanden. Mithilfe von

Molecular-Dynamics Simulationen zeigen wir, dass ein Gas aus Lennard-

Jones Partikeln gleichgerichtet werden kann, sofern es eine raumliche Asy-

metrie und Dissipation im System gibt. Ausserdem demonstrieren wir, dass

i

es eine nichtmonotone Abhangigkeit des Partikelflusses von Systemparame-

tern wie z.B. der Thermostattemperatur, der Kanalasymmetrie und Parti-

keldichte gibt, und finden drei Bereiche mit einer jeweils charakteristischen

Dynamik.

ii

Abstract

This work consists of two parts, in the first one the idea of using lattice

kinetic methods i.e. extensions of the Lattice Boltzmann method to areas

different than fluid dynamics is explored. Particularly, a new semi-implicit

second order correction scheme to the kinetic Kohn-Sham lattice model in-

troduced in Ref.[1] is developed. The new approach is validated by perform-

ing realistic exchange-correlation energy calculations of atoms and dimers

of the first two rows of the periodic table finding good agreement with the

expected values. Additionally, we simulate the ethane molecule where we

recover the bond lengths and compare the results with standard methods.

Finally the current applicability of pseudopotentials within the lattice kinetic

Kohn-Sham approach is also discussed. Continuing with the idea of study-

ing lattice kinetic methods beyond hydrodynamics, a lattice kinetic scheme

to solve the Wigner equation was developed. The approach was validated

for the case of quantum harmonic and anharmonic potentials, showing good

agreement with theoretical results. It was also applied to the study of the

transport properties of one and two dimensional open quantum systems with

potential barriers. The computational viability of the scheme for the case of

three-dimensional systems is also illustrated.

In the second part, the subject of the rectification of a two-dimensional

thermal gas in a channel of asymmetric dissipative walls is studied. We

use molecular dynamics simulations to show that a gas of Lennard-Jones

particles can be rectified provided there is spatial asymmetry and dissipation

in the systems. Further, we find that there is a non-monotonic dependence

of the particle flux on systems parameters such as thermostat temperature,

iii

channel asymmetry and particle density, with three distinct regimes.

iv

Related publications

This thesis contains content of the following published or for peer-review

submitted articles:

• S. Solorzano, M. Mendoza, and H. J. Herrmann, “Second order kinetic

Kohn-Sham lattice model,” Phys. Rev. A., vol. 93, p. 062504, 2016.

(arXiv:1605.05032v1)

• S. Solorzano, M. Mendoza, S. Succi and H. J. Herrmann, “Lattice

Wigner equation,” submitted to Phys. Rev. E.

• S. Solorzano, N. A. M. Araujo, and H. J. Herrmann, “Thermal gas

rectification using a sawtooth channel,” Phys. Rev. E., vol. 96, p.

032901, 2017 (arXiv:1706.04429)

For each of these three articles, the author contributed the most effort,

including: design, implementation, and execution of all simulations; analysis

and interpretation of all data generated from simulations; all conceptual

and mathematical derivations; execution of all numerical analyses; literature

research; writing of the manuscript; and design and creation of all figures.

v

Chapter 1

Introduction

1.1 Overview and motivation

This work is composed of two parts. The first one is of quantum nature and

explores the relations between Lattice Kinetic methods, Density Functional

Theory and Wigner Function formalism. The second part is classical and

explores the possibility of using thermal fluctuations, spatial asymmetry and

dissipation to rectify the motion of a gas of particles.

1.1.1 Part I

According to some accounts, Quantum Mechanics is no more than linear

algebra with some physics into it. This is clearly a crass oversimplification

however, there is a bit of truth to it. This impression is well deserved if

the only thing one learns or aims to do is just to diagonalize matrices as

it is usually done at introductory levels. Physical understanding requires

two ingredients to progress, deeper ideas and appropriate tools to handle

these ideas. In the following, we overview two very interesting examples of

the first ingredient, the Density Functional Theory (DFT) and the Wigner

Formulation of quantum mechanics. Later, we comment on how they can

both be handled with the highly versatile tool of Lattice Kinetic Models.

The starting point for the description of microscopic systems is usually

Schrodinger’s equation. It can describe for instance, the dynamics of the

1

CHAPTER 1. INTRODUCTION

protons and electrons that made up matter, ultimately defining electrical,

optical, magnetic and thermal characteristics of materials. In general the

set up of the Schrodinger equation is relatively simple, its solution however

is not. In order to provide a perspective on the DFT approach to the many

electron problem, we provide a concise historical account of the methods

that have been developed to solve the many particle Schrodinger equation.

One of the first systematic approaches to solve Schrodinger’s equation for

a system of many electrons dates back to D.R.Hartree [2] 1928 method. It

was later improved by V. A. Fock [3](1930) by the use of trial wave func-

tions that respected the Fermi exclusion principle and is currently known

as the Hartree-Fock method. In essence, it is a variational approach that

enforces the physically correct symmetry of the wave function. Around the

same time Hylleraas [4, 5] studied the Helium atom problem obtaining good

agreement with experiments. Instead of a single wave function as in the

Hartree-Fock method, he used a superposition of them [6] effectively per-

forming a Configuration Interaction calculation. Also in 1930, Fermi devel-

oped the idea of effective potentials to study scattering problems, providing

the foundation to the modern concept of pseudopotentials [7]. Pseudopoten-

tials are a theoretical as well as a computational tool that allows to reduce

the number of studied electrons by including additional potential terms in

the Schrodinger equation while preserving most of the physics of the system.

The approximation is based on the observation that most of the chemical

and macroscopic properties of materials are determined by the outer most

electrons [8]. Various types have been developed over the years. Some of the

most popular ones, due to their high accuracy and transferability properties,

are Vanderbilt ultrasoft pseudopotentials [9], the family of pseudopotentials

designed by J. Hutter et. al. [10, 11] that include relativistic effects useful

for calculations involving transition elements; and also pseudopotentials for

high-throughput calculations [12].

Fast forward to the second half of the 20th century and a wealth of tech-

niques like DMRG techniques for 1D spin chains [13], Quantum Montecarlo

(QM) methods [14], Many Bodies Perturbation Theory (MBPT) [15], Dy-

2

1.1. OVERVIEW AND MOTIVATION

namical Mean Field Theory (DMFT) [16] and Density Functional Theory

(DFT) [17, 18] where developed to study many electron systems. From a

fundamental perspective, the most important difference between these tech-

niques is the way they incorporate the many body effects. QM, MBPT and

DMFT directly include many body effects whereas DFT only does so in

the independent particle approximation. This may seem as flaw for DFT

methods however, the method is exact for ground state properties, is com-

putationally tractable up to millions of atoms [19] and its output is used as

an input for more accurate methods [20]. This shows that DFT methods

are still a key piece for understanding many electron systems.

Quantum mechanics is without doubt one of the most successful theories

in physics. It is interesting however, that it has at least nine equivalent

formulations [21]. The most common ones are the Schrodinger, Heisenberg

and Feynmann formulations that place the emphasis on wave functions, op-

erators and classical trajectories respectively. In addition to these, there

is also the Wigner formulation or more generally the quantum mechanics

in phase space approach. In broad terms, phase space quantum mechanics

puts the emphasis on complex valued functions of the classical position an

momentum variables. Historically, the Wigner formulation was first intro-

duced in a paper [22] by E. Wigner in 1932 where he was exploring quantum

corrections to thermodynamic equilibrium. Nonetheless, by 1927 H. Weyl

had already given a prescription on how to associate Hilbert space opera-

tors with complex valued functions i.e a quantization rule. This prescription

was used and partially extended by von Neuman in 1932 [23], being in it-

self a precursor of the star product (?) developed by Groenewald [24] in

1946. The star product allows to connect the classical Poisson bracket to

a similar phase space structure known as Moyal bracket that provides the

time evolution of any phase space distribution [25]. Nowadays the idea of

quantum mechanics in phase space has matured [26, 27]. It is known that

there is not a unique way of building a quantum mechanics phase space rep-

resentation, and thus there are many different and equivalent formulations

that differ from each other in the way they associate Hilbert space opera-

3


tors to phase space functions [28]. That is, for example, Wigner function

(symmetric ordering) [22], Mehta function (standard ordering) [29], Kirwood

function (anti standard ordering) [30, 31], Glauber-Sudarshan P and Q func-

tions (normal/antinormal order) [32, 33] and Hussimi function (generalized

order) [34]. Beyond the historical perspective and the formal elegance of

quantum mechanics in phase space in general and of the Wigner function

in particular, the approach has been successfully used in the study of many

areas in physics e.g. Nonlinear dynamics [35, 36], chaotic systems [37, 38],

quantum optics [39] and even ultra cold atoms [40]. More recently, the

Wigner formalism has been useful in the study of noncommutative gauge

theories [41] and it is expected that it can contribute to the growing field

of quantum computing and communication as it is suited to study quantum

decoherence and environmental interactions [23].

Lattice Boltzmann (LB) methods are among the most flexible and efficient

methods for fluid dynamics calculations. The origin of the method can be

traced back to the cellular gas models. These models aim to describe the

complex macroscopic behaviour of fluids using simple microscopic models,

following the idea of Kadanoff [42] that the macroscopic behaviour is mostly

insensitive to the microscopic details. Some of the seminal works are those

of Wolfram [43], d’Humires [44] who developed a 3D model using a 4D hy-

perlattice and Frisch et. al. [45] who discovered the importance of lattice

symmetry in order to recover the Navier-Stokes equations. Even though

these first attempts of discrete fluid dynamics simulations where promising,

their short comings: lack of Galilean invariance, statistical noise, exponen-

tial complexity and spurious invariants where soon evident [46]. The first

answer to these problems particularly to the statistical noise problem was

given by McNamara and Zanetti [47] who proposed to switch from “hard”

occupation numbers to ensemble average populations. Shortly after, a fur-

ther improvement was made by Higuera and Jimenez [48] who proposed to

linearise the kinetic collision term leading to the LBGK model after the

Bhatnagar-Gross-Krook collision operator [49]. Similar ideas where inde-

pendently proposed by Qian [50] and Chen [51]. A few years afterwards, it

4

1.1. OVERVIEW AND MOTIVATION

was shown that the LB models could also be derived from the continuous

Boltzmann equation in the BGK approximation [52, 53] and recently that it

is an exact consequence of the Boltzmann equation [54]. Over the years, it

has become clear that LB methods for fluid dynamics are higly flexible for

introducing new physics in the models as well as for handling curved bound-

aries [55] and complex domains such as blood vessels [56, 57]. Furthermore,

the methods are highly amenable for parallel computing as their parallel

content in the sense of Amdahl law is 90% and its ratio of communica-

tion to computation is relatively low [46]. Demonstrations of these benefits

have been presented as early as 1997 for simulations of turbulent flows [58]

to recent implementations using nearly 300.000 processors [59]. Perhaps as

impressive as the previous accomplishments of LB methods in its original do-

main of application, are its extensions to curved manifolds [60, 61], graphene

physics [62, 63, 64], electrodynamics [65], quantum mechanics [1, 66] and

even relativistic physics [67, 68].

It is clear that Lattice Boltzmann methods are flexible enough to go beyond

fluid dynamics. How flexible are they? would then be the next question

to ask. As it was expressed in Ref. [69], Lattice Kinetic methods could

be generators of broad families of partial differential equations (PDEs) by

“uplifting” the existing theory from its original space, to a phase space

in which the kinetic moments are used to recover the original macroscopic

equations. In this work a few steps into that direction are taken and the idea

of using Lattice Kinetic Methods for Density Functional Theory and Wigner

Equation calculations is explored. This is interesting not only because it will

help to elucidate how flexible Lattice Kinetic Methods really are, but also

because it opens the door to the thought provoking idea of whether or not

certain theories can be considered as kinetic theories with adequate quasi

particles and collisions rules.

1.1.2 Part II

History has it that Brownian motion was discovered by the botanist Robert

Brown while observing pollen grains on water under the microscope. This

5


apparently simple phenomena was soon explained by Einstein [70] and von

Smoluchowski [71]. Shortly after, the idea of harnessing random fluctuations

that is, Brownian motion, was introduced by von Smoluchowski in the form

of a trap door model that aims to emulate a Maxwell demon [72]. The idea

was taken to its modern form by Feynmann, who framed it in the form of a

ratchet and pawl mechanism [73] and explained that it could not work due to

thermal fluctuations. Just like Maxwell demons, ratchet mechanism can not

violate the second law of thermodynamics in equilibrium conditions. How-

ever, transforming random fluctuations in directed motion i.e motion recti-

fication, is possible provided the system is not in equilibrium and that there

is spatial symmetry breaking [74, 75]. Now a days, there are many exam-

ples of systems that function by exploiting Brownian motion e.g. biological

processes [76, 77], particle segregation systems [78, 79, 80, 81] and trans-

port [82, 83, 84]. Also possible is particle motion rectification on sawtooth

channels in the presence of pulsating potential both at a macroscopic [85]

and microscopic levels [79]. Asymmetric objects immersed in granular gases

also display rectification phenomena [86, 87, 88, 89, 90, 91]. As well as active

matter systems from microscopic bacteria to humans [92, 93, 94, 95, 96].

Although these systems rectify the motion of the particles or objects im-

mersed within the particle bath, they still require either a pulsating potential

or active particles. Examples, that in some sense relax these requirements,

include rectification using differentiated noise sources [97, 98] or asymmetric

piston models [99, 100, 101] that show rectification effects, even when work-

ing at a single temperature, provided there is friction and the particle-piston

collisions are different on both sides of the piston. So far there are only few

examples [102, 103] in which the motion of a single particle in a single di-

mension is rectified without external driving forces. In the present work, a

novel example of collective particle motion rectification in two dimensions is

given. We show that the motion of a gas of Lennard-Jones particles can be

rectified without external pulsating potentials only by means of dissipation

and broken spatial asymmetry.

6

1.2. STRUCTURE AND ORGANIZATION

1.2 Structure and organization

This work is organized as follows:

• Part I: Lattice Kinetic Methods for DFT and Wigner function formal-

ism

– Chapter 2 presents the general formulation of the lattice kinetic

method that will be used in chapters 3 and 4. The discussion is of

a general character as the particular specializations for DFT and

Wigner function formalism will be addressed in the corresponding

chapters.

– Chapter 3 starts by introducing the necessary background on den-

sity functional theory (DFT), imaginary time propagation (ITP)

and the kinetic approach to DFT. Afterwards, the second order

correction method is presented and validated with realistic calcu-

lations of atoms and molecules. Finally, the use of pseudopoten-

tials is also addressed.

– Chapter 4 first presents the relevant background on the Wigner

formalism. Afterwards, the Lattice Wigner Method is explained

in detail and validated using quantum harmonic and anharmonic

potentials. Finally, the method is applied to simple one and two

dimensional open driven systems and the viability in three di-

mensions is explored.

• Part II: Thermal gas rectification

– In chapter 5 it is shown that the motion of a thermal gas of

Lennard-Jones particles can be rectified using broken spatial sym-

metry and dissipation. To this end, first the Smoluchowski-Feynmman

ratchet is discussed to show that dissipation and broken spatial

symmetry are required to observe rectification phenomena. Af-

terwards the background on canonical molecular dynamics, which

is the tool used to perform the simulations, is given. Next, the

7


ratchet model is introduced and the dependency of the particle

flux on various system parameters is discussed.

8

Chapter 2

General formulation of lattice

kinetic methods

In this section the general approach to Lattice Kinetic methods is presented.

The discussion is kept general in order to highlight that this family of meth-

ods can go, in principle, beyond the fluid dynamics applications. First it is

shown how the standard Lattice Boltzmann equation is derived, afterwards

the discretization of the velocity space via quadratures will be explained and

then the role of the equilibrium function and source terms is discussed.

2.1 Lattice kinetic methods

2.1.1 From Boltzmann to Lattice Boltzmann

The standard Lattice Boltzmann equation (LBE) is derived from the Boltz-

mann equation (BE) following the formalism presented in [104]. Consider

the Boltzmann equation in the relaxation time approximation

∂f

∂t+ v · ∇f = − 1

τk(f − f eq) + S, (2.1)

where f(x,v, t) is a distribution function in phase space, f eq(x,v, t) is the

equilibrium distribution of the considered system, S(x,v, t) is a general

source term and τk is the kinetic relaxation time.

9

CHAPTER 2. GENERAL FORMULATION OF LATTICE KINETICMETHODS

Observe that Eq. (2.1) can be formally written as an ordinary differential

equationdf

dt+f

τk=

g

τk, (2.2)

where g = f eq + τkS and ddt

= ∂∂t

+ v · ∇ is the time derivative along the

characteristic line v. Eq. (2.2) can be formally integrated in time over the

interval [0, δt] leading to the result

f(x+vδt,v, t+δt) =1

tke−δt/tk

∫ δt

0

et′/tkg(x+vt′,v, t+t′)dt′+e−δt/tkf(x,v, t)

(2.3)

if δt is small enough, g can be linearly approximated in the interval [0, δt] as

g(x + vt′,v, t+ t′) =

(1− t′

δt

)g(x,v, t) +

t′

δtg(x + vδt,v, t+ δt) +O(δt2).

(2.4)

If Eq. (2.4) is substituted in Eq. (2.3) and the integral is solved, the result

is

f(x + vδt,v, t+ δt)− f(x,v, t) = (e−δt/tk − 1)(f(x,v, t)− g(x,v, t))

+

(1 +

tkδt

(eδt/tk − 1)

)(g(x + vδt,v, t+ δt)− g(x,v, t)). (2.5)

Finally if eδt/tk is Taylor expanded and terms of order O(δt2) or higher are

neglected the result is the simultaneous space and time discretization of

Eq. (2.2) namely,

f(x + vδt,v, t+ δt)− f(x,v, t) = −δtτk

(f(x,v, t)− f eq(x,v, t)) + δtS. (2.6)

In the processes of deriving the Lattice Boltzmann equation (Eq. (2.6))

from the Boltzmann equation (Eq. (2.2)) a number of approximations where

made. It could be argued that more sophisticated integration schemes or

softer approximations could lead to additional or different results. This

question was addressed in Ref. [54] and the conclusion is that Eq. (2.6) is an

exact consequence of Eq. (2.1). In different terms, better integration schemes

in Eq. (2.3) and more terms in the Taylor expansion of eδt/tk eventually lead

to the LB equation.

