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cate/physa

Physica A 360 (2006) 1–16

www.elsevier.com/lo

A

0

d

Diffusion in the presence of a pole: From theodel

continuous Gaussian to a discrete lattice m

E. Bakalis, C. Vlahos, M. Kosmas�

Chemistry Department, University of Ioannina, 45110 Ioannina, Greece

Received 15 January 2005

Available online 28 June 2005

bstract

used to

walker,

X2 is in

ormulate

or their

t and the

termined

ence of a

sponding

to higher

nsities of

values in

The continuous Gaussian model with delta function potentials which has been

describe the effects that interacting centers (poles) bring to the diffusion of a random

provides an exact solution only in dimension d ¼ 1. Its application in dimension d

need of introduction of cutoffs coming from the finite sizes of poles. We therefore f

an analogous discrete model in the d-dimensional lattice space where no cutoffs

regularization are necessary. The discrete model is amenable to numerical treatmen

Green’s function, its first moments and properties derived from them can be de

numerically to all dimensionalities. In the one-dimensional (1-D) case, in the pres

single center, the outcomes of the Gaussian model are compared, with the corre

results of the discrete lattice model. The possibility of extension of the lattice model

dimension permits the study of the two-dimensional (2-D) case too. The inte

interactions considered before to obtain positive values are extended to negative

order to describe attractions also.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Diffusion; Gaussian model; Discrete model; Dimensionality; Interacting pole

378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved.

oi:10.1016/j.physa.2005.05.088

�Corresponding author. Tel.: +302651098442; fax: +30 2651098798.

E-mail address: mkosmas@cc.uoi.gr (M. Kosmas).

1. Introduction

rmationcurs and[2]. Therandomcentersr three-oted tos wherediscern

itions ofce givescts fromc micro-y moreents of

ffect theeeds thement ofl comingve beentinuum,ach nextfunction[7], thelds such] or therials by

otentialsand in ansidera-ng theseures areque wayermore,termina-t the fullsary forwith theus seekfor the

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E. Bakalis et al. / Physica A 360 (2006) 1–162

The study of diffusion of small molecules in materials can provide infoabout the microscopic structure of matrices of media in which diffusion octhis gains importance and interest both in nanotechnology [1] and biologypotentials met in nanoscale materials and biological systems which alter thewalk character of trajectories of the diffusant are due to sets of interactinglocated at specific positions, in the form of linear arrays, surface patterns odimensional constructions [3]. Although much literature has been devdescription of diffusion in disorder media with random environmentpositions of the interacting points are randomly distributed [4], we have tofrom the beginning the two classes of systems with and without fixed posscattering centers. Averaging over the positions of scatterers in all spasolutions which contain only the density of scatterers [5] but misses the effevarious configurational architectures of interacting centers met in specifiscopic structures. The solutions to the latter problems are generallcumbersome, but they are necessary for description of specific arrangeminteracting points met in real systems.

The potentials from organized structures of interacting centers, which adiffusive motion of molecules, are complicated and their consideration nunderstanding of simple initial models. Attractive for this purpose is employthe simplest possible one-center potential for creation of the overall potentiafrom a larger set of interacting points. Delta function potentials which haapplied, in the past, in many fields [6] are suitable for this purpose. In the conthe Gaussian model which uses a Gaussian function for the probability of estep to describe random walks and diffusion has been enriched with deltapotentials for description of real systems. Beyond the field of diffusionGaussian model with delta function potentials has been applied to other fieas the field of polymers to describe the excluded volume problem [8,9interactions of chains with a surface [10] and the field of magnetic matemeans of the Ising model [11].

Serious problems of the continuous Gaussian model with delta function pare those met at higher dimension. Increasing the dimension above d ¼ 1perturbation theory scheme (see Section 2) divergences occur because of cotion of the interacting centers of finite size as dimensionless points. For savidivergences, cutoffs, which need regularization and renormalization procednecessary, in order to be handled properly [12]. The trouble is that no uniexists to define and use of the cutoffs and their elimination. Furthregularization and renormalization techniques are mainly confined in the detion of critical exponents without evaluating multiplicity constants [13]. Budeterminations of the functions in all regions of their definitions are necesproper description of real systems. The weakness of the continuous modelsimple delta function potentials to be extended to higher dimension madethe solution in the discrete lattice space and formulate a discrete analogdescription of the random motion in the presence of interactive centers.

Parallel to the study in the continuum, walks on lattices consisting of a finitehe deltaon the

tures oftices notials aree effectsions andms. Thealks onotentials

d in thence of ns

d yields. Theirforwardbecomesment. Aide time-ng-timeompleteorms. Int to theis evenly exactons. Thet, whichause of

plied tonsity ofor exacted for aD cases.study ofse worksnalyticalthe twopercubicaveragewill see

ial at theof u foranother

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E. Bakalis et al. / Physica A 360 (2006) 1–16 3

number of steps and having a discrete nature [14] have been modeled. Tfunction potential of the continuous model becomes a Kronecker deltadiscrete lattice and it can easily be used to construct complicated strucinteracting centers as well. For evaluation of properties of walks on latcutoffs are necessary even if extra soft potentials like delta function potenincluded. The discrete lattice model offers thus the opportunity to study thon diffusion of one or more interacting centers in one, two or three dimensstudy diffusion effects from complex patterns of obstacles met in real systeeffort comes to augment the already large literature of models studying wlattices which as far as we know did not include the simple one-body soft pof the type of the delta of Kronecker.

From the computational point of view, two kinds of problems are faceGaussian continuous space model. The formulation of diffusion in the presedelta functions at positions rj ðj ¼ 1; 2; . . . ; nÞ has been done [7] anprobabilities whose time Laplace transforms are linearly connecteddetermination is equivalent to the inversion of a matrix, which is straightfor the first members of the series with one or two interacting centers, butmore difficult for a larger number of centers and in need of a numerical treatrelated problem is the analytical inversion of the Laplace transforms to provdependent properties. Although it is possible in certain conditions like the lolimit, it is not possible in all regions of the parameters of the system. Csolutions are also in need of numerical inversion of the time Laplace transfthe lattice discrete space models generating functions, which are equivalenLaplace transforms of the continuous space, are possible but their inversionmore cumbersome compared to their continuous counterparts. Unfortunatesolutions to specific problems in the field are very rare and use approximatiinteresting thing is that lattice models are amenable to numerical treatmencan be done by an iteration relation. This becomes a necessity beccomputational difficulties met in analytical treatment of the problem.

The formulation which we give in the lattice space is general. It can be apcases of any number of scatterers at fixed positions, for any kind and inteinteractions and for any dimension. The iteration relation, which is suitable fenumerations, is presented in Section 3 and explicit examples are presentsingle scattering center located at a specific lattice vertex for the 1-D and 2-Close is the one-dimensional discrete walk which has been used for theeffects from an impenetrable surface [15]. Although the potential used in thois of the hard-core type with an interactive part and more difficult for an atreatment than the delta function potential, similar results are taken fromkinds of potentials. Recently the study of diffusion on a d-dimensional hylattice in presence of a forbidden site has also been done and thedisplacement of diffusant in the long-time limit has been given [16]. As wetraps and forbidden sites can be taken from the simple delta function potentspecific limit of huge repulsions. Present effort first extends to negative valuesthe study of useful properties in the attractive region and second deals with

point missing from the literature which is the comparison in detail between models ina lattice.ly in thextendednsion d,problemrison ofdiscretehe sameutcomescessary.he studyction 3.

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E. Bakalis et al. / Physica A 360 (2006) 1–164

the continuum like the Gaussian model and their discrete counterparts onWe will show that the analytic solution of the Gaussian model is possible on1-D case, and the discrete numerical treatment of the lattice model can be eto higher dimension. Although the two formulations are given for any dimetheir comparison is possible only in the 1-D case. We study in depth the 1-Din the presence of a single pole by both methods in Sections 2 and 3. Compathe analytic solution in the continuum with the numerical results of thelattice model shows that the structure of properties from the two models is twhich offers a confidence about the correctness of the effort. However, the oare not identical, and hence a careful use and switch of the two [17] are neThe ability of the lattice model to be extended to higher dimension permits tof the 2-D case by means of the numerical technique, presented in SeDiscussion and Conclusion are included in Sections 4 and 5, respectively.