10

2.1. LATTICE KINETIC METHODS

2.1.2 Velocity space discretization

In order to explain the velocity space discretization, it is convenient to first

introduce the velocity moments Π(g)nα1,α1,...,αnof an arbitrary phase space

function, g(x,v, t), as

Π(g)nα1,α2,...,αn=

∫dvvα1vα2 · · · vαng(x,v, t), (2.7)

where n indicates the order of the moment and vα denotes the α component

of the velocity variable. The relevance of the moments steams, as it will be

seen in subsequent chapters, from the fact that these are the quantities that

contain the physical information (particles, energy or momentum densities)

of the problem under consideration.

The velocity space is discretized using quadratures [105] instead of a regular

grid. This approach avoids first, the necessity of fixing a cutoff in veloc-

ity space, which also results in an inaccurate computation of the moments

Eq. (2.7) and second, it provides a better discretization of the ∇ opera-

tor [106], that due to the structure of Eq. (2.1) appears repeatedly in the

concerning problems.

Formally, discretization by quadratures requires that the moments Eq. (2.7)

in the velocity space of the functions f , f eq and S can be exactly calculated

up to a fixed order NΠ. This is achieved with the aid of a quadrature defined

as a finite set vi, ωiNqi=1 of Nq vectors, vi, and corresponding weights, ωi,

that satisfy the relations

Π(n)α1α2...αn

(f) =

∫dvvα1vα2 ...vαnf(x,v, t) (2.8a)

=

Nq∑i

vi,α1vi,α2 ...vi,αnωif(x,vi, t) (2.8b)

=

Nq∑i

vi,α1vi,α2 ...vi,αnfi(x, t) (2.8c)

for n ≤ NΠ. vi,αn denotes the αn component of the i-th velocity vector and

fi(x, t) ≡ ωif(x,vi, t). Similar expressions must also hold for the moments

of f eq and S using the same quadrature vi, ωiNqi=1. The quantities fi, feqi

11


and Si are respectively known as (discrete) “distributions”, “equilibrium

distributions” and “source distributions”.

The problem of finding quadratures that satisfy Eq. (2.8) can be solved

by observing that they are finite and therefore can not reproduce infinitely

many moments i.e., for a fixed quadrature, Eq. (2.8) can not be satisfied

for an arbitrary large value of n. This imply that only a limited number

of moments of f will be reliably accessible, and that it is not necessary to

work with the full analytical forms of f eq and S. Instead, an expansion in

orthonormal polynomials will be assumed for f eq, S and implicitly for f .

More specifically, given a family of polynomials Pn(v) orthonormal under

the weight function ω(v)∫dvω(v)Pn(v)Pm(v) = δm,n,

f can be represented approximately as

f(x,v, t) ≈ ω(v)

Np∑n

an(x, t)Pn(v), (2.9)

whereNp is the maximum order of the polynomials used in the representation

and the expansion coefficients are given by

an(x, t) =

∫d3vf(x,v, t)Pn(v). (2.10)

The representation Eq. (2.9) simplifies the problem of finding the quadra-

tures, notice that any combination of the form viα1viα1 · · · viαn can be exactly

represented as a linear combination of the Pn,

viα1viα1 · · · viαn =∑m

βmPm(v). (2.11)

If Eq. (2.9) and Eq. (2.11) are substituted in Eq. (2.8a) and Eq. (2.8b), it

can be concluded that the requirement of Eq. (2.8) is then equivalent to

solving the set of constrained algebraic equations

Nq∑i=0

ωiPn(vi)Pm(vi) = δn,m ∀n,m ≤ Np (2.12)

vi ∈ Zd ∀i

wi ≥ 0 ∀i,

12


for vi, ωi. Technically the constrain vi ∈ Zd is not necessary, however, not

imposing it requires the use of interpolation schemes whenever x + viδt is

not a node of the spatial lattice implied by the space-time discretization of

the Boltzmann equation (Eq. (2.6)). The lattice-Boltzmann equation that

defines the time evolution of the distribution functions fi(x, t) is thus given

by

fi(x + viδt, t+ δt)− fi(x, t) = −δtτk

(fi(x, t)− f eqi (x, t)) + δtSi, (2.13)

where by construction the f eqi and Si are given by

f eqi (x, t) = ωi

Np∑n

aeqn (x, t)Pn(vi) (2.14)

Si(x, t) = ωi

Np∑n

sn(x, t)Pn(vi). (2.15)

A simple, yet effective, approach to construct lattices in any number of

dimension is to find quadrature rules in one dimension and then use ten-

sor products to extend the lattices in two or three dimensions. A family

of orthonormal polynomials that lends itself to such a program is that of

the Hermite polynomials. This is because the Hermite weight function in d

dimensions is the product of d one dimension weight functions. From the

theory of numerical integration [107] it is known that in one dimension a

possible set of quadrature points (vectors) vi is that of the roots of the

polynomial of highest degree for which the quadrature rule is required. Up

to 4th degree Hermite polynomials, this way of proceeding leads to sets of

vectors vi that satisfy the constrain vi ∈ Zd upon a suitable renormaliza-

tion. For polynomials of higher degree the fact that their roots can not, in

general, be expressed as radicals, precludes this way of proceeding. The gen-

eral approach to solve Eq. (2.12) is then heuristic, first the desired discrete

lattice is fixed, and then Eq. (2.12) is solved for the values of the weights.

Typically the weight function and family of polynomials have extra param-

eters, when that is the case Eq. (2.12) becomes a nonlinear problem. The

specification of the generated lattices using the approach just described is

13


given in Appendix.A.6. It is important to notice that for the particular case

of fluid dynamics, there are a number of ways [108, 109, 110] to system-

atically generate lattices that satisfy Eq. (2.8) and that take into account

additional constrains. Finally the lattice Boltzmann convention of desig-

nating the quadratures or velocity lattices by the scheme DnQm where n

denotes the dimensionality and m the number of velocity vectors or quadra-

ture points will be used throughout.

2.1.3 Equilibrium and source term distributions

It is evident that Eq. (2.13) dictates the time evolution of the distributions

fi, but it is not clear what physical system is actually being modelled. This

information is encoded in the equilibrium distribution, source terms and

also possibly on the boundary conditions. For example, in the traditional

application of Lattice Kinetic methods to fluid dynamics, f eq is a Gaussian

(Maxwellian) distribution such that the moments of fi follow the Navier-

Stokes equations. In general f eqi and S can be physically motivated, but it

is not a requirement that they correspond to a physical equilibrium function

or external source. Instead, they are engineered such that the moments

of fi follow a prescribed dynamic thus defining the system being modelled.

It is this freedom that allows the lattice kinetic methods to be extended

to a variety of situations ranging from Wave equations [111] to Maxwell

equations [65], relativistic hydrodynamics [112] and even fluid dynamics in

curved manifolds [113].

2.1.4 Algorithmic details

Given the explicit expressions for Si and f eqi that define what system is being

modelled and a quadrature vi, ωiNqi=1 that is able to satisfy the necessary

moments (Eq. (2.8)), the way in which a Lattice Kinetic scheme proceeds

is rather simple. The time propagation of the fi is accomplished using

Eq. (2.13) which is evaluated in two stages known as collision and streaming.

14


In the former one the quantity

f ∗i (x, t) = fi(x, t)−δt

τk(fi(x, t)− f eqi (x, t)) + δtSi(x, t) (2.16)

is evaluated, and in the later it is streamed

fi(x + viδt, t+ δt) = f ∗i (x, t), (2.17)

observe that the collision stage is purely local in space, whereas the stream-

ing stage only involves information flow. In order to repeat the streaming-

collision cycle Si and f eqi need to be updated, in most cases these two

functions are functionals of fi i.e Si = Si[Π0(f),Π1(f), ...Πk(f)] and f eqi =

f eqi [Π0(f),Π1(f), ...Πk(f)] where the different moments are calculated ac-

cording to Eq. (2.8c). The effect of the boundaries can be introduced at

any convenient point, depending on whether they affect the distribution

functions or the moments. The process is illustrated in Fig.2.1

Figure 2.1: Illustration of the Lattice Kinetic algorithm. At every lattice

site there are three different distributions, the distribution function itself,

the equilibrium distribution function and the source term distribution. In

the first step, these distributions are mixed according to Eq. (2.16) leading

to f ∗i . In the second step f ∗i is streamed according to Eq. (2.17). In the

last step Π0(f),Π1(f), ...Πk(f) are calculated and used to update f eqi and Si

before the cycle starts over again.

15

Chapter 3

Lattice kinetic approach to

Density Functional Theory

3.1 Density functional theory and imaginary

time propagation

3.1.1 Density functional theory

Density functional theory can be thought simultaneously as a theory and

method to study the properties of many interacting quantum particles. It

is mostly applied to ground state properties of atomic, molecular and solid

state systems although there are also examples in fields such as cold atomic

gases [114] where the effective particle-particle interaction is short ranged.

From a formal point of view, the problem of finding the properties of a

physical system like a molecule or solid is related to the problem of solving

the stationary Schrodinger equation

Hψi = εiψi, (3.1)

where ψi is the many particle wave function with associated eigenenergy εi.

In the Born approximation the Hamiltonian, H, of a molecular or atomic

17

CHAPTER 3. LATTICE KINETIC APPROACH TO DENSITYFUNCTIONAL THEORY

system is generally given by

H = T + Vex + Vee (3.2)

=∑i

− ~2

2m∇2 −

∑i,n

e2zn|Rn − ri|

+1

2

∑i 6=j

e2

|ri − rj|. (3.3)

where m denotes the electron’s mass, ri are the “coordinates” of the i-th

electron, e is the value of the fundamental electric charge and Rn is the

location of the n-th atomic nucleus with z number zn. Observe that the first

term in the r.h.s of Eq. (3.2) is the kinetic energy of the system of electrons,

the second term is related to the external potential due to the atomic nuclei

and the last term is the electron-electron Coulomb interaction.

The difficulty in solving Eq. (3.1) with the Hamiltonian given by Eq. (3.2)

steams from the electron-electron interaction term and from the requirement

that ψ must be antisymmetric on the exchange of any two electrons i.e. the

Pauli exclusion principle.

The DFT approach consist of replacing the original many interacting parti-

cles problem defined by H, with an auxiliary problem HKS of independent

particles that has the same ground state properties as H [115, 116, 117]. The

problem reformulation is carried out using the Hoenberg-Kohn theorems [17]

and the Kohn-Sham ansatz [18].

Let the particle (electron) density associated to an arbitrary state φ be given

by n(r) = 〈φ|n|φ〉 where n =∑

i δ(r − ri) is the particle density operator.

The Hoenberg-Kohn theorems assert that given a physical system described

by a Hamiltonian of the form Eq. (3.2) i) there exists a one to one relation

between the external potential Vex and the ground state particle density n0

that is independent of the electron-electron interaction. And ii) that there

exist an energy functional E[n] that depends on the particle density and

is such that its minimum is the actual ground state energy of the system.

Furthermore, it also holds that the density that minimizes E[n] corresponds

to the ground state density of the system. The Kohn-Sham ansatz starts

from an independent particle system

HKS = T + V KSex , (3.4)

18

3.1. DENSITY FUNCTIONAL THEORY AND IMAGINARY TIMEPROPAGATION

that is assumed to have the same ground state particle density as the original

problem. Under that assumption, the Hoenberg-Kohn theorems show that

the form of V KSex is given by

V KSex = Vex + VH + Vxc, (3.5)

where Vex is the external potential of the original problem and VH is the

Hartree potential. Vxc = δExc[n]δn

is known as the exchange correlation poten-

tial, and is defined via the functional derivative of the exchange correlation

energy functional Exc[n] with respect to the particle density. The important

aspect of Exc[n] is its universality i.e. it is independent of the external po-

tential and therefore the same for all possible problems that have the same

electron-electron interaction.

The crucial consequence of the Kohn-Sham approach is that it is possible to

find the ground state particle density of a many interacting particles problem

by solving the much simpler problem of non interacting particles Eq. (3.4).

The downside however, is that in spite of its universal nature and the fact

that all many particle effects are accounted by Exc[n], only approximations

to it are known. Currently there are many different exchange correlation

potentials. Depending on their degree of sophistication and accuracy, they

are catalogued at different levels on the Jacob Ladder [118] of exchange

correlation potentials. For the present work the BLYP exchange correlation

potential from Becke [119] and Lee et. al [120] will be used throughout.

For practical purposes, once a particular form of Exc[n] has been chosen the

problem is to solve the Kohn-Sham equations[−1

2∇2 + Vex + VH + Vxc

]ψi = εiψi. (3.6)

Since VH and Vxc depend directly on the particle density n, Eq. (3.6) must

be solved in a self consistent way that is, an initial guess n(0) is made,

from there V(0)H and V

(0)xc are calculated and used to solve the Kohn-Sham

equations leading to ψ(0)i . Since the system is non interacting, the ground

state wave function is a single Slater determinant and the updated particle

density can be calculated as n(1) =∑

i |ψ(0)i |2 which can then be used to

iterate the process until |n(i)−n(i+1)| is smaller than a prescribed tolerance.

19


3.1.2 Imaginary time propagation

The imaginary time propagation is a technique to find the lowest lying eigen-

functions and eigenenergies of a quantum mechanical problem described by

Eq. (3.1). The method is based on the fact that the time evolution of any

state ϕ, is given by the time dependent Schrodinger equation

i~∂ϕ

∂t= Hϕ, (3.7)

as

ϕ(t) = e−iHt/~ϕ(0) =∑i

aie−iεit/~ψi (3.8)

where

ϕ(0) =∑i

aiψi (3.9)

is the spectral decomposition of ϕ in the eigen basis of H. Observe that if

the time variable is changed as t → −iτ , the imaginary time evolution of

ϕ(0) is given by

ϕ(τ) = eHτ/~ϕ(0) =∑i

aie−εiτ/~ψi. (3.10)

Since the evolution is no longer unitary, the norm of ϕ(0) is not preserved.

In fact, depending on the nature of the spectrum of H, |ϕ(τ)| will either

shrink to zero, or grow without bound. The former case is associated to free

states εi > 0 whereas the later to bound states εi < 0. It is precisely this

behaviour that allows to extract both the eigen functions ψi and the cor-

responding eigen energies εi. To see this, assume without loss of generality

that the system is bounded, for example a molecule, and that the eigenen-

ergies are sorted by increasing value that is, ε0 ≤ ε1 ≤ · · · ≤ 0. In this case

Eq. (3.10) shows that the term a0e−ε0τ/~ψ0 in the spectral expansion of ϕ(τ)

will be the one growing the fastest due to the factor e−ε0τ/~. If after a fixed

time of evolution ϕ(τ) is renormalized, ϕ(τ) → ϕ(τ)/|ϕ(τ)|, then it holds

that ϕ(τ) ≈ ψ0. In the case of an unbounded system a similar argument

holds, but the largest contributing term to ϕ(τ) is the one that decreases

20


the slowest. Assuming that ϕ(τ) ≈ ψ0, ε0 can be found as follows

ε0 = − 1

δτln

(|ϕ(τ + δτ)||ϕ(τ)|

). (3.11)

Once ϕ0 is found, the exited states and energies can be found by repeating

the process, but starting from the state ϕ(1)(0) = ϕ(0)− 〈ϕ(0)|ψ0〉.

3.1.3 Kinetic approach to DFT

The kinetic approach to density functional theory is based on the observation

that the time dependent Kohn-Sham equations in imaginary time can be

recovered as a special macroscopic limit of the Boltzmann equation in the

BGK approximation [49]. The connection is established in two steps, the

first one is to observe that upon a Wick rotation the time dependent Kohn-

Sham equation is formally equivalent to a diffusion equation. The second

step is to show that an equilibrium distribution function, f eq, and source

term S can be tailored such that the moments of the distribution function

follow a diffusive dynamic consistent with that of the Wick rotated Kohn-

Sham equations.

The first step readily leads to

∂ϕ

∂τ=

~2m∇2ϕ− 1

~V KSϕ, (3.12)

where the T + V form of the Hamiltonian was used. Notice that formally,

Eq. (3.12) is a diffusion equation for the field ϕ, with a diffusion constant~

2mand a source/sink term given by 1

~VKSϕ.

For clarity of explanation, the result of the second step will be first presented

and afterwards it will be shown how it yields the desired diffusive behaviour.

To fix the notation let the α1α2...αn component of the n-th moment of the

distribution function, equilibrium distribution function, and source term be

21


respectively defined as

Π(n)α1α2...αn

=

∫dvvα1vα2 ...vαnf(x,v, t), (3.13a)

Π(n)eqα1α2...αn

=

∫dvvα1vα2 ...vαnf

eq(x,v, t), (3.13b)

Σ(n)α1α2...αn

=

∫dvvα1vα2 ...vαnS(x,v, t), (3.13c)

where vα is the α component of the phase space velocity vector v.