2. The continuous Gaussian model and properties of solution in the presence of an

y a briefand a

s can beaussiann is thating on as modelfind theequal toð0Þ ¼ 0s a path

(2.1)

½RðsÞ�Þ is

(2.2)

interacting pole in one dimension

Since the solution to the continuous model has been given before [7] onloutline of it will be given here, in terms of a path integral formalismperturbation theory expansion. We will see in this way how these techniqueapplied to the diffusion problem, but also the weakness of the continuous Gmodel to be extended to higher dimension. Another reason for this repetitiothe path integral formalism is close to the picture of random walker travellattice via many different routes and permits the extension of the continuouto the discrete analog as well. The un-normalized probability PuðR; tÞ todiffusing particle at position R at time t, if its initial position is at R ¼ 0, isthe sum of the contributions from all different trajectories RðsÞ starting at R

and ending at RðtÞ ¼ R considered as continuous curves. It can be written aintegral:

PuðR; tÞ ¼

ZPuð½RðsÞ�ÞD½RðsÞ�d½RðtÞ � R�d½Rð0Þ � 0� ,

where D½RðsÞ� is the measure of the differential of the path integral and Puð

the probability of each trajectory RðsÞ. Its form is

Pu½RðsÞ� ¼ Po½RðsÞ� exp �

Z t

0

dsXns

j¼1

V ½RðsÞ � rj �

8<:

9=; ,

where

Po½RðsÞ� ¼ A exp �1

4Do

Z t

0

dsdRðsÞ

ds

� �2

8<:

9=;

is the ideal probability of a trajectory which includes an integration of the square ofiffusionterm ofositionsiffusant.otentialg underequal toters arecontact.ns whileðR; tÞ ofðR; tÞ isbabilityt

mber ofa largeralues ofe Pu butral casesicable tonly oneas

…

(2.3)

�1

21

ð2:4Þ

ðf̄ ðsÞ ¼

iagrams

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E. Bakalis et al. / Physica A 360 (2006) 1–16 5

derivative of position vector along the contour of the path [18], Do is bare dcoefficient and A is the normalization constant. The second exponentialEq. (2.2) includes the sum of interactions from all ns centers located at prj ðj ¼ 1; 2; . . . ; nsÞ and determines changes to the evolution of motion of dA form of V ðRÞ amenable to analytic treatment is the delta function pV ½RðsÞ � rj� ¼ ujd½RðsÞ � rj� which being a generalized function has meaninthe t integral and permits touch of diffusant with poles when RðsÞ becomesany rj . Intensities of interactions between the particle and the active cendescribed by uj which can be considered as the gain of energy for eachA positive u means smaller probability of contact which describes repulsionegative u leads to an increased probability and describes attractions. Pu

Eq. (2.2) is un-normalized and a normalization factor CðtÞ ¼RdR Pu

necessary to give the corresponding normalized non-ideal proPðR; tÞ ¼ PuðR; tÞ=CðtÞ. Expanding the exponential of Eq. (2.2) we take tha

Pu½RðsÞ� ¼ Po½RðsÞ�X1l¼0

ð�1Þl

l!

Xj

uj

Z t

0

ds dd½RðsÞ � rj�

8<:

9=;

l

.

The terms of increasing order in uj coming from action of an increasing nudelta functions represent trajectories with one ðl ¼ 1Þ, two ðl ¼ 2Þ ornumber ðl42Þ of visits of the diffusant to positions of poles. For small vintensity parameter uj the first members of the series are adequate to describfor increasing uj larger and larger number of terms are necessary. For sevewe can sum all terms of the series producing closed analytic solutions applall values of u from full attractions to full repulsions. Such is the case with ointeracting pole at rP which in a diagrammatic language can be represented

Pu(R,t) = -u ++u2 −u3 +… +un

n

Generally the n-loop diagrams have the following form:

n

¼1

4pDo

� �dn=2 Z t

0

dt1

td=21

exp �r2P

4Dot1

� �Z t�t1

0

dt2

td=22

� � �

Z t�t1�t2�����tn�2

0

dtn

td=n�

�

Z t�t1�����tn�1

0

dtn

td=2n

exp �ðR� rPÞ

2

4Doðt� t1 � � � � � tn�1Þ

� �

and involve time integrals of convolution type, which in the Laplace spaceR10

dt e�stf ðtÞÞ can be converted into products. Notice that multiplicities of d

are equal to l! and exactly cancel the 1=l! of the expansion of Pu½RðsÞ�.integralso whichds exactbut forthe 1-Dd. Smallstudies

r way toture thet is whatlattice in

rams of

(2.5)

ound byechanicspression

ð2:6Þ

reducedd of theoperties

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E. Bakalis et al. / Physica A 360 (2006) 1–166