If the moments of the equilibrium distribution function and source term are

chosen as

Π(0)eq = Π(0), (3.14a)

Π(1)eq = 0, (3.14b)

Π(2)eqij = C2

sΠ(0)δij, (3.14c)

Π(n)eq = 0 for n > 2, (3.14d)

Σ0 = S, (3.14e)

Σ(n)eq = 0 for n > 0, (3.14f)

where C2s is a characteristic speed of the system and S is a function of space

and time, then the 0-th moment of the distribution function evolves in time

according to the diffusion equation

∂Π0

∂t= τkC

2s∇2Π0 + S. (3.15)

To see how the moments definition Eq. (3.14) lead to Eq. (3.15), the tool

of choice is the Chapman-Enskog expansion [121]. Let the the distribution

function and time derivative be expanded as

f = f (0) + εf (1) + ε2f (2) + · · · , (3.16)

∂

∂t= ε

∂

∂t1+ ε2

∂

∂t2+ · · · , (3.17)

and let the spatial derivative and source term be rescaled as ∇ = ε∇1 and

S = εS(1), where ε is regarded as a small quantity. Substituting these

22


relations in Eq. (2.1) and equating terms of equal order in ε the following

set of equations are found

f (0) = f eq, (3.18)

∂t1f(0) + v · ∇1f

(0) = − 1

τkf (1) + S(1), (3.19)

∂t1f(1) + ∂t2f

(0) + v · ∇1f(1) = − 1

τkf (2). (3.20)

Taking the 0-th moment of Eq. (3.19) and (3.20) we found

∂t1Π(0,0) +∇1 · Π(1,0) = − 1

τkΠ(0,1) + Σ(0,1), (3.21)

∂t1Π(0,1) + ∂t2Π(0,0) +∇1 · Π(1,1) =1

τkΠ(0,2). (3.22)

Where Π(i,j) is the i-th moment of f (j) and Σ(0,1) is the 0-th moment of the

rescaled source term. Eqs. (3.16), (3.18) and the constraint Eq. (3.14a)

implies that the previous equations simplify as

∂t1Π(0) +∇1 · Π(1,0) = Σ(0,1), (3.23a)

∂t2Π(0) +∇1 · Π(1,1) = 0. (3.23b)

Eq. (3.18) and constraint Eq. (3.14b) implies that Π(1,0) = 0. Thus Eq. (3.23)

can be written as

∂t1Π(0) = Σ(0,1), (3.24a)

∂t2Π(0) +∇1 · Π(1,1) = 0. (3.24b)

Π(1,1) can be calculated by taking the first moment of Eq. (3.19), due to the

constraints Eq. (3.14b) and (3.14f) only the term v · ∇1f(0) will contribute,

explicitly

Π(1,1)k = −τk

∫vkv · ∇1f

(0)d3v

= −τk∂i∫vkvif

(0)d3v

= −τk∂iΠ(0)i,j . (3.25)

23


∇1 · Π(1,1) is then calculated as

∇1 · Π(1,1) = ∂iΠ(1,1)i (3.26)

= −τk∂i∂kΠ(0)i,k (3.27)

= −τkC2s∇1 · ∇1Π(0) (3.28)

= −τkC2s∇2

1Π(0), (3.29)

where the constraint Eq. (3.14c) was used. Eq. (3.24b) then reads

∂t2Π(0) − τkC2s∇2

1Π(0) = 0. (3.30)

Finally, Eq. (3.15) is obtained by multiplying Eq. (3.24a) by ε and Eq. (3.30)

by ε2 and adding them taking into account the scaling of the spatial deriva-

tive and source term, as well as the expansion of the temporal derivative.

If the identifications Π0 ≡ ϕ, τkC2s ≡ ~

2mand S ≡ −V KSex

~ ϕ are made, then

Eq. (3.15) can be rewritten as

∂ϕ

∂t=

~2m∇2ϕ− V KS

ex

~ϕ, (3.31)

which is the Wick rotated time dependent Kohn-Sham equation Eq. (3.12).

At this point is important to notice that in the kinetic approach to DFT the

relation between the quantum mechanical wave functions and the kinetic

distribution functions is not direct, instead the quantum mechanical wave

functions emerge as the zeroth moment of the distribution function.

From the previous discussion on imaginary time propagation, it follows that

for any initial condition ϕ(0), that has a non vanishing projection on the

ground state ψ0, the imaginary time evolution guarantees that as the time

increases and the wave function is renormalized, only the ψ0 contribution is

obtained. It is clear then, that by sequentially removing lower lying eigen-

states from the initial condition ϕ(0) all possible states will be eventually

found. Therefore, the Lattice Kinetic approach contains the same physical

information as the Schrodinger equation.

Even though all the physical information regarding a given quantum system

is contained in Eq. (2.1) and Eq. (3.14), for computational purposes it will

24


prove convenient to make it explicit. This is done by introducing an ex-

tended kinetic model in which each Kohn-Sham orbital ψl l = 1, 2, . . . , N is

associated to its own distribution function fl, and the different distributions

interact with each other via a dynamical orthonormalization potential Wl.

That is, each of the fl evolves according to

∂fl∂t

+ v · ∇fl = − 1

τk(fl − f eql ) + S +

1

τkWl, (3.32)

where the moments of Wl are

Ω(0) = −∑i<l

〈ψl|ψi〉〈ψi|ψi〉

ψi, (3.33a)

Ω(1) = 0, (3.33b)

Ω(2) = −τkC2s

∑i<l


ψi, (3.33c)

Ω(n) = 0 for n > 2. (3.33d)

Since the structure of the moments of Wl is the same as that of f eql , each

Π(0)l will be given by

Π(0)l = ψl −

∑i<l


ψi. (3.34)

Thus, Π(0)k has no contributions from ψl with l < k. Since ϕl → ψl before

ϕk → ψk for l < k it follows that effectively ϕk has no components along

any of the ψl eigenstates and thus the next lowest available eigenstate is the

one that is going to be selected by the imaginary time evolution. Notice

also that as the different orbitals start converging, the effect of Wl becomes

weaker due to the orthonormality of the wave functions and once the different

orbitals have converged it plays no further role. In other terms, Wl is only

used to drive the different fl in such a way that they converge to different

Kohn-Sham orbitals. Finally, the relation between the kinetic approach to

DFT and the Kohn-Sham equations is summarized in Fig.3.1

25


Figure 3.1: Diagram of the relation between the time dependent Kohn-Sham

equations and the Kinetic approach to DFT via the diffusion equation. From

the time dependent Kohn-Sham equation the diffusion equation is reached

by means of the Wick rotation. Starting from the Boltzmann equation, it

can be reached by the right definition of the moments of the equilibrium

function, Πeq, and source term Σ.

3.2 Lattice Kinetic model for DFT

Mendoza et.al proposed a Lattice Kinetic model that evolves according to

Eq. (2.13) and satisfies both the explicit constraints Eq. (3.14) on the equi-

librium and source term distribution functions and the requirement that

the different Kohn-Sham orbitals need to be orthonormal. In the discrete

setting of that model, it can be show that the 0-th order moment (Kohn-

Sham orbital), of the lattice distribution function fil evolves according to

the diffusion equation

∂Π(0)

∂t= δt

(τk −

1

2

)C2s∇2Π(0) + S +O(δt), (3.35)

which is equivalent to Eq. (3.31), up to terms of order O(δt), if the identi-

fications Π0 ≡ ψ, δt(τk − 1

2

)C2s ≡ ~

2mand S ≡ −V

~ ψ are made. The extra

“−1/2” term in the identification of ~/2m compared to that of Eq. (3.15) is

due to the spacial discretization and is well known in the Lattice Boltzmann

community.

3.2.1 Formal correction of discrete lattice effects

In the following it is shown how to improve the proposal of Mendoza et.al

using a semi implicit correction term. First the Lattice Kinetic model is

26

3.2. LATTICE KINETIC MODEL FOR DFT

formally improved by the introduction of a correction term. Afterwards,

the equilibrium and source term distributions for the model are explicitly

constructed and finally the semi implicit implementation of the correction

term is explained.

The second order Lattice Kinetic model is defined as

fi(x+viδt, t+δt)−fi(x, t) = − 1

τk(fi(x, t)−f eqi (x, t))+δtSi+

δt2

2DiSi, (3.36)

where Di = ∂∂t

+vi ·∇ and δt2

2DiSi is the forcing correction term that cancels

spurious discretization effects. In simple terms, the addition of δt2

2DiSi works

because it exactly compensates a similar term that manifests itself at the

second scale level in the Chapman-Enskog expansion of the standard Lattice

Boltzmann equation with source term Eq. (2.13). To see this in more detail

first consider the Taylor expansion of the l.h.s of Eq. (3.36)

δtDifi +δt2

2D2i fi = − 1

τk(fi − f eqi ) + δtSi +

δt2

2DiSi, (3.37)

observe that the convective terms of the original Boltzmann equation are

already present in the first order terms of Eq. (3.37), while the second order

ones appear due to the fact that the Lattice-Boltzmann equation is a discrete

approximation. The diffusion equation is recovered by following a similar

path as that used in the continuous Boltzmann equation case. That is, a

multi-scale expansion of the distribution functions and time derivatives in

a small parameter ε (that in fluids dynamics plays the role of a Knudsen

number) is assumed

fi = f(0)i + εf

(1)i + ε2f

(2)i + · · · , (3.38)

∂

∂t= ε

∂

∂t1+ ε2

∂

∂t2+ · · · , (3.39)

whereas space derivatives and source term are rescaled as ∇ = ε∇1 and

Si = εS(1)i respectively.

Replacing Eq. (3.38) and Eq. (3.39) into Eq. (3.37) and collecting terms of

27


equal order in ε, yields the result

O(ε0) : f(0)i = f eqi , (3.40)

O(ε1) : D1ifeqi = − 1

τkδtf

(1)i + S

(1)i , (3.41)

O(ε2) :∂f eqi∂t2

+D1if(1)i +

δt

2D2

1ifeqi = − 1

τkδtf (2) +

δt

2D1iS

(1)i . (3.42)

Eq. (3.41) is further substituted in the l.h.s of Eq. (3.42) and the result is

summed over all discrete velocities leading to the relation

∂Π0

∂t2+

(1− 1

2τk

)∇1 ·

∑vif

(1)i = 0, (3.43)

where the moments definition of f eq and S Eq. (3.14) were used. The first

moment of the first correction to the distribution function,∑

vif(1)i , is eval-

uated by taking the product of Eq. (3.41) with vi and summing over all

velocities. Substituting this result back in Eq. (3.43) yields

∂Π(0)

∂t2= δt

(τk −

1

2

)C2s∇2

1Π(0). (3.44)

In a similar manner if Eq. (3.41) is summed over all velocities the result is

∂Π(0)

∂t1= S(1). (3.45)

Finally when Eq. (3.44) is multiplied by ε2 and added to Eq. (3.45) multiplied

by ε it is found that Π(0) evolves according to

∂Π(0)

∂t= δt

(τk −

1

2

)C2s∇2Π(0) + S +O(δt2). (3.46)

The result is that the modified model (Eq. (3.36)) together with the same

moments definition (Eq. (3.14)) as in the continuous case lead to a second

order improvement in the recovered equation for the zeroth order moment of

the distribution function, that is for the Kohn-Sham orbitals. The inclusion

of the term δt2

2DiSi in Eq. (3.36) is not intuitive, however it can be seen that

if it had not been added then the additional term δt2∂S(1)

∂t1would be present

in Eq. (3.43) and then propagated to Eq. (3.46). It is important to notice

28

3.2. LATTICE KINETIC MODEL FOR DFT

that the inclusion of the correction term is only possible because, as shown

before, it does not change the macroscopic limit of the Lattice-Boltzmann

equation i.e. the zeroth moment of the distribution function still follows a

diffusive dynamics.

The extension to a system of coupled equations for many orbitals is given

by

fil(x + viδt, t+ δt)− fil(x, t) =

− 1

τk(fil(x, t)− f eqik (x, t)) +

δt2

2DiSil + δtSil +

1

τkWil, (3.47)

where the index l is associated to the l-th orbital.

3.2.2 Equilibrium and source distributions construc-

tion

Up to this point, the equilibrium and source term distribution functions are

only implicitly known in terms of their moments structure Eq. (3.14). How-

ever, to have a practical Lattice Kinetic scheme their explicit form is needed.

Following the discussion of Sec.2.1.2, the form Eq. (2.9) is assumed for the

equilibrium distribution function. Then, using the quadrature definition of

the moments Eq. (2.8c) and the explicit moment definition Eq. (3.14) the

set of linear equations

Π(0) =∑i

∑n

ωiaeqn Pn(vi)

Π(1)αi

= 0 =∑i

vαi∑n

ωiaeqn Pn(vi) (3.48)

Π(2)αiαj

= C2sΠ(0)δij =

∑i

vαivαj∑n

ωiaeqn Pn(vi)

Π(k)αij ···αk = 0 =

∑i

vαi · · ·vαk∑n

ωiaeqn Pn(vi) ∀k > 2

for the unknown coefficients an is defined. Taking the Pn as Hermite poly-

nomials and using the D3Q19 and D3Q111 lattices, the explicit forms of f eqi

29


and Si are

f(D3Q19)i (x, t) = Π0(x, t)wi

(1 +

1

2C4s

(C2s −D)(3C2

s − vi · vi)), (3.49a)

S(D3Q19)i (x, t) = S(x, t)wi

(1 +

1

2C4s

(C2s )(3C2

s − vi · vi)), (3.49b)

f(D3Q111)i (x, t) = Π0(x, t)wi

(1 +

1

2C4s

(C2s −D)(3C2

s − vi · vi)+ (3.49c)

1

8C2s

(C2s − 2D)(15C4

s − 10C2svi · vi + (vi · vi)2)

)(3.49d)

S(D3Q111)i (x, t) = Swi

(1 +

1

2C4s

(C2s )(3C2

s − vi · vi)+ (3.49e)

1

8C2s

(C2s − 2D)(15C4

s − 10C2svi · vi + (vi · vi)2)

), (3.49f)

where the diffusion constant is given by D = (τk− 12)C2

s = ~2m

and the value

of C2s depends on the lattice.

3.2.3 Semi implicit correction

From a computational perspective the implementation of Eq. (2.13) (model

1) requires no special discussion as it conforms to standard Lattice Boltz-

mann schemes. However, in our approach, (model 2) Eq. (3.36), there are

various ways to implement the correction term DiSi. These can be explicit

DiSi =1

δt(Si(x, t)− Si(x− viδt, t− δt)) , (3.50a)

DiSi =1

δt(Si(x + viδt, t)− Si(x, t− δt)) , (3.50b)

or implicit

DiSi =1

δt(Si(x + viδt, t+ δt)− Si(x, t)) . (3.51)

Given that explicit implementations do not require solving a system of

equations at every iteration of the algorithm, we performed tests using

Eqs. (3.50a) and (3.50b), in both cases we found that the procedure was

numerically unstable leading to wild oscillations of the measured quantities.

30

3.3. APPLICATION OF THE IMPROVED LATTICE KINETICAPPROACH

To use the implicit form of DiSi and avoid the necessity of solving systems

of equations, Eq. (3.51) was approximated using the formally known imag-

inary time evolution of the different orbitals. That is, for the n-th orbital

DiSi,n = DiV~ ϕn, and its discretized version is given by

DiSi,n =1

δt

(V

~ϕn(x + viδt, t+ δt)− V

~ϕn(x, t)

). (3.52)

An approximation of Eq. (3.52) can be obtained if ϕn(x, t + δt) can be

estimated. From the imaginary time evolution it is known that

ϕn(x, t) =∑j≥n

cjψje−εjt

~ , (3.53)

where ψj and εj are the eigenfunctions and eigenenergies of the Kohn-Sham

Hamiltonian and cj are the projection coefficients of ϕ in the basis ψj.Therefore, we consider the approximation

ϕn(x, t+ δt) ≈ ϕn(x, t)e−εnδt~ . (3.54)

Notice that this approximation improves after every iteration and is exact

once the steady state has been reached, this follows from the time projection

technique that progressively drives all the cj → 0 for j 6= n.

Finally it is worth noticing that the actual implementation of the correction

term does not add any extra complexity to the scheme. It corresponds to an

extra scalar-matrix-vector multiplication of the same kind used to calculate

the original source term.

3.3 Application of the improved lattice ki-

netic approach

3.3.1 Model comparison

The improved scheme was used to calculate the exchange and correlation

energies of H, He, Be and Ne atoms as well as the bond lengths of H2 and

LiH dimers. We compared it with the model 1 using a D3Q19 lattice and

31


the BLYP exchange correlation potential [119, 120] . The physical length

of the simulation box is given by Lp = L∆x where L is the number of grid

points in one direction, ∆x is the distance between two successive sites and

the resolution of the system is defined as ∆x−1.

Results from the He atom and H2 molecule are shown in Fig.3.2 a,c). Both

models show that for a fixed resolution ∆x−1, as the number of lattice sites

i.e the physical size of the simulation box increases, the value of the measured

quantities tend to converge to a limiting value, and as the resolution level is

improved, the limiting value approaches the expected BLYP values Ref. [20,

122]. These two behaviours are consistent with the fact that as the physical

system size and resolution increase, the boundary effects are reduced and the

system better approximates an atom or molecule in free space. Furthermore,

the limiting values of model 2 are closer to the expected BLYP values than

those of model 1 for a fixed ∆x, and as the resolution improves both models

tend to agree.

The degree to which model 2 is more accurate than model 1 with respect

to the expected BLYP values depends on the considered atom or molecule

and measured quantity (Ex, Ec, bond length, etc). For example, Fig.3.2

b,d) show respectively the relative error, ∆Ex and ∆Ec, of the exchange

and correlation energies as a function of ∆x−1 when the system size is fixed

at L = 150. In the case of Ex, ∆Ex ∝ ∆x1.5 for the first model and

∆Ex ∝ ∆x1.6 for the second. In contrast the behaviour of ∆Ec is non

monotonic. It is worth noting that ∆Ex ranges from 4% to less than 1%

whereas ∆Ec is always smaller than 0.3%.At this point the difference between

the two models seems small, however, this is due to the fact that Ex and

Ec are the integrals of non trivial functions of the density and the density

gradient, where the later has to be numerically calculated. To better observe

the difference between both models, the ground state energy of the H atom,

which only requires the norm of the wave function at two consecutive time

steps, was calculated. Its relative error ∆EH as a function of the resolution

is shown in Fig.3.3, where it can be observed that ∆EH in the second model

(m2) is one order of magnitude smaller than in the first model (m1). That is

32


Figure 3.2: Model comparison: The top and medium left panels are respec-

tively the calculated exchange and correlation energies of the He atom as

a function of system size for three different resolutions using models 1 and

2. Top and medium right panels show the relative error as a function of

∆x−1 for a system size of L = 150. The bottom panel shows the equilibrium

length of the H2 molecule. The solid blue line is the DFT result using the

BLYP functional reported in Ref. [20, 122]

33


0.5 1.0 1.5 2.0 2.5 3.0Log(∆x−1)

−10

−9

−8

−7

−6

−5

−4

Log(|E

−Eex|)

m1m2

Figure 3.3: Relative error of the ground state energy of the H atom as

a function of the resolution for m1 and m2. The circles correspond to the

simulation data and the dashed lines are the linear fits. In both cases ∆EH ∝∆x1.6

Atom −Ex −Ex BLYP −Ec −Ec BLYP

H -0.301 -0.301

He -1.0197 -1.0255 -0.0437 -0.0438

Be -2.6741 -2.6578 -0.0965 -0.0945

Ne -12.0532 -12.1378 -0.3827 -0.3835

Molecule Bond length Bond length (BLYP)

H2 1.3867 1.4000

LiH 3.005 3.016

Table 3.1: Exchange and correlation energies of different atoms, and bond

lengths of different molecules calculated using model 2 with D3Q19 lattice

compared to the known BLYP values.

an indication that m2 indeed solves the kinetic Kohn-Sham equation more

accurately than m1. Finally the results for simulations of the other systems

are summarized in Table.3.1.

34


3.3.2 Ethane molecule test

As a test of the proposed model, the C2H6 (ethane) molecule was simulated.

The carbon atoms were initially located such that their center of mass was

in the center of the simulation box and they were aligned along the z axis.

The H atoms were randomly located, three of them closer to the upper

carbon atom, and the remaining ones closer to lower carbon atom (Fig 3.4

left). This set up mimics the common scenario in which there is only partial

information available. The final configuration, obtained after 3.6 days of

run time on a single core, is shown in Fig 3.4 where the qualitatively cor-

rect shape of the ethane molecule and electronic density distribution can be

observed, compared to the initial configuration. The relative errors of the

bond lengths and angles with respect to the expected ones [122] are 1.3%

for the C − C bond length, a mean relative error of 2.1% for the H − C

bond length and a 7% for the H − C − H angles. Except for the angles,

the accuracy is comparable to that of a Carr-Parinello Molecular dynam-

ics (CPMD) simulation performed with identical initial conditions using a

wavefunction cutoff of 100Ry. It achieves a 2.7% C − C bond length error,

2.0% mean H − C bond length error and a 0.05% mean H − C −H angle

error. The CPMD simulation took about three hours, which is a small frac-

tion of the computational time spent by our model. However, CPMD uses

pseudopotentials while our model considers the bare Coulomb potential.

3.3.3 Pseudopotentials

Pseudopotentials are a way to reduce the computational cost of atomistic

simulations that works under the approximation that core electrons are

mostly inert [7] and play a minimal role in most of the chemistry. Although

pseudopotentials are designed to be highly accurate and transferable, it is

not always clear, a priori, how do they couple to different simulation meth-

ods. For instance pseudopotentials are known to be problematic or not

directly applicable in diffusion Monte Carlo and Green functions approaches

[123, 124, 125].

In order to assess how pseudopotentials couple to our method, tests were

35


Figure 3.4: (Color online). On the left side, the initial configuration of the

atoms that conform the ethane molecule is depicted. On the right side, the

final configuration is shown along with the electron density, the regions of

high(low) electron density are indicated by red(blue) surfaces

performed using the dual-space Gaussian pseudopotentials (DSGPP) intro-

duced in Ref. [10]. The DSGPP were chosen because they are optimized

for the BLYP exchange correlation potential used in this work and because

their real space representation is compatible with the real space nature of

our method.

The first non trivial example that includes both local and non local contri-

butions of the DSGPP is the BH molecule where the two inner electrons of

the boron atom are neglected. In this case the B atom is described by the

pseudopotential Vpp = Vloc +Hnonloc where

Vloc(r) =−Zione|r−RB|

erf

(|r−RB|√

2rloc

)+ exp

(|r−RB|2

2r2loc

)(3.55)

×[C1 + C2

|r−RB|2

r2loc

+ C3|r−RB|4

r4loc

+ C4|r−RB|6

r6loc

],

and

Hnonloc(r, r′) =

2∑i=1

Y0,0(r)p0i (r)h0

i p0i (r′)Y ∗0,0(r′) (3.56)

+∑m

Y1,m(r)p11(r)h1

1p11(r′)Y ∗1,m(r′).

36


The values of the constants Ci i = 1, ..4, rloc and hi as well as the functional

form of the projectors p0i (r),p1

1(r) can be found in Ref. [10]. The DSGPP

for boron was implemented and used to calculate the bond length of the

BH molecule. It was found that when pseudopotentials are used within our

approach the scheme becomes unstable (black dashed line Fig.3.5). The

instabilities were partially controlled by artificially resetting the wave func-

tions and electron density to their initial values after a fixed number of it-

erations while keeping the current position of the ions (blue dashed-dot line

Fig.3.5), but eventually instabilities arise. Different initial conditions and

resolution levels also suffer from instabilities (green doted line Fig.3.5). The

reason why our approach becomes unstable may be related to the overall

nonlinear nature of the system and the nonlocal component of the pseu-

dopotential that requires the evaluation of projection integrals of the form∫p0i (r′)Y ∗l,m(r′)ψ(r′), that may not be sufficiently resolved due to the fact

that ψ(r′) is only known at a limited number of lattice points.

10000 20000 30000 40000 50000Iterations

2.0

2.5

3.0

3.5

Bond

leng

tha0

BH Lx=10∆x = 0.1BH Lx=10∆x = 0.1BH Lx=10∆x = 0.06BLYP value

Figure 3.5: (Color online). BH bond length calculated using pseudopoten-

tials. The different lines show that the use of pseudopotentials within our

approach leads to numerical instabilities.

37


3.3.4 Lattice performance

The performance and accuracy of model 2 using either the D3Q19 or D3Q111

lattice was investigated by simulating the He atom as well as H2 and LiH

dimers. The generally observed trend is that for small resolutions the re-

sults obtained using both lattices differ, but as the resolution increases the

difference is reduced and the results converge.

Although both lattices lead to practically the same results, the D3Q111

lattice allows to chose a higher value for the diffusivity ~/m, that can be

used to control the convergence rate of the procedure. As an example, the H2

molecule was simulated using the D3Q111 lattice for three different values

of ~/m Fig.3.6(inset). It can be seen that in all cases the system converges

to the same value of the bond length, but for ~/m = 1 the approach is faster

than for ~/m = 1/3 or ~/m = 0.69. The D3Q111 lattice has almost six

times more velocity vectors than D3Q19, and the time of a single iteration

using the D3Q111 lattice was measured to be about 1.5 times longer than

that of the D3Q19 lattice. After accounting for this, the comparison between

the speed of convergence using both lattices for two different system sizes is

presented in Fig.3.6. It can be observed that the use of the D3Q111 lattice

allows for a faster convergence. However, for large resolutions, since both

lattices lead to the same accuracy, using D3Q111 presents no advantage in

terms of computational time. For instance for the Be atom we found that

the D3Q111 lattice with ~/m = 1 converges equally fast as D3Q19.

3.4 Summary

In this section a new and more accurate Lattice Boltzmann scheme to solve

the kinetic Kohn-Sham equations has been introduced and validated. The

scheme uses a novel way of implementing a semi implicit second order cor-

rection to the forcing term, that makes use of the known asymptotic be-

haviour of the simulated orbitals. This approach avoids the instabilities of

the explicit implementations and the computational load of solving implicit

systems of equations.

38

3.4. SUMMARY

0 200000 400000 600000Iterations

1.0

1.5

2.0

2.5

3.0

3.5

Bond

Leng

th(a

.u.)

Functionalh/m = 1h/m = cs2 = 0.69h/m = 1/3

0 1000 2000 3000 4000 5000 6000 7000Time/t0

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

Bond

Leng

th(a

.u.)

FunctionalD3Q19 L=80D3Q19 L=100D3Q111 L=80 h/m = 1D3Q111 L=100 h/m = 1

Figure 3.6: Effect of the numerical value of the diffusivity ~/m on the conver-

gence speed using the D3Q111 lattice in the simulation of the H2 molecule.

The use of pseudopotentials within our approach requires further work to

eliminate the associated instabilities and computational demands, not only

for the tested case, but also for general pseudopotentials. Possible ap-

proaches in that direction include subgrid refinements.

The results of the ethane molecule simulation show that our method can

39


reproduce the bond lengths of complex molecules, but that further work

is required to achieve an overall performance similar to that of established

methods such as CPMD, including the full integration with pseudopoten-

tials. It was also confirmed that the D3Q19 and D3Q111 lattices lead to the

same results for high enough resolution, giving an advantage to the D3Q19

lattice in terms of computational resources.

40

Chapter 4

Lattice Wigner Model

4.1 Wigner formalism

The Wigner formalism is a kinetic formulation of quantum mechanics phys-

ically equivalent to the Schrodinger representation [21]. It is however, very

different, as it treats both position and momenta as independent real vari-

ables instead of operators, similar to classical Hamiltonian dynamics and

kinetic theory.

The Wigner function is defined as

W (q,p, t) =1

(2π~)dW(ρ)

=1

(2π~)d

∫ ∞∞

dyρ(q− y/2,q + y/2)eip·y/~, (4.1)

where ρ(x,x′) is the real space representation of the density matrix of the

quantum system under consideration, d is the dimensionality of the sys-

tem and the Weyl transform, W(·), of a quantum mechanical operator O is

defined as:

O(q,p) =W(O) =

∫eip·y/~〈q− y/2|O|q + y/2〉dy. (4.2)

In general, W (q,p) is real and normalised in phase-space, i.e∫dp dqW (p,q, t) = 1. (4.3)

41

CHAPTER 4. LATTICE WIGNER MODEL

However, due to quantum interference effects, it is not positive semi definite,

and consequently, it cannot be regarded as a proper distribution function,

but rather as a quasi-distribution.

Expectation values of a physical observable O are obtained through the

prescription:

tr(ρO) =

∫dpdqO(q,p)W (p,q, t). (4.4)

The moments of the Wigner function with respect to the momentum variable

are defined as

Π(W )nα1,...,αn=

∫dppα1 ...pαnW (q,p), (4.5)

where n indicates the order of the moment and pαi denotes the αi component

of the momentum variable. The first two moments Π(W )0 and Π(W )1αi

can

be identified with the particle density ρ(x,x) and momentum density respec-

tively, whereas the sum of the diagonal terms of Π(W )2αiαj

is proportional

to the kinetic energy density.

The time evolution of the Wigner function can be obtained as the Weyl

transform of the Liouville-von Neumann equation, namely:

∂ρ

∂t=

1

i~[H, ρ],

where H = p2

2m+ V (x) is the Hamiltonian of the system.

The result (see Appendix A.1) is known as the Wigner equation and it reads

as follows:∂W

∂t+

p

m· ∇W + Θ[V ]W = 0, (4.6)

where Θ[V ]W can be written as

Θ[V ]W =

∫ ∞−∞

δ[V ](q,p− p′)W (q,p′)dp′, (4.7)

δ[V ](q, p) =

i

2π~2

∫ ∞−∞

(V (q− y/2)− V (q + y/2))eiy·p/~dy, (4.8)

or alternatively

Θ[V ]W = −∑|s|∈Nodd

(~2i

)|s|−11

s!

∂sV

∂qs∂sW

∂ps, (4.9)

42

4.2. LATTICE WIGNER MODEL

where s is a vector of non negative integers, |s| =∑d

i=1 si,∂s

∂as≡ Πd

i=1∂si∂asi

for a = q, p. Finally, it is important to notice that the different terms of

the Wigner equation Eq. (4.6) can be linked to the different terms of the

Liouville-von Neumann equation.

The convective term arises solely from the kinetic energy term in the Hamil-

tonian, whereas the force term Θ[V ]W originates from the potential energy

contribution. To be noted that spatial derivatives of the potential at various

orders couple to corresponding derivatives in momentum space, multiplied

by the corresponding power of the Planck’s constant ~. Such higher-order

terms are responsible for the “quantumness” of the Wigner representation

and the occurrence of negative values due to quantum interference effects.

4.2 Lattice Wigner model

In this section we introduce the lattice Wigner scheme. For this, it is con-

venient to work in the dimensionless form of Eq. (4.6). Upon the change

of variables q → l0x, p → m(l0/t0)v, t → t0τ where x, v, τ are the new

dimensionless variables and l0, t0 are characteristic length and time scales,

respectively, Eq. (4.6) and Eq. (4.9) can be written as:

∂W

∂τ+ v · ∇xW + Θ[V ]W = 0, (4.10)

and

Θ[V ]W = −∑|s|∈Nodd

(H

2i

)|s|−11

s!

∂sV

∂xs∂sW

∂vs, (4.11)

where H = ~t0ml20

, V = Vm(l0/t0)2 are the dimensionless reduced Planck constant

and potential terms, respectively.

Using the formalism introduced in Ch.2 the lattice kinetic discretization of

Eq. (4.10) is given by

Wi(x + viδt, t+ δt)− Wi(x, t) = −δt(Θ[V ]W )i. (4.12)

Following the lattice Boltzmann nomenclature, the Wi and (Θ[V ]W )i are

termed respectively “distributions” and “source distributions”. Observe that

43


the time evolution of the Wi distributions is given by Eq. (4.12) and that at

every spatial lattice point x there are Nq distributions (see sec. 2.1.2). From

these distributions, the moments such as density Π(W )0 = ρ or momentum

density Π(W )1α = ρuα, can be calculated at every time step using Eq. (2.8).

It is important to notice that, although a discretization by quadratures re-

quires no cutoff in velocity space, it does nonetheless involve a ceiling on the

highest moment for which Eq. (2.8) holds. In other words, it is a truncation

in discrete momentum space.

It is in principle possible to use Eq. (4.12) to track the time evolution of the

moments of the Wigner function under the action of a specified potential.

However, it was shown in Ref. [126] that the resulting structure of the forcing

term leads to numerical instabilities. To address this problem, the lattice

Wigner model is introduced as

Wi(x + viδt, t+ δt)− Wi(x, t) = δtΩi + δtSi +δt2

2DiSi,

Ωi = − 1

τw(Wi(x, t)− W eq

i (x, t)), (4.13)

where Si = −(Θ[V ]W )i, Di ≡ ∂∂t

+vi·∇ and W eq is an artificial “equilibrium”

distribution such that Π(W eq)nα1,α1,...,αn= Π(W )nα1,α1,...,αn

for n ≤ NΠ. τw > 0

and NΠ ∈ N are model parameters.

Compared to Eq. (4.12), Eq. (4.13) exhibits two additional terms. As it will

be shown, the term δt2

2DiSi [127, 61] also eliminates first-order discretization

artefacts in the present case. The term Ωi, is a regularizing artificial collision

term. Since Ω is a relaxation-type collision term, its use is allowed because

it preserves the positive semi definite character of the density matrix that

underlies the Wigner function [128, 129]. Its role is to improve the stability

of the numerical scheme by inducing selective numerical dissipation without

directly affecting the dynamics of the first n ≤ NΠ moments of the Wigner

equation. This can be seen as follows, let us consider the Taylor expansion

up to second order of the l.h.s of Eq. (4.13), namely

DiWi +δt

2D2i Wi = Ω + Si +

δt

2DiSi, (4.14)

44


by solving for DiWi and recursively substituting back in the second term of

the l.h.s of Eq. (4.14), it is found that

DiWi+

δt

2Di

(−δt

2D2i Wi + Ω + Si +

δt

2DiSi

)= (4.15)

Ω + Si +δt

2DiSi.

From Eq. (4.15), it can be seen that had the term δt2

2DiSi not been intro-

duced in the definition of the model, there would be an uncompensated term

of order δt. Further, if the velocity moments of Eq. (4.15) are calculated,

it can be seen that all the contributions involving Ωi vanish, provided that

the order of the moment is not larger than NΠ. Thus, up to terms of order

O(δt2) and n ≤ NΠ, the resulting set of equations

∂

∂tΠ(W )nα1,α1,...,αn

+∇ · Π(W )n+1α1,α1,...,αn+1

= Π(S)nα1,α1,...,αn+O(δt2), (4.16)

is consistent, with the moments of Eq. (4.10). In summary, Eq. (4.13) ap-

proximately solves the Wigner Equation by solving the corresponding trun-

cated hierarchy of equations Eq. (4.16).

So far, the series representation of the source term Eq. (4.9) was assumed. It

is clear that its equivalence to the natural integral representation Eq. (4.8)

depends on the potential smoothness [26]. Furthermore, in passing from

Eq. (4.8) to Eq. (4.9) it is also assumed that the potential can be Taylor

expanded and that radius of convergence is the whole domain of integration.

This extra condition may not be fulfilled by some smooth functions. This

rises the question on whether the Lattice Wigner model depends on the

chosen representation of the source term. To address this question, first

notice that only the moments of the source term are relevant to the dynamics

of the moments of the Wigner function Eq. (4.16), thus it is sufficient to

show that Eq. (4.9) and Eq. (4.8) have the same moments. First consider

45


the moments of the source term using the series representation,

Π(S)n =

∫pn

∑|s|∈Nodd

(H

2i

)|s|−11

s!

∂sV

∂xs∂sW

∂vs

dp, (4.17)

since only the Wigner function depends on the momentum variable the pre-

vious expression can be simplified as

Π(S)n =∑|s|∈Nodd

(H

2i

)|s|−11

s!

∂sV

∂xs

∫pn∂sW

∂vsdp, (4.18)

using iterated integration by parts it is found that∫pn∂sW

∂vsdp = (−1)λ

n!