The determination of values of diagrams, pass through evaluation of the t

of Eq. (2.4). These integrals exist only when the dimension d is less than twindicates the dimension where the present Gaussian continuous model yielsolutions. For the 1-D case these integrals exist and can be determined,dimension 2 and 3 they do not. The pole with its zero volume affects onlyproblem [19] while in higher dimension finite time cutoffs have to be includescale cutoffs are necessary for the existence of integrals for dX2 giving rise tobased on renormalization group techniques used previously [9,12]. Anothecircumvent this problem is to employ lattice analogs which have in their nameaning of the smallest minimum length which is the lattice site length. Thawe have in mind formulating the problem in the d dimensional hypercubicthe next section.

For the 1-D problem the combined Fourier–Laplace transform of diagEq. (2.3) exist and have the form

FLðn� LoopÞ ¼1

ð4DosÞðnþ1Þ=2exp �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs=DoÞ

pjxPj � ikxP

� �Dok2

þ s,

which yields for the probability

~~Puðk; sÞ ¼

Zþ10

dt e�st

Zþ1�1

dx e�ikxPuðx; tÞ

the expression:

~~Puðk; sÞ ¼1

ðDok2þ sÞ�

u

2ffiffiffiffiffiffiDo

pexp �jxpj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs=DoÞ

p� ikxP

� �ðDok2

þ sÞffiffispþ u=2

ffiffiffiffiffiffiDo

p� � .

This expression is in accordance with the propagator of previous works fmeans of equivalent differential equations used in diffusion or quantum m[20]. By applying inverse Fourier and Laplace transforms on ~~Puðk; sÞ the exof probability in real space takes the form [21]

PuðX ; tÞ ¼1ffiffiffiffiffiffiffiffiffiffi4Dotp

expf�X 2gffiffiffipp �U expf½U þ jX Pj þ jX � X Pj�

2

�

� ½jX Pj þ jX � X Pj�2g erfcðU þ jX Pj þ jX � X PjÞ

,

where

erfcðxÞ ¼2ffiffiffipp

Z1x

do expf�o2g

and the quantities U ¼ ut=ffiffiffiffiffiffiffiffiffiffi4Dotp

, X ¼ x=ffiffiffiffiffiffiffiffiffiffi4Dotp

and X P ¼ xP=ffiffiffiffiffiffiffiffiffiffi4Dotp

arevariables. They describe the intensity of interactions and distances of the entrajectory and the pole from the origin, respectively. Three interesting pr

taken from Puðx; tÞ are the average number hvi of contacts between the diffusant and2 s of the

utcomes

þUÞ ,

(2.7)

,

ð2:8aÞ

Þ

(2.8b)

pf�X 2Pg

ð2:8cÞ

withoutth theseaphs of4(a). Indetailedte lattice

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E. Bakalis et al. / Physica A 360 (2006) 1–16 7

the pole and the two moments hxi and hx i. They can be expressed in termnormalization function and the two quantities

x‘ðtÞ ¼

Z1�1

x‘Puðx; tÞdx ð‘ ¼ 1; 2Þ .

We write them explicitly in order to be able to give a comparison with the oof the discrete model:

CðtÞ ¼

Zþ1�1

dx Pðx; tÞ ¼ 1� erfcðX PÞ þ expfðX P þUÞ2 � X 2Pg erfcðX P

hvi ¼ �1

CðtÞ

qCðtÞ

qu¼

ffiffiffiffiffiffiffiffiffit

pDo

r

�expf�X 2

Pg �ffiffiffippðX P þUÞ expfðX P þUÞ2 � X 2

Pg erfcðX P þUÞ

1� erfcðX PÞ þ expfðX P þUÞ2 � X 2Pg erfcðX P þUÞ

hxi ¼ xðtÞ=CðtÞ ¼ xPexpfðX P þUÞ2 � X 2

Pg