(n− s)!Π(W )(n−s), for n ≥ s (4.19)

thus the final result is

Π(S)n =n∑

s∈Nodd

(−1)s(~2i

)s−1(n

s

)∂sV

∂xsΠ(W )(n−s). (4.20)

For the Integral representation of the source term, the starting points are

Eq. (4.7) and Eq. (4.8) and the moment definition that leads to

Π(S)n =

∫ ∫ ∫W (q, p)

(i

π~2

)[V (q + y)− V (q − y)

]pne−i2(j−p)y/~dpdydj.

(4.21)

The previous expression can be reorganized as

Π(S)n =

∫ ∫W (q, p)

(i

π~2

)[V (q + y)− V (q − y)

]e−2jy/~

∫pnei2py/~dpdydj,

(4.22)

using the Fourier property∫xne−iνxdx = 2πinδ(n)(ν) it follows that

Π(S)n =

∫ ∫W (q, p)

(i

~

)[V (q + y)− V (q − y)

]e−2jy/~

(~2i

)nδ(n)(y)dydj.

(4.23)

The terms involving the y variable are integrated as follows∫ [V (q + y)− V (q − y)

]e−2jy/~δ(n)(y)dy =

(−1)n(dn

dyn[V (q + y)− V (q − y)]e−2jy/~

)|y=0 =

(−1)nn∑s=0

(n

s

)(1− (−1)k)

∂sV

∂xs

(−2ij

~

)n−s, (4.24)

46


finally replacing Eq. (4.24) back in Eq. (4.23) leads to Eq. (4.20). It can

be seen that the general moments of the source term are independent of

the chosen representation. The only depend on the local existence of the

derivatives, not on the more restrictive requirement that the function has a

Taylor expansion with infinite radius of convergence.

Since the Wigner function is bounded over the phase space [130] and only a

limited number of moments are required, due to the truncation in Eq. (4.16),

an expansion in orthonormal polynomials can be assumed for W , S and W eq,

from which the expressions of the corresponding distributions can be derived.

From sec. 2.1.2 the formal result is

W (x,v, t) ≈ ω(v)

Np∑n

an(x, t)Pn(v), (4.25)

where Pn(v) is a family of polynomials orthonormal under the weight

function ω(v) and Np is the maximum order of the polynomials used in the

representation. The expansion coefficients are given by

an(x, t) =

∫d3vW (x,v, t)Pn(v). (4.26)

It is interesting to note that, since the expansion coefficients are linear com-

binations of the moments of the distribution, this procedure is similar to

Grad’s method [131], although not restricted to Hermite polynomials.

In practice, Hermite polynomials are a convenient choice, as they permit the

systematic generation of lattices in any number of dimensions [108, 109, 110].

For example, in one dimension and using Hermite polynomials, Hn(v; cs)

with weight function ω(v; cs) = 1√2πc2s

e− v2

2c2s and parameter cs > 0 [132], the

expressions for Wiand W eqi are, by construction, given by

Wi = ωi

Np∑n

an(x, t)Hn(vi; cs), (4.27)

W eqi = ωi

NΠ∑n

an(x, t)Hn(vi; cs), (4.28)

47


whereas Si can be explicitly calculated (see Appendix. A.3) leading to

Si = −ωi∑n,s

an(x, t)

√(n+ s)!

n!

(−H/i)s−1

csss!

∂sV

∂xsHn+s(vi; cs), (4.29)

where H is the dimensionless reduced Planck constant. It should be noted

that, in general, the condition NΠ < Np must hold, for otherwise Ωi becomes

trivially zero.

4.3 Validation

We validate our model, first for the harmonic oscillator and then for the case

of the anharmonic potentials with up to sixth order.

4.3.1 Harmonic potential

As a first example to illustrate the lattice Wigner method described in the

previous sections, we consider the quantum harmonic oscillator described by

the following Hamiltonian Eq. (4.30):

H =p2

2+

1

2x2. (4.30)

We track the time propagation of the Wi distributions from the initial condi-

tions, for different choices of the spatial resolution and number of moments

NΠ.

The initial condition consists of an equally weighted superposition of the first

two eigenstates of the quantum harmonic oscillator, |φ〉 = 1√2(|ψ0〉 + |ψ1〉).

Since this state is non stationary, it shows time oscillations all along the

evolution.

The Wigner function corresponding to |φ〉 can be calculated from the defi-

nition Eq. (4.1), the result being:

W|φ〉(x, v) =e−

v2+x2

H

(√2√Hx+ v2 + x2

)πH2

. (4.31)

48

4.3. VALIDATION

Observe that if Hermite polynomials are used, W|φ〉(x, v) is already of the

form Eq. (4.25). It follows that the distributions, Wi, are given by

Wi = ωie− x2

2c2s√2πc2

s

(H2(vi; cs)

2+c2s + 2csx+ x2

√2c2s

H1(vi; cs)

)(4.32)

where the specific values of wi, vi and cs for different lattices are given in

the Appendix A.6 and H was taken to be numerically equal to 2c2s.

The results of our simulation using the D1Q3 lattice with a lattice spacing

δx = 0.06 and a equilibrium function with NΠ = 3, are shown in Fig. 4.1.

On the upper panel, it can be seen that both the zeroth and first order

moments (ρ, ρu) are correctly propagated and agree with the theoretical

values at different times.

From Eq. (4.25), it is clear that, given the expansion coefficients an(x, t), it

is possible to reconstruct an approximation of the Wigner function. These

coefficients can be obtained from Eq. (4.27) as linear combinations of the

moments of the Wigner function, which, in turn, can be calculated by means

of quadratures.

The results for the quantum harmonic oscillator are presented on the lower

panel of Fig.4.1, which shows the phase-space representation of the Wigner

function. From this figure, a prototypical shape is clearly recognized, in-

cluding the expected non classical regions of negative values.

To quantitatively characterize the present method, we have studied the ef-

fects of the spatial resolution, lattice configuration and number of preserved

moments (NΠ)

To this end, the root mean square error between the theoretical density and

the simulated one after a full oscillator period, T0,

∆ =

√1

Nx

∑x

(ρtheory(x)− ρsim(x))2, (4.33)

was evaluated for different conditions.

In Fig.4.2 a), the effect of using different lattices and resolution levels is

49


Figure 4.1: (Color on line) a) First moment of the Wigner function (den-

sity) for different times being fractions of the oscillation period T0. The

inset shows the corresponding time evolution for the second moment (ve-

locity density). The symbols denote the simulation and the solid lines the

analytical solution. b) Phase space reconstruction of the quantum harmonic

oscillator Wigner function at τ = T0. The dashed contour line shows where

the Wigner function vanishes.

50

4.3. VALIDATION

1.5 1.6 1.7 1.8 1.9 2.0 2.1log(1/δx)

−5.0

−4.5

−4.0

−3.5

log(∆)

a) D1Q3D1Q8D1Q16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15NΠ

10-6

10-5

10-4

10-3

10-2

10-1

100

∆

b)D1Q8D1Q16

Figure 4.2: (Color online) a) Root mean square error of the density, ∆, for

different velocity lattices and spatial resolution using an equilibrium function

that preserves the first three moments. b) Effect on ∆ of using different

values of NΠ for two different lattices with δx = 0.008, Np = 8 and Np = 16

for D1Q8 and D1Q16 respectively.

shown; two features are apparent, namely that the error ∆ decreases quadrat-

ically as a function of 1/δx and that, at a given value of the resolution δx,

schemes with higher number of preserved moment provide better results.

The effect of changing the value of NΠ is presented in Fig.4.2 b). From this

figure, an ideal range for NΠ can be identified. If NΠ is low, the order of

truncation of Eq. (4.16) leads to a crude approximation which in turn yields

large values of ∆. On the other end, if NΠ is equal to Np, the model becomes

unstable (this is why we have chosen NΠ < Np for both lattices), because

then the collision term Ωi, Eq. (4.13), vanishes, which implies no artificial

dissipation, hence the onset of stability issues discussed in Ref. [126]. The

anomalous point NΠ = 13, in the D1Q16 case on Fig.4.2 b), may be due to

51


compensated high order modes that reduce the artificial dissipation leading

to a larger than expected error. That was only observed for the particular

case of the harmonic oscillator (and won’t be the case for the anharmonic

potential).

4.3.2 Anharmonic potential

As a second example, we simulate the anharmonic quantum oscillator de-

scribed by the Hamiltonian

H =p2

2+

1

2x2 + αx4 + βx6 , (4.34)

where the parameters α and β determine the strength of the anharmonic

terms.

As discussed earlier on, anharmonic terms involve genuinely quantum effects

in the forcing expansion described in Eq. (4.9).

Similar to the previous example, the initial condition is taken to be the equal

superposition of the first two eigenstates of the Hamiltonian Eq. (4.34) |φ〉 =1√2

(|ψ0〉+ |ψ1〉). Here, |ψ0〉, |ψ1〉 are obtained by direct diagonalization of

Eq. (4.34), using a truncated basis set of 50 eigenvectors, ϕn, from the

quantum harmonic oscillator.

The density matrix for this system is given by

ρ(x, x′) =

∑m,n

cnc∗mϕm(x)ϕn(x′), (4.35)

where the coefficients cn are easily obtained from the diagonalization proce-

dure. Given ρ, the corresponding Wigner function W|φ〉(x, v) was calculated

(see Appendix A.4) with the help of the results in Ref. [24], leading to the

following expression:

W|φ〉(x, v) =1

2πH

∑n≤m

2

1 + δm,n<(cnc

∗mkn,m), (4.36)

52

4.3. VALIDATION

where <(·) denotes the real part, the coefficients km,n are given by

km,n =2(−1)min (m,n)

√min (m,n)!

max (m,n)!e−

x2+v2

H

(2

H(x2 + v2)

) |m−n|2

L|m−n|min (m,n)

(2

H(x2 + v2)

)e(i(m−n) arctan (v/x)) (4.37)

and Lmn is the m order n degree associated Laguerre polynomial.

In order to find the corresponding Wi, it is used that each term of Eq. (4.36)

can be written as the product of a polynomial and a Gaussian function in the

velocity space. Once the Gaussian is factored out, the result in Eq. (4.38)

is readily cast into the form of Eq. (4.25), namely

W (x, v) =e−

v2

H

√2πH

∑n≤m

2

1 + δm,n(crnmk

rnm − cinmkinm), (4.38)

In the above, cnm = (crncrm + cinc

im) + i(cinc

rm − cimcrn) and kmn = ev

2/H

2√πHkmn,

where the superscripts r and i denote real and imaginary parts, respectively.

Since Hermite quadratures are in use, the Wi follow directly.

The results for the anharmonic oscillator Eq. (4.34) with parameters α = 0.1,

β = 0.05 are summarized in Fig.4.3 a), from which it is apparent that for

mild anharmonicities, the method is able to properly evolve the given initial

condition. In Fig.4.3 b) the error as a function of the used lattice and

resolution is reported; the general trend is an error decrease at increasing

resolution; it decreases as NΠ increases and ∆ tends to saturate relatively

fast. In Fig.4.3 c), the behaviour of ∆ as function of NΠ is shown. Similarly

to the harmonic case, as NΠ increases, ∆ decreases, until it reaches and

optimal value (NΠ = 8) and then saturates.

In order to study stronger anharmonic cases, not only larger resolutions,

but also more terms in the representation Eq. (4.27) of the Wigner function

are required because as the strength of the anharmonicity increases, so does

the number of terms in Eq. (4.38). In order to account for them, both the

number of polynomials in Eq. (4.27) and the size of the velocity lattice needs

to be increased.

53


−10 −5 0 5 10x

0.000.050.100.150.200.250.300.35

ρ

a)τ= T0/3

τ=3T0/4

τ= T0

1.6 1.8 2.0 2.2 2.4log(1/δx)

−4−3−2−1012

log(∆)

b)

D1Q16 NΠ = 1D1Q16 NΠ = 15D1Q20 NΠ = 1D1Q20 NΠ = 15

1 3 5 7 9 11 13 15 17 19NΠ

10-4

10-3

∆

c) D1Q16D1Q20

Figure 4.3: (Color online) Results for the anharmonic oscillator with pa-

rameters α = 0.1, β = 0.05. a) Comparison between the density obtained

using the Lattice Wigner method and the one obtained directly from the

Schrodinger equation. (b) Error as a function of resolution, used lattice and

number of projections NΠ. It can be seen that as the resolution increases the

error saturates and that the error decreases upon increasing the number of

projections. (c) Effect on ∆ of using different values of NΠ for two different

lattices with δx = 0.008.

The effect of the relaxation time τw was also studied. By definition, this

parameter controls dissipative effects and consequently, it is not expected to

affect the results. However, numerically it was found that this is the case

only in the range 0.56 ≤ τw ≤ 5, which is similar to the allowed range of τw

in the closely related lattice Boltzmann schemes. This is possibly due to a

marginal coupling between high order moments and the ones relevant to the

Wigner dynamics.

54

4.4. COMPUTATIONAL COST

4.4 Computational cost

For an arbitrary problem, it is a priori not known how many polynomi-

als are required to give an accurate representation of the Wigner function,

Eq. (4.25). The number of polynomials, Np, determines the smallest lattice

that is able to support the orthogonality constraints, Eq. (2.12), and also

the computational cost of solving the respective problem. The scaling of the

cost can be estimated by observing that a single update of the complete set

of lattice points involves four basic steps: 1) the calculation of the expansion

coefficients an in Eq. (4.27), 2) the update of the source term distributions

in Eq. (4.29), 3) the update of W eq in Eq. (4.28), and 4) the update of Wi

according to Eq. (4.13).

The number of floating point operations (+,−,×, /) required at each step

scales respectively as O(NpNq), O(NsNpNq), O(NpNq) and O(1), where Ns

is the number of terms in Eq. (4.29) that are consistent with a cutoff at s

in H.

Under a worst-case scenario, i.e. the largest possible NΠ, NΠ ∼ Np and

Nq ∼ Np, the total cost of updating a single site scales as:

O(NsN2p +N2

p ), (4.39)

In 1D, Ns is effectively O(1) and therefore the cost per site update scales as

O(N2p ). This bound was tested and the corresponding results are reported in

Fig.4.4, from which it is seen that the cost of updating a single site scales like

N2p . The difference with respect to the theoretical value can be accounted

for by the time to access data, which becomes dominant as the size of the

problem is increased.

In 2D, Ns scales O(s2), and since the number of polynomials and lattice

vectors also scale quadratically, the update cost per site is expected to grow

as O(s2N4p ).

55


101

Np

10-6

10-5

t(s)

Figure 4.4: (Color online) The symbols show the time it takes to update

a single site as a function of the number of polynomials. The dashed line

shows the scaling t ∼ N2p . The simulations were performed for the quantum

harmonic oscillator, in every case NΠ was set to the highest value compatible

with numerical stability.

4.5 Lattice Wigner application

4.5.1 1D system

As an application of the proposed model, next we study the dynamics of

the zeroth and first moment of the Wigner function for a system subject to

the combined action of an external drive and potential barriers. As a model

of an homogeneous system, we assume that the initial state is given by the

following thermal density matrix:

ρ =∑|p〉〈p|e−βp2/2m, (4.40)

where |p〉 are plane waves, m is the mass of the particle and β is the inverse

temperature.

The system is taken to be of finite length L, which implies quantization of

the allowed momenta. However, L is assumed sufficiently large to justify a

continuum limit.

56

4.5. LATTICE WIGNER APPLICATION

The potential barriers extend throughout the domain according to:

V (x) =v0

2(erf ((x+ δ/2)ξ)− erf ((x− δ/2)ξ)) , (4.41)

where v0, δ and ξ define the height, width and stiffness of the barrier,

respectively. The barriers were symmetrically distributed at the points

xi = ±Di, i = 0, 1, . . . , Nb where D is the inter barrier distance.

The system is driven by the potential Vd(x) = −ax, where a determines

the strength of the forcing, and is assumed to be open, i.e. each end of the

domain is connected to a fixed reservoir, also described by Eq. (4.40).

Similar to the previous examples, a lattice Wigner representation of the form

Eq. (2.14) is required for the initial condition. In this case, the Wigner

transform of Eq. (4.40) is given by

W (x, v) =1

2πHe−

v2β2 , (4.42)

where β = βm(l0/t0)2.

Comparing Eq. (4.42) with the form of the Hermite polynomials weight

function, ω(v; cs) = 1√2πc2s

e− v2

2c2s , and using Eq.(4.25,2.14) it follows that if

β and H are fixed respectively to 1/c2s and cs then only the a0 expansion

coefficient, that corresponds to the constant Hermite polynomial, is required.

That is, the representation of the initial condition is optimal and the the

distributions Wi are proportional to the weights of the lattice configuration

Wi = ωi1√2π. (4.43)

Finally, it is important to observe that the barrier potential Eq. (4.41)

has infinitely many non-zero derivatives, as opposed to the harmonic and

anharmonic potentials. This implies that a cutoff in Eq. (4.29) needs to

be chosen. For the present simulations, the parameters characterising the

barriers were fixed as v0 = 0.4β−1, δ = 2 and ξ = 1. In this case, the

cutoff is taken at s = 9, since the next contribution, s = 11, is six orders of

magnitude smaller than the first order contribution.

The first two moments of the Wigner function were studied for different

values of the driving force Vd, number and location of the barriers. Fig. 4.5 a)

57


Figure 4.5: (Color online) a): Reconstructed steady state Wigner function.

b): The second moment of the Wigner function as function of Vd in steady

state, for different number of barriers Nb. c) σ as a function of the number of

barriers for different inter barrier distances. The system size is set to 400 (in

dimensionless units) and all simulations were performed with a resolution

δx = 0.004, with NΠ = 14, using a D1Q16 lattice.

shows the reconstructed steady state Wigner function, W (x, v), of a system

with Vd = 10−4 and four barriers randomly located across the domain.

Similar results were obtained for different configurations of barriers and driv-

ing force. The first visible feature is that W (x, v) shows a number of “cuts”

along the v axis at given values of x. These cuts are located at the potential

barriers. Along the barriers, the Wigner function attains lower values as

compared to the nearby regions. This implies that the density in the cuts is

smaller compared to the surroundings.

A second feature is that the Wigner function is nearly translationally invari-

58


ant in the interstitial region between two subsequent cuts, as long as the

cuts are sufficiently far apart, which implies that the density ρ is uniform

between cuts.

Further, from Fig. 4.5 a) it seems that the Wigner distribution is symmetric

along the v = 0 axis, although this is not the case. The driving potential

slightly shifts the distribution, leading to a finite and spatially uniform first

moment (ρu), which is consistent with the continuity equation ∂ρ∂t

+∇ρu = 0,

at steady state. Finally, it can be seen that the Wigner function is nowhere

negative, first, because the reservoir naturally tends to wash out quantum

coherence and second, because the ratio between the height of the barriers

and the thermal energy is about 0.4, whereas in applications such as resonant

tunnelling diodes, such ratio is about ten [133]. Similarly to the case of strong

anharmonicities, to treat systems with higher energy barriers, more terms

i.e. polynomials in the representation of the Wigner function (Eq. (4.25))

are needed, along with the corresponding increase in the velocity lattice size.

From Fig. 4.5 b) it can be seen that the relation between the velocity density

ρu and the forcing potential Vd is linear for a fixed number of barriers, uni-

formly and symmetrically distributed across the domain. Further, Fig. 4.5

b) also implies that, as the number of barriers increases, the electric conduc-

tivity, σ, decreases.

In other words, the capacity of the system to transport momentum from one

end to the other, declines with number of barriers. To quantify this relation,

simulations with a fixed number of barriers, Nb, but different interbarrier

distances, D, were performed. The results, reported in Fig.4.5 c), show

that the overall tendency is a decreasing σ at increasing Nb. However, this

decrease shows a dependence on the interbarrier separation D. For D = 2

and D = 2.5, σ is nearly constant, whereas for D ≥ 3 it decreases rapidly

with Nb. Furthermore, σ saturates above D ≥ 5.

The above picture can be understood as follows: once the barriers are suffi-

ciently close together, they overlap and the resulting potential is no longer

a set of disjoint barriers, but rather a single larger barrier Fig.4.6 a).

In this case, it is known that all incoming plane waves with energy below

59


D a b c

5 53.3± 0.2 38.2± 0.2 0.898± 0.002

5.5 53.7± 0.2 37.6± 0.2 0.896± 0.002

6 53.6± 0.2 37.7± 0.2 0.898± 0.003

6.5 53.4± 0.2 37.8± 0.2 0.899± 0.003

8 53.3± 0.2 37.9± 0.2 0.898± 0.003

Table 4.1: Individual fitting parameters of Eq. (4.44) for different interbar-

rier distances

the barrier are exponentially attenuated as a function of the barrier length,

whereas those with energy above the barrier manage to penetrate, if only

with a non-zero reflection probability. It follows then that the number of

states that can cross the barrier diminishes as the length of the barrier

increases thereby limiting the amount of momentum transported across the

system, thus leading to an overall decrease of σ.

When the separation between the barriers is sufficiently large, the system

can be approximated as a sequence of disjoint barriers. If the system was

closed, this would imply that, T being the transmission coefficient for a

single incoming plane-wave on a single barrier, the transmission coefficient

for n barriers, would be TN = TN(1−T )+T

(See Appendix A.5), without any

dependence on the interbarrier separation. Since this holds for every plane

wave contributing to the thermal density matrix, the system as a whole is

expected to follow a similar trend.

From the previous picture, it can be inferred that the σ − n relation must

have a similar form for the D ≥ 5 settings. The semi empirical formula

σ = a+b cn(1−c)+c , where a, b, c are parameters depending on the interbarrier

separation, offers a good fit to the casesD = 5, 5.5, 6, 6.5, 8. From Table4.5.1,

it is apparent that the parameters a, b and c are constant within error bars.

Therefore, for D ≥ 5, the relation between σ and n and D, is effectively

independent of D and given by

σ = a+ bc

n(1− c) + c, (4.44)

60


Figure 4.6: (Color online) a) The total potential as a function of the inter-

barrier separation. For D = 2 the barriers are close enough such that the

total resulting potential acts as a single barrier. As the interbarrier separa-

tion increases, the resulting potential exhibits the structure shown for the

D = 2.5, 3 cases. b) The symbols show the behaviour of ρu, averaged

over 50 random samples, as a function of the driving potential. The dashed

lines show the behaviour of ρu in the uniformly distributed case with an

interbarrier distance D = 8.

with a = 53.4± 0.2, b = 37.8± 0.2, c = 0.899± 0.002.

The intermediate case 2.5 < D < 5, when the barriers do not form a single

monolithic barrier and the system can no longer be regarded as a superposi-

tion of disjoint subsystems, requires a deeper analysis which is left for future

work.

We have also studied the momentum transport in the presence of a random

distribution of barriers.

Simulations were performed for a fixed number of barriers Nb, randomly

located across the domain. The minimum distance between any two barriers

was constrained to be larger than 2 lattice sites, in order to avoid excessive

overlap, leading to an effective single larger barrier instead of two distinct

ones. The results are presented in Fig. 4.6 b), where for every instance 50

random realizations were considered.

The main observation is that the relationship between the current ρu and

Vd is, on average, the same as with uniformly distributed barriers, with an

interbarrier distance D > 5. This result can be understood as follows; since

61


the barriers are constrained to be far apart, most configurations behave as a

collection of subsystems. This, in turn, implies that σ only depends on the

number of barriers Eq. (4.44) and, as a consequence, the average relation

between ρu and Vd does not depart significantly from the case of a regular

distribution of barriers.

4.5.2 2D system

The transport properties of a square shaped two-dimensional system of side

length L, were also studied. Open boundary conditions were used at the

x = 0 and x = L ends, while periodic boundary conditions are used at the

y = 0 and y = L ends. The system is driven by an external potential of the

form Vd(x) = −ax, where a controls the strength of the external driving.

The barriers are described by the potential

V (x) = v0e− |x|

2

2ξ2 , (4.45)

where v0 determines the height of the barrier and ξ its stiffness.

The initial state is also given by Eq. (4.40), where |p〉 is assumed to be

two dimensional. Following calculations similar to the 1D case, the initial

condition for the lattice Wigner model is given by

Wi = ωi1

2π. (4.46)

The cutoff of Eq. (4.11) was set to s = 9 and the simulations where carried

out on a 256× 256 grid, using the D2Q16 lattice (see Appendix for details).

Similarly to the 1D case, regular and a random settings for the location of

the potential barriers were considered. Fig.4.7 a) shows a sample result for

a simulation with 16 randomly placed barriers. The location of the poten-

tial barriers can be easily identified through the blue color spots, denoting

density depletion. Further, it can be seen that the streamlines bend around

the potential barriers, similarly to the way fluid streamlines turn around

obstacles in porous or campylotic media [134, 113].

The relation between the flux Φ (2D analogue of ρu in 1D) and the driving

potential is presented in Fig. 4.7 b). From this figure, it is seen that the

62


Figure 4.7: (Color online) a) Density map for a 2D system with 16 randomly

located barriers. The effect of the barriers can be observed on the regions

that get depleted (blue color) and on the streamlines that bend around them.

b) Behaviour of Φ as a function of the driving strength. The red dashed line,

blue dot-dashed line and green dotted line correspond to systems where the

barriers are arranged in regular grids of 2× 2, 3× 3 and 4× 4 barriers with

an interbarrier distance of D = 9. The circle, square and triangle symbols

represent respectively the mean flux of 50 random samples of 2 × 2, 3 × 3

and 4× 4 randomly located barriers. The solid lines are a guide to the eye

showing the trend of Φ as a function of Vd for the case of random barriers.

relation σ versus Φ and Vd is linear when the barriers are regularly organised

on a square grid, and that σ decreases at increasing number of barriers.

Furthermore, when the barriers are randomly placed, the average behaviour

63


of Φ is close to the regular case, as it was also observed in 1D. However, as the

number of barriers increases, specific realizations can deviate significantly

from the regular grid behaviour, this can be seen from the error bars of the

red triangles in Fig. 4.7 b).

Finally, for the purpose of showing the viability of the present method also

in three spatial dimensions, we have simulated a three-dimensional open

quantum system. The simulation was performed on a 20 × 20 × 20 lattice,

with a D3Q125 velocity set, which was chosen because it includes terms of

order H2 in the force expansion Eq. (4.29). The boundary conditions are

open (thermal density matrix) at the planes normal to F (See Fig.4.8) and

periodic on the remaining boundaries.

Figure 4.8: (Color online) The figure shows the density ρ and streamlines of

ρu of an open driven system in 3D. The drive is given by a constant force,

F , along the x direction.

In addition to the driving potential generating a force in the x direction,

a random potential is included. It is modelled as a smooth Gaussian with

varying amplitude at different locations in the domain. From Fig.4.8, it

is seen that the streamlines tend to circumvent the regions of low density,

64

4.6. SUMMARY

where the potential is high, and concentrate in the regions of high density,

thus effectively avoiding “impurities”. A systematic analysis of the transport

properties of this three-dimensional open quantum system is left for future

work.

4.6 Summary

In this work, a new numerical method to track the time evolution of the

Wigner function has been introduced. The stability problem previously de-

scribed in Ref. [126], is handled through the inclusion of an artificial collision

term, designed in such a way as to preserve the dynamics of the relevant mo-

ments of the Wigner function. Reducing the momentum space to a compar-

atively small set of representative momentum vectors, opens up interesting

prospects for the simulation of one, two and also three dimensional quan-

tum systems. Preliminary results for 1D systems with regular and random

potentials provide evidence of linear transport laws which are independent

of the barrier configuration for dilute systems. In the 2D case, we find the

same transport laws at low barrier density, while for higher concentrations,

deviations from the linear behaviour are observed (as shown in Fig.4.7 b)

when the barriers are randomly located. Finally, we also presented a pre-

liminary simulation of a 3D open quantum system, to illustrate the ability

of the model to handle the three-dimensional Wigner equation.

The computational cost of the method scales polynomially with the number

of basis functions. However, the simulations show that just a few equilibrium

moments and comparatively small lattices, are often sufficient to obtain

reasonably accurate results.

The present work opens up a number of research directions for the future.

Technically, the performance can be improved by choosing alternative fam-

ilies of lattice configurations and orthonormal polynomials, or by directly

designing orthonormal polynomials that fit the specific problems under in-

vestigation. Since our model is computationally viable also in 3D, problems

like the heat transport properties of three-dimensional semiconductor struc-

65


tures, which are highly relevant to the next generation electronics [135],

could be studied. In addition, the method could also be used as a practical

tool to explore fundamental issues, such as the relation between quantum

entanglement and the Wigner function in diverse systems [136, 137].

66

Chapter 5

Thermal gas rectification using

a sawtooth channel

5.1 Ratchet systems

The subject of ratchet systems has been studied in a number of contexts,

from classical and quantum physical systems to biological examples. In the

following we illustrate the main physical ideas of such systems following the

analysis of the Smoluchowski ratchet done by Feynmman.

The set up of the Smoluchowski-Feynmman ratchet is shown in Fig. 5.1.

It consists of two chambers filled with a gas temperature T , and an axle.

On the right chamber the axle is connected to a set of paddles, on the left

chamber it is connected to a ratchet pawl mechanism and in the middle to a

load. The intended working mechanism of the contraption is as follows: On

the right chamber, the gas particles will collide with the paddles and given

the symmetry of the system, it is expected that the particle-paddle collisions

will not generate a rotation of the axle in any preferred direction. However,

it is also expected that the fluctuations will be such that in some occasions

there is an excess of collision that lead to a small motion in a preferred

direction. If the direction of rotation is that of the “forward” or “easy”

direction of the ratchet in the left chamber, then the axle will rotate and in

the process it will perform work on the load rising it. From the second law of

thermodynamics such a mechanism can not function for it could extract work

67

CHAPTER 5. THERMAL GAS RECTIFICATION USING ASAWTOOTH CHANNEL

from a single heat reservoir. From a mechanical point of view the system

would not be able to function because if it assumed that the ratchet-pawl

interaction is conservative, on average, the pawl would not be able to lock

in on the ratchet tooth [73]. If the interaction is made dissipative then the

system will heat up and thermal fluctuations will preclude its functioning

unless the excess heat is taken away by a reservoir at lower temperature,

case in which no violation of the second law occurs.

In summary, the previous example shows that the two necessary elements

for a system to exhibit a ratchet or rectifying behaviour are broken spatial

symmetry and dissipation (which leads to non equilibrium conditions) [74,

75]. Notice, that the example also shows that spatial asymmetry alone is not

enough to observe a ratchet effect, this is due to the intrinsic time symmetry

of the mechanical equations of motion [73].

Figure 5.1: Ratchet and pawl mechanism: the mechanism has two parts,

on the right chamber there are a set of paddles attached to an axle that

is connected to the ratchet and pawl system on the left chamber. (Picture

taken from [73] )

68

5.2. CANONICAL MOLECULAR DYNAMICS

5.2 Canonical molecular dynamics

5.2.1 Molecular dynamics

Given the practical impossibility of analytically solving the Newton equa-

tions of motion that describe systems of many interacting particles for all but

a small number of examples, the use of alternative methods is necessary. Be-

sides the tools from statistical physics, one of the most successful approaches

to understand systems of many classical particles is that of molecular dy-

namics simulations. The method was introduced in the 1950 and it has been

applied to a number of systems in fields ranging from physics [138, 139, 140]

to biology [141, 142]

The general idea of molecular dynamics is to integrate in time the Newton

equations of motion of a system of particles. More specifically, it is assumed

that the system is composed by N particles that interact with each other via

a potential of the form V (x1,x2, . . . ,xN). This leads to the set of equations

xi = − 1

mi

∇xiV (x1,x2, . . . ,xN) i = 1, 2, . . . , N (5.1)

that are integrated in time using any one of a number of available methods

such as Verlet [143], Leap frog [144] or predictor-corrector algorithms [145].

The choice of method depends on the particular system under considera-

tion, for instance Verlet methods are suitable for conservative systems be-

cause they are known to be simplectic integrators i.e. they conserve phase

space volume and are therefore guaranteed to be energy conserving [146].

Similarly, predictor-corrector algorithms are better suited for modified first

and second order equations of the general form x = g(x, x) [147]. Since

the system of interest in this work is not conservative and includes terms

proportional to x, a predictor-corrector algorithm of fifth order is used.

5.2.2 Predictor corrector integration

Predictor-corrector algorithms for molecular dynamics are based on the idea

that dynamical quantities such as position, velocity and acceleration at a

future time t+δt can be predicted using a forward in time Taylor expansion,

69


and then corrected using the information from the equation of motion. More

specifically, let the scaled time derivatives of the position x be defined as

r0 = x, r1 = δt ddt

x, r2 = δt2

2d2

dt2x, r3 = δt3

6d3

dt3x, r4 = δt4

24d4

dt4x then the

predicted values rpi , i = 0, 1, 2, 3, 4 at a future time t + δt that would be

obtained from a Taylor expansion are given byrp0(t+ δt)

rp1(t+ δt)

rp2(t+ δt)

rp3(t+ δt)

rp4(t+ δt)

=

1 1 1 1 1

0 1 2 3 4

0 0 1 3 6

0 0 0 1 4

0 0 0 0 1

r0(t)

r1(t)

r2(t)

r3(t)

r4(t)

(5.2)

It is clear that Eq.(5.2) alone can not lead to the correct dynamics of the

system because it is missing the information from the forces. This is ac-

counted by the correction step. Given the predicted location and possibly

velocity of the particles, the force at time t + δt is calculated and from it

the corrected acceleration, rc2, is found. Using rc2 the correction term ∆r2 is

defined as

∆r2 = rc2 − rp2, (5.3)

and the full correction step is given byrc0(t+ δt)

rc1(t+ δt)

rc2(t+ δt)

rc3(t+ δt)

rc4(t+ δt)

=

rp0(t+ δt)

rp1(t+ δt)

rp2(t+ δt)

rp3(t+ δt)

rp4(t+ δt)

+

c0

c1

c2

c3

c4

∆r2 (5.4)

where the coefficients ci where derived by Gear [145] and for the specific case

of a fifth order predictor corrector are given by

c0 c1 c2 c3 c4

19/90 3/4 1 1/2 1/12

Table 5.1: Fifth order predictor corrector Gear coefficients

70

5.2. CANONICAL MOLECULAR DYNAMICS

5.2.3 Canonical molecular dynamics

Assuming there are no dissipative forces, the molecular dynamics approach

as previously described is only capable of performing microcanonical sim-

ulations. Observe that in principle the system of particles remains al-

ways in a manifold of constant energy. To perform a simulation at some

fixed temperature T there exist a number of techniques such as stochastic

methods [148, 149], constraint methods [150], and extended system meth-

ods [151, 152]

Here we use the so called Nose Hoover thermostat [152]. This approach is

based on the idea of coupling the actual system of particles to an additional

degree of freedom which has the role of a reservoir. The dynamics of the

combined systems undergoes a microcanonical evolution, but the dynamics

of the system of particles is canonical. Notice that this approach is analo-

gous to the way the canonical ensamble is typically introduced in statistical

physics.

In more detail, the reservoir degree of freedom is denoted by s, its contribu-

tion to the total energy has both a “kinetic” and a potential energy term.

The former is given by 12Qs2 where Q is known as a thermal inertia, and the

later is described by the potential

V (s) = (3N + 1)KbT ln (s), (5.5)

where N is the number of particles in the system and Kb is the Boltzmann

constant. The coupling between the system of particles and the reservoir is

given by the relations

xi = p/mi, (5.6)

pi = fi − ξpi, (5.7)

ξ =s

s, (5.8)

ξ =1

Q

(N∑i=1

p2i

mi

− (3N + 1)KbT

). (5.9)

Notice that the net effect of the reservoir degree of freedom is to act as

a dynamic “friction coefficient” that can slow down or speed up particles

71


in order to reach the appropriate mean kinetic energy. An example of the

velocity distributions obtained with this method is show in Fig. 5.2.

0 5 10 15 20 25 30 35 40 45v

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

P(v

)

T=2.5

T=10.5

T=100.5

Figure 5.2: Results of a test simulation to verify the proper working of the

Nose-Hoover thermostat. The obtained velocity distributions were checked

against the expected Maxwell Boltzmann distribution for different tempera-

tures. In all cases the difference between the mean and standard deviation of

the velocity distribution of the simulations was within 1% of the theoretical

ones

72

5.3. MODEL AND METHODS

5.3 Model and methods

We consider a two-dimensional sawtooth channel of linear length L consist-

ing of a sequence of N equal cells, as represented in Fig. 5.3, with periodic

boundary conditions along the horizontal axis. The geometry of the chan-

nel is characterized by four lengths: the length of each cell l = L/N , the

aperture size h, and the horizontal position a and height d of the edge. To

systematically study the dependence on the asymmetry of the channel, we

fix h and define the adimensional asymmetry coefficient α as,

α = 1− 2a

l, (5.10)

where α ∈ [−1, 1]. For α = 0 the channel is symmetric with respect to the

vertical axis, while for α = ±1 the cells look triangular. We classify channels

of negative and positive α as left and right asymmetric, respectively.

We consider a gas of particles interacting pairwise, where the force of particle

j on particle i is conservative and given by Fij = −∇iULJ. ULJ is the 12-6-

d

Figure 5.3: Schematic representation of the channel of size L, with N = 4

cells. The shape of each cell is characterized by four lengths: the linear

length l, the aperture size h, and the horizontal position a and height d of

the peak. The depicted channel is classified as right asymmetric (see text)

and the arrows indicate the corresponding direction of particle flow. The

dashed lines delimit the region where particles are initially released.

73


Lennard-Jones potential,

ULJ (rij) = 4ε

[(σ

rij

)12

−(σ

rij

)6]

, (5.11)

where rij = |rj − ri|, and ri and rj are the positions of particles i and j,

respectively. ε corresponds to depth of the potential well which is located

at rm = 21/6σ. The force Fiw of the wall on a particle i is described as

the superposition of two contributions: a conservative force, Fciw, and a

dissipative one, Fdiw. The conservative force is described as a Lennard-Jones

interaction with the closest point on the wall, with the same ε and σ of the

particle/particle interaction. The dissipative force is given by,

Fdiw = −γ (ri · niw) niw , (5.12)

where niw = ri−rwriw

is the unit vector pointing from the closest point on the

wall rw to the particle i and γ ≥ 0 is a friction constant. The particle/wall

interaction is conservative for γ = 0 and dissipative otherwise. Interactions

with the wall are truncated at a cutoff distance dc = 2.5σ and if the par-

ticle is within the cutoff distance of multiple points the contributions are

superimposed. The particle/wall interaction model was chosen to study the

effect of the wall geometry on the particles dynamics. It is assumed that the

particles locally bounce off the wall, thus the use of a cutoff distance and

the nearest point prescription. dc = 2.5σ has been found to be a reasonable

cut-off Ref. [147]. The analysis of more complex particle/wall interaction

models where the walls are directly modelled as a fixed set of particles is left

for future work.

We performed canonical molecular dynamics simulations, using the Nose-

Hoover thermostat [147, 140, 152]. Accordingly, the equation of motion of

particle i is,

ri =1

mi

(∑j 6=i

Fij(rij) + Fiw

)+ FNH , (5.13)

FNH = −ξx ,

74

5.4. RESULTS

where rij = rj − ri, mi is the particle mass and FNH is the force per unit

of mass, resulting from the coupling with the thermostat [152] and ξ is the

variable that describes the thermostat, its dynamics is given by Eq.(5.9) and

the thermal inertia was set to Q = 0.05 throughout the work.

For simplicity, we set mi ≡ m and consider reduced units, such that: mass is

in units of m, distance in units of σ, and energy in units of ε. The equations

of motion are integrated using a fifth-order predictor-corrector algorithm,

with a time step dt = 10−5 and we run the simulation up to t = 165.

To generate the initial configurations, all particles were released within the

region delimited by the dashed lines in Fig. 5.3, with an initial velocity

drawn from a uniform distribution of zero mean. Particles are thermalized at

the thermostat temperature within the dashed region, considering periodic

boundary conditions along the horizontal direction and reflective top and

bottom boundaries, without interacting with the channel walls. At t = 15,

the constraint imposed by the dashed lines is removed and particles move

inside the channel, following the dynamics described by Eq. ( 5.13).

5.4 Results

To characterize the effect of the asymmetry of the channel walls on the

overall flux, we fixed N = 24, l = 15, d = 3, and h = 6, and performed

simulations for different values of the asymmetry coefficient α ∈ [−1, 1]. To

reduce statistical noise, instead of directly measuring the outlet flux φ(t), we

introduce an integrated quantity B(t), which we call balance. B(t) is defined

as the difference between the cumulative number of particles crossing the

rightmost boundary from the left to the right and the ones crossing it in the

opposite direction, up to time t. In the continuum limit,

φ(t) =1

hB(t) . (5.14)

Asymptotically, we expect that the balance scales linearly in time, and so

we estimated the flux from a linear regression fit of the curve B(t) in the

linear regime.

75


Figure 5.4: (Color online) Snapshots of the time evolution of a system with

700 particles at T = 5 with γ = 1 and α = 1. The top row shows the t = 15

configuration of the system. The successive rows show the configurations at

t = 20, t = 30, t = 150 respectively

The time evolution of a particular realization of the studied system can be

observed in Fig. 5.4. On the top row the initial condition can be observed.

The particles velocities along the horizontal axis are color coded, blue(red)

correspond to right(left) velocity. It can be seen that the distribution is

random and that there are approximately as many particles with positive

velocity as there are with negative. In the subsequent rows, it can be seen

how as time progresses the number of particles travelling to the left dimin-

ishes, and in the last row it can be observed that the particles going to the

right are the majority.

On a quantitative basis, figure 5.5(a) shows the balance as a function of time

for different values of α. Clearly, spontaneous flow emerges as a result of the

asymmetry of the channel walls. For right symmetric channels (α > 0), the

flow is from the left to the right, while for left symmetric channels (α < 0),

the flow is in the opposite direction. Figure 5.5(a) also shows examples

for α = 0, 0.8, 1, for a particle-wall cutoff distance of 5σ. These examples

show that the observed rectification of the particle motion is still observed for

a larger cutoff distance, but the quantitative values of the flux are obviously

different. To analyse the transition at α = 0, we define Π as the fraction of

samples where B(t) > 0 for large values of t (t = 165). The dependence of

76

5.4. RESULTS

0 20 40 60 80 100 120 140t

-200

-100

0

100

200B(t)

a)

−1.0 −0.5 0.0 0.5 1.0α

0.0

0.2

0.4

0.6

0.8

1.0

Π(α

)

b)

Figure 5.5: (Color online) a) Time evolution of the balance B(t) for differ-

ent values of α = −1.0,−0.8,−0.6,−0.4,−0.2, 0, 0.2, 0.4, 0.6, 0.8, 1.0 (from

bottom to top). Results are averages over 500 samples of systems of 100 par-

ticles at a thermostat temperature T = 2.5 and γ = 1. The black dashed

lines correspond to simulations with α = 0, 0.8, 1 and dc = 5σ which show

that the rectification phenomena is not unique to the choice dc = 2.5σ b)

Fraction of samples Π for which B(t) > 0, for T = 2.5,ρ = 0.143, and

N = 250 (blue circles), N = 500 (green squares), N = 1000 (red triangles).

Π on α is in Fig. 5.5(b), for three different sizes of the channel. One sees

that Π is 0.5 for α = 0 and it seems to converge to a step function as the

system size increases.

The dependence of the flux on the density is shown in Fig. 5.6(a), for α = 1

and T = 2.5. One clearly observes an optimal density (ρopt ≈ 0.45) at

which the flux is maximized. The data for ρ < ρopt suggests two different

regimes (see inset of Fig. 5.6(a)): a low-density regime, for ρ < 0.1, and

an intermediate-density regime, for 0.1 < ρ < ρopt. It is expected that the

flux of particles is a monotonic increasing function of the density up to the

point where it either saturates or starts to decrease. For systems where the

particle/particle collisions are more frequent than the particle/wall collisions,

kinetic theory [153] suggests that the flux is linear in the density. However,

for low densities, most of the particle collisions are with the wall; This implies

that the flux is mostly determined by the chance of a particle to bounce off

the wall and eventually cross the boundary at either end of the channel thus

77


0.00 0.25 0.50 0.75 1.00ρ

0.0

0.5

1.0

1.5

2.0φ

a)

10-2 10-1 10010-2

10-1

1001.0

1.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8ρ

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

v x

b)

Figure 5.6: (Color online) a) Flux φ as a function of the density ρ, for

T = 2.5 and α = 1. The two initial regimes are shown in the inset in

a double logarithmic plot, where the dashed lines represent the power-law

fits, φ ∼ ρβ, with β = 1.47 ± 0.07 and β = 1.01 ± 0.04 for the low and

intermediate density regimes, respectively. b) Average horizontal component

of the particle velocity (vx) as a function of the density, for the same set of

parameters.

increasing or decreasing the net flux. In this case the monotonic increase of

the flux is not necessarily linear. The numerical data suggests a power law

scaling: φ ∼ ρβ. Assuming a power-law scaling, we estimate β = 1.47±0.07,

for the low-density regime, and β = 1.01±0.04, for the intermediate-density

one.

Figure 5.6(b) shows the average horizontal component of the velocity (vx) as

a function of the density. For the first regime, vx increases with the density

thus, the enhancement of the flux with the density stems from an increase in

the number particles per cell and possible collective effects affecting the par-

ticle velocity. Similar flux enhancement was reported in the context of comb

systems, where the comb tooth would take the role of the sawtooth [154].

However, in Ref. [154], the flux enhancement is observed for high densities

and is related to the saturation of the traps. Here, instead, we observe a

flux increase for much lower densities suggesting a different mechanism.

By contrast, for the intermediate-density regime, vx does not significantly

change with the density and so the flux only increases due to an increase in

78

5.4. RESULTS

0.0 0.2 0.4 0.6 0.8 1.0

α

0

5

10

15

20

25

30

φ/ρβ

ρ=0. 013

ρ=0. 018

ρ=0. 024

ρ=0. 030

ρ=0. 068

ρ=0. 091

ρ=0. 106

ρ=0. 152

ρ=0. 213

ρ=0. 244

ρ=0. 275

ρ=0. 382

Figure 5.7: (Color online) a) Flux φ rescaled by ρβ as a function of α for

T = 2.5 and different values of the density. The values reported on the

left(right) legend were rescaled using β = 1.47(β = 1.01).

the number of particles per cell, yielding a linear scaling. For ρ > ρopt, a

third regime is observed, for which the flux simply decreases with the density

due to crowding effects. That is, as the density increases, the available space

for a particle to move diminishes. Thus, the motion of a single particle is

strongly constrained by the presence of others. This in turn implies that

for a particle to move over an extended region, it requires a concerted re-

arrangement of several other particles. As the density increases, the chances

that this concerted motion leads to a majority of particles moving in a

preferred direction diminishes. Instead, it is more likely that the particles

re-arrange by moving with no preferred direction leading to a decrease in

the flux. Notice that this is consistent with the fact that vx approaches zero

(Fig. 5.6(b)) which implies that roughly the same number of particles travel

in each direction.

79


−10 −5 0 5 10 15 20 25

T− Tmax

0

2

4

6

8

10

12

φ/ρ

γ=1. 0

γ=0. 9

γ=0. 8

γ=0. 7

γ=0. 6

γ=0. 5

0.5 0.6 0.7 0.8 0.9 1.0

γ

2

4

6

8

10

12

Tmax

Figure 5.8: (Color online) Scaled flux, φ/ρ, as a function of the thermostat

temperature, for α = 1, ρ = 0.0917, and different values of the friction

constant, γ. The thermostat temperature was shifted by Tmax, defined as

the optimal temperature at which a maximum is observed in the flux. The

dependence of Tmax on γ is shown in the inset.

To study the dependence on α, we plot in Fig. 5.7, the flux rescaled by

ρβ, using the estimated values of β for the corresponding regime. We ob-

serve a data collapse for each regime, suggesting that the power-law scal-

ing is resilient over the entire range of α values. The low density regime

shows an optimal value of α whereas the intermediate regime shows in-

stead a nearly constant behaviour. We think that, for high-enough density,

the rate of particle-particle collisions is significantly higher than the one of

particle-wall collisions and thus the geometry of the walls does not play a

significant role on the overall dynamics. By contrast, for low density, the

rates of particle-particle and particle-wall collisions are comparable and a

competition between the two is observed, leading to the maximum in the

flux. Although there is some dispersion on the data collapse, it is clear

80

5.5. SUMMARY

that there are two distinct regimes. The origin of the dispersion can be due

to finite-size effects or scaling corrections. The study of the nature of the

transition and crossover between these two regimes requires further study,

that is beyond the scope of this work. It is interesting to notice that the

results for low densities in Fig. 5.7 are similar to those obtained by A. Sar-

racino [102]. Namely, the quantity that indicates the presence of motion

rectification (φ in our case and 〈V 〉 in Ref. [102]) shows a qualitatively sim-

ilar non-monotonic behaviour as a function of the asymmetry parameter. In

both cases, it vanishes for the symmetric case and initially grows with the

asymmetry, having a maximum for intermediate values, and then decreasing

towards a saturation value for large values of the asymmetry parameter.

The dependence of the flux on the thermostat temperature is shown in

Fig. 5.8, for different values of γ. For the entire range of values of γ, a

maximum is observed at an optimal temperature, Tmax, that increases with

γ (see inset). Also, the optimal flux grows with dissipation (increasing γ).

Note that, for γ = 0 the overall flux vanishes and thus a dissipative inter-

action with the walls is necessary to rectify the thermal motion of the gas.

This is consistent with the work of Prost et al. that suggests that time-

reversal symmetry of trajectories needs to be broken to obtain rectification

from asymmetric walls [155].

Finally, to quantify the flux for a specific system, let us consider a channel of

total length L = 216µm, single cell length l = 9µm, α = 0, p = 1.8µm and

h = 3.7µm at room temperature, with colloidal particles of σ = 6× 10−7 m,

m = 2.49 × 10−16 kg, and γ = 2 × 10−14 kg/s [156, 157]. If we assume

ε/kb = 0.01414 K (where kb is the Boltzmann constant), we obtain a flow

velocity φ/ρ ≈ 42µm/s.

5.5 Summary

In this work, we systematically study the dependence of the rectification of

the motion of a thermal gas on a channel of asymmetric dissipative walls.

We found that the overall flux enhances with the friction constant of the

81


particle/wall interaction and that it shows a nonmonotonic dependence on

three other model parameters, namely, the thermostat temperature, chan-

nel asymmetry, and particle density. For the dependence of the flux on the

density of particles, we found three different regimes. For low density, the

flux scales superlinearly with the density, as collective effects lead also to

an increase in the horizontal component of particle velocity. For interme-

diate density, the horizontal component of particle velocity saturates at a

constant value and the overall flux scales linearly with the density. Finally,

above an optimal value of the density, the flux monotonically decreases due

to crowding effects. Future work might consider different geometries and a

generalization to the three-dimensional case. The effect of different dissipa-

tion mechanisms as well as particle shapes are still open questions.

82

Chapter 6

Conclusion and outlook

In this work we have studied transport ideas from very different yet interest-

ing perspectives. We have shown that Lattice Kinetic methods (LKM) are

flexible enough to handle diverse scenarios. More specifically, we improved

the original application of LKM to the field of density functional theory.

This was achieved by a mixed use of lattice kinetic and physical ideas i.e.

the introduction of correction terms to the original Lattice Kinetic scheme

was possible by using an approximation to the imaginary time evolution

of wave functions hence, the semi-implicit nature of the method. The new

approach was tested on a number of realistic calculations and it showed

good agreement with the known results. For a balanced assessment of the

approach, it is also important to say that the incorporation of pseudopoten-

tials was problematic and that further research on that direction is required.

A very interesting path for future research is to use the diffusive character

of the Schrodinger equation not in imaginary time, but in the real time. It

is easy to see that by considering the real and imaginary parts of the wave

function, the Schrodinger equation naturally splits into two coupled diffusion

equations that are amenable to study using lattice kinetic methods.

As a further case study for the applicability of LKM, its use for the Wigner

function formalism was studied. In this case, a new approach to perform

the time evolution of the moments of the Wigner equation was developed.

The proposed approach improved the previously known stability problems

of similar attempts. This was achieved by introducing a collision term that

83

CHAPTER 6. CONCLUSION AND OUTLOOK

although, is not present in the original problem formulation, does not affect

the dynamics of the moments of the Wigner function and also keeps the pos-

itive semidefinite character of the density matrix that underlies the Wigner

function. The idea was tested for the quantum harmonic and anharmonic

problems where good agreement with the theory was found. The approach

was also used to study transport on simple open driven one dimensional

systems with potential barriers, there a linear relation between the particle

current and the driven potential was found. Also studied was the relation

between the conductivity and the number of barriers in the system. Here it

was found that if the barrier’s separation was large enough, then the conduc-

tivity was independent of the separation both in regular or random barriers

arrangements. The two dimensional analogue of the system just described

was also considered, in this case it was found that for random arrangements

of barriers there could be deviations from the linear behaviour between par-

ticle current and driving potential. Directions for future work include the

extension of the method to handle magnetic fields, as well as interacting

particles.

In the second part of this work, a novel example of Brownian motion rectifi-

cation was introduced. It was shown that it is possible to rectify the motion

of a gas of Lenard-Jones particles by using dissipation and broken spatial

symmetry. The dependence of the flux of particles on various system pa-

rameters was studied and two distinct regimes where found for the quantity

φ/ρβ, one at low and the other at intermediate densities. We expect these

results to be useful in the future in microfluidic applications or lab on a chip

settings.

84

Appendix

A.1 Wigner function derivation

The time evolution of the Wigner function can be obtained by transforming

the Liouville-von Neumann equation

∂ρ

∂t= − i

~[H, ρ], (A.1)

using the prescription of Eq.(4.1), where

H =p2

2m+ V (x), (A.2)

is the system Hamiltonian. For simplicity the derivation is made using a

pure state (ρ = |ψ〉〈ψ|) in a single dimension. The general result for mixed

states follows from the linearity of the theory and the extension to more

dimensions is straightforward.

Equation Eq.(A.1) can explicitly be written as

∂ρ

∂t= − i

~[p2

2m, ρ]− i

~[V , ρ]. (A.3)

The first commutator on the right hand side of Eq.(A.3) reads:

〈q−y2|[ p

2

2m, ρ]|q+y

2〉 =

1

2m

(〈q − y

2|p2|ψ〉〈ψ|q +

y

2〉 − 〈q − y

2|ψ〉〈ψ|p2|q +

y

2〉),

(A.4)

and in real space this is

〈q−y2|[ p

2

2m, ρ]|q+y

2〉 = −~22

m

(∂2ψ(q − y

2)

∂y2ψ∗(q +

y

2)− ψ(q − y

2)∂2ψ∗(q + y

2)

∂y2

).

(A.5)

85

APPENDIX

The Wigner transformation of the first commutator in the r.h.s of Eq.(A.3)

is then given by

− ~πm

∫ ∞−∞

(∂2ψ(q − y

2)

∂y2ψ∗(q +

y

2)− ψ(q − y

2)∂2ψ∗(q + y

2)

∂y2

)eipy/~dy.

(A.6)

Integrating by parts the preceding equation yields

~πm

∫ ∞−∞

(∂ψ(q − y

2)

∂y

∂(ψ∗(q + y2)eipy/~)

∂y−∂ψ∗(q + y

2)

∂y

∂(ψ(q − y2)eipy/~)

∂y

)dy

=ip

πm

∫ ∞−∞

(∂ψ(q − y

2)

∂yψ∗(q +

y

2)−

∂ψ∗(q + y2)

∂yψ(q − y

2)

)eipy/~dy. (A.7)

Changing the derivatives in the y variable for derivatives in the q variable

gives:

− ip

2πm

∂

∂q

∫ ∞−∞

ψ(q − y

2)ψ∗(q +

y

2)eipy/~dy = −ip~

m

∂W

∂q(A.8)

The second commutator of the r.h.s of Eq.(A.3) is evaluated as follows

〈q− y

2|[V , ρ]|q+

y

2〉 = 〈q− y

2|V |ψ〉〈ψ|q+

y

2〉 − 〈q− y

2|ψ〉〈ψ|V |q+

y

2〉 (A.9)

which in real space representation reads

〈q − y

2|[V , ρ]|q +

y

2〉 = V (q − y

2)ψ(q − y

2)ψ∗(q +

y

2)− ψ(q − y

2)V (q +

y

2)ψ∗(q +

y

2)

(A.10)

=(V (q − y

2)− V (q +

y

2))ψ∗(q +

y

2)ψ(q − y

2).

(A.11)

The Wigner transformation of Eq.(A.11) is:

1

2π~

∫ ∞−∞

(V (q − y

2)− V (q +

y

2))ψ∗(q +

y

2)ψ(q − y

2)eipy/~dy. (A.12)

If the potential is smooth enough such that it can be expanded as

V (q + y) =s=∞∑s=0

(y/2)s

s!

∂sV

∂qs, (A.13)

86

A.1. WIGNER FUNCTION DERIVATION

then Eq.(A.12) is transformed into

− 1

π~

∫ ∞−∞

∑s∈Nodd

(y/2)s

s!

∂sV

∂qsψ∗(q +

y

2)ψ(q − y

2)eipy/~dy. (A.14)

Using the fact that ys = (~i)s ∂

seipy/~

∂psEq.(A.14) can be simplified as

− 1

π~∑s∈Nodd

(~2i

)s1

s!

∂sV

∂qs

∫ ∞−∞

ψ∗(q +y

2)ψ(q − y

2)eipy/~dy =

− 2∑s∈Nodd

(~2i

)s1

s!

∂sV

∂qs∂s

∂ps1

2π~

∫ ∞−∞

ψ∗(q +y

2)ψ(q − y

2)eipy/~dy =

− 2∑s∈Nodd

(~2i

)s1

s!

∂sV

∂qs∂s

∂psW. (A.15)

Finally combining Eq.(A.8) and Eq.(A.15) with the corresponding prefactors

in Eq.(A.3) results in

∂W

∂t+p

m∂qW (q, p)−

∑s∈Nodd

(~2i

)s−11

s!

∂sV

∂qs∂sW

∂ps= 0,

∂W

∂t+p

m∂qW (q, p) + Θ[V ]W = 0,

(A.16)

where

Θ[V ]W = −∑s∈Nodd

(~2i

)s−11

s!

∂sV

∂qs∂sW

∂ps. (A.17)

Alternatively, the Wigner transform of the potential term in (A.3)

V [p, q] =1

2π~i

~

∫ ∞−∞〈q − y/2[V , ρ]q + y/2〉eipy/~dy, (A.18)

can be evaluated as follows, using the result from (A.11)

V [p, q] =1

2π~i

~

∫ ∞−∞

(V (q − y

2)− V (q +

y

2))ψ∗(q +

y

2)ψ(q − y

2)eipy/~dy,

(A.19)

or more generally

87

APPENDIX

V [p, q] =1

2π~i

~

∫ ∞−∞

(V (q − y

2)− V (q +

y

2))ρ(q−y

2, q+

y

2)eipy/~dy. (A.20)

Using the inverse representation of the Wigner function∫ ∞−∞

dpe−ipy/~W (q, p)dp = ρ(q − y/2, q + y/2), (A.21)

V [p, q] takes the form

V [p, q] =1

2π~i

~

∫ ∞−∞

(V (q − y

2)− V (q +

y

2))W (q, p′)eiy(p−p′)/~dydp′.

(A.22)

The final result is then:

∂W

∂t+p

m∂qW (q, p)

+1

2π~i

~

∫ ∞−∞

(V (q − y

2)− V (q +

y

2))W (q, p′)eiy(p−p′)/~dydp′ = 0.

(A.23)

or if δV is defined as

δ[V ](q, p) =i

2π~2

∫ ∞−∞

(V (q − y/2)− V (q + y/2))eiyp/~dy, (A.24)

then

∂W

∂t+p

m∂qW (q, p) +

∫ ∞−∞

δ[V ](q, p− p′)W (q, p′)dp′ = 0, (A.25)

where the identification

Θ[V ]W =

∫ ∞−∞

δ[V ](q, p− p′)W (q, p′)dp′ (A.26)

can be made.

88

A.2. GENERALIZED HERMITE POLYNOMIALS

A.2 Generalized Hermite polynomials

The physicist Hermite polynomials Hn are defined as

Hn = (−1)nex2 dn

dxne−x

2

(A.27)

and satisfy the orthogonality relation∫RHn(x)Hm(x)e−x

2

dx =√π2nn!δmn, (A.28)

with weight function ω(x) = e−x2.

Upon the change of variables x→ v√2cs

the following relation holds∫RHn

(v√2cs

)Hm

(v√2cs

)e− v2

2c2sdx√2cs

=√π2nn!δmn. (A.29)

Using the definition of the general weight function ω(v; cs) = 1√2πc2s

e− v2

2c2s ,

taking m = n and dividing by the normalization constant it is found that∫R

[1√n!2n

Hn

(v√2cs

)][1√n!2n

Hn

(v√2cs

)]ω(v; cs)dx = 1. (A.30)

The generalized Hermite polynomials Hn(x; cs) orthonormal with respect to

the weight function ω(v; cs) are then defined as

Hn(x; cs) =1√n!2n

Hn

(v√2cs

). (A.31)

From Eq.(A.31) the equivalent expression to Eq.(A.27) is given by

ω(v; cs)Hn(x; cs) = (−1)n(2cs)

n/2

√n!2n

dn

dxnω(v; cs). (A.32)

A.3 Wigner Forcing term calculation

In this section the calculation of the the Wigner forcing term in the Lattice

Kinetic approach is presented for the case of Hermite polynomials. The

derivation is carried out in one dimension, and the general result is provided

at the end.

89

APPENDIX

In dimensionless variables, the Wigner forcing term is given by

Θ[V ]W = −∑s∈Nodd

(H

2i

)s−11

s!

∂sV

∂xs∂sW

∂vs, (A.33)

and the Wigner function is assumed to be represented as

W (x, v, t) = ω(v; cs)

Np∑n

an(x, t)Hn(v; cs), (A.34)

where Hn(v;Cs) are the Hermite polynomials with weight function

ω(v; cs) =1√2πc2

s

e− v2

2c2s . (A.35)

If W is replaced in Eq.(A.33), it is clear that the terms that need to be

evaluated are of the form

∂s

∂vs(ω(v; cs)Hn) . (A.36)

From the generalized Hermite Polynomials definition Eq.(A.32), it follows

that∂s

∂vs(ω(v; cs)Hn) = (−1)n

(2cs)n/2

√n!2n

dn+s

dxn+sω(v; cs). (A.37)

Using Eq.(A.32) again, the result reads

∂s

∂vs(ω(v; cs)Hn) = (−1)s

√(n+ s)!

n!(c2s)

ω(v; cs)Hn+s. (A.38)

Using this result, the Wigner forcing term finally reads

Θ[V ]W = ω(v; cs)∑n,s

an(x, t)

√(n+ s)!

n!

(H/i)s−1

Csss!

∂sV

∂xsHn+s(vi; cs). (A.39)

Given that Eq.(A.39) is already expressed as the weight function ω(v; cs)

times Hermite polynomials, the source distribution follows directly from the

prescription ω(vi; cs)→ ωi and Si = Θ[V ]W (vi).

For dimension D > 1, the procedure that was used in the one dimensional

case can also be used to find Θ[V ]W in terms of the expansion coefficients

90

A.4. GENERAL WIGNER FUNCTION CALCULATION

of the Wigner function. That is, if the Wigner function is assumed to be

represented as

Θ[V ]W = ω(v; cs)∑n

anHn(v) (A.40)

where n = (n1, n2, . . . , nD) is a D dimensional set of non-negative integer

indices ni and Hn is a D dimensional tensor Hermite polynomial defined as

Hn = Hn1 ⊗Hn1 · · · ⊗ HnD , then Θ[V ]W can be written as

Θ[V ]W = ω(v; cs)∑

|s|∈Nodd,n

anK(v,n, s)∂s

∂xsV, (A.41)

where the kernel K is given by

K(v,n, s) =

(H

2i

)|s|−1 [1

(2c2s)|s|/2

∏l sl!

]∏l

Hnl+sl(v). (A.42)

In this case the source term distribution also follows from the prescription

ω(vi; cs)→ ωi and Si = Θ[V ]W (vi).

A.4 General Wigner function calculation

The calculation of the Wigner function can be conveniently performed if the

underlying density matrix is written in terms of Hermite functions (eigen-

functions of the quantum harmonic oscillator) that are defined as

ϕn =1√2nn!

(1

πH

)1/4

e−x2

2HHn

(x√H

). (A.43)

Let the density matrix of a given physical system be given by

ρ(x, x′) =∑n,m

cnc∗mϕn(x)ϕm(x). (A.44)

By definition the Wigner function is given by

W =1

2πH

∑n,m

cnc∗m

∫ϕn

(x− y

2

)ϕm

(x+

y

2

)eivyH dy, (A.45)

=1

2πH

∑n,m

cnc∗mKnm(x, v), (A.46)

=1

2πH

∑n≤m

2

1 + δm,n<(cnc

∗mkn,m), (A.47)

91

APPENDIX

where the Knm are found using Groenewold’s formula [24] and are given by

km,n =2(−1)min (m,n)

√min (m,n)!

max (m,n)!e−

x2+v2

H

(2

H(x2 + v2)

) |m−n|2

L|m−n|min (m,n)

(2

H(x2 + v2)

)e(i(m−n) arctan (v/x)) (A.48)

where Lmn is the m order n degree associated Laguerre polynomial.

At this point it is important to mention that there is a mistake, possibly

due by a typo, in Ref [24] where it is stated that Lmn is a Legendre polyno-

mial instead of a Laguerre one. In the following it is shown that the right

polynomial is the later one.

The starting point is the second line of Eq (5.16) on page 457 of Ref [24],

from which the relevant part is

min (a,b)∑k=0

(−1)k

(m− k)!(n− k)!k!

(~2

(p2 + q2)

)min(m,n)−k

, (A.49)

for simplicity let z =(~

2(p2 + q2)

)and let S denote the sum on Eq.(A.49).

Without loss of generality let m > n thus

S =n∑k=0

(−1)k

(m− k)!(n− k)!k!zn−k, (A.50)

using the change of variables u = n− k, S can be written as

S = (−1)nn∑u=0

(−1)u

(m− n+ u)!(n− u)!(u)!zu, (A.51)

with the help of the identity S = Sm!m!

, S is transformed into

1

m!

[n∑u

(−1)u(

m

n− u

)zu

u!

]=

(−1)n

m!Lmn . (A.52)

Note that the expression in square brackets corresponds to the definition of

Laguerre (see Ref.[158]) and not Legendre polynomials.

92

A.5. MULTIPLE BARRIERS EFFECTIVE TRANSMISSIONCOEFFICIENT

A.5 Multiple barriers effective transmission

coefficient

Let T1, T2 be the transmission coefficients of two barriers for a given incoming

state ψ. The effective transmission coefficient T12 is given by the probability

that the incoming state passes through the barriers without reflection, with

two reflections, with four reflections..., since each of these events are exclusive

the total probability is given by the sum of the individual events probabilities

i.e.

T12 = T1T2 + T1T2R1R2 + T1T2(R1R2)2 + · · · =∞∑i=0

T1T2(R1R2)i, (A.53)

where Ri = 1− Ti. Since |R1R2| < 1 the series can be summed leading to

T12 =T1T2

1−R1R2

, (A.54)

taking the reciprocal of the last equation and using again that Ri = 1− Ti,it follows:

1

T12

=1

T1

+1

T2

− 1, (A.55)

that can be casted in the form

1− T12

T12

=1− T1

T1

+1− T2

T2

, (A.56)

Thus for N identical barriers the result is

TN =T

N(1− T ) + T. (A.57)

A.6 Lattice specification

Here the full specification of the lattices used through this work is given. 1D

lattices where generated as described in the main text. 2D lattices and in

general n dimensional Hermite based lattices can be constructed by taking

n times the tensor product of the set of vectors and weights of a fixed

93

APPENDIX

1D Lattice. For example the D2Q4 lattice is given by Table.A.6. It is

important to notice that this way of building higher dimensional lattices

does not exhaust all possible lattices.

Table A.1: D1Q4 Lattice with Cs = 0.60625445810016454

vi wi

0 0.63664690312607816284434609283846

-1,1 0.18141458774368577505004149208377

-3,3 0.00026196069327514352778546149699


vi wi

0 0.37428019874212190129215011724318

-1,1 0.24105344284458452784844296921093

-2,2 0.06434304152476086575379872184362

-3,3 0.00713156628791277339406557854605

-4,4 0.00032523057375714836476726255033

-5,5 6.6163470389851878681133911638949×10−6

-7,7 3.0508847488049822958363118638543×10−9


vi wi

0 0.32444899174631946866086595194671

-1,1 0.23309081165504033632566413700874

-2,2 0.08642582836940192624063184539752

-3,3 0.01653989847863324979993319254793

-4,4 0.00163342485156222352004541584861

-5,5 0.00008333063878279730921268566542

-6,6 2.1783167706100240902344965225688×10−6

-7,7 3.1805869765623071575130276965860×10−8

-9,9 1.0779356826917937931616055896767×10−11

94

A.6. LATTICE SPECIFICATION


vi wi

0 0.24603212869787232483785340883852

-1,1 0.20342468717937742901117526034797

-2,2 0.11498446042457243913866706495342

-3,3 0.04443225067964028999644337006636

-4,4 0.01173764938741580915572505702913

-5,5 0.00211976456798849884644315007219

-6,6 0.00026170845228301249011385925086

-7,7 0.00002208877826469659955769726449

-8,8 1.2745253026359480126112714680367×10−6

-9,9 5.0275261810959383411581297192576×10−8

-10,10 1.3556297819769484757032262002820×10−9

-11,11 2.5012031341031852003279373539252×10−11

-12,12 3.1243604817078012360750317883072×10−13

-13,13 2.9655118189640940365948400709026×10−15

-15,15 5.5758174181938354491800200913102×10−19

95

APPENDIX


vi wi

0 0.21731931022112109059537537887018

-1,1 0.18735357499686018912399983787195

-2,2 0.12004746243830897823022161375249

-3,3 0.05717041140835294313179190148076

-4,4 0.02023564183037203154174508370450

-5,5 0.00532341082536521716813053993040

-6,6 0.00104085519989277817787032717819

-7,7 0.00015125787069729717011289372472

-8,8 0.00001633702528266419558030012576

-9,9 1.3114608069909806258412013955825×10−6

-10,10 7.8246508661616857867473666191193×10−8

-11,11 3.4697808952346470123102609636720×10−9

-12,12 1.1435779630395075964965610648417×10−10

-13,13 2.8013182217362421082623834683491×10−12

-14,14 5.0995884226301388644438757982605×10−14

-15,15 6.9079520892785667788901676695952×10−16

-16,16 6.8680470174442627832690379600090×10−18

-17,17 5.7551467186859264824045886746476×10−20

-19,19 8.3761764243303081227469285304101×10−24


vi wi

(0,0) w20

(0,± 1),(± 1,0) w0w1

(± 1,± 1) w1w1

(0,± 3)(± 3,0) w0w3

(± 3,± 1),(± 1,± 3) w1w3

(± 3,±3) w23

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Lattice Kinetic Methods & Thermal Rectification Phenomena

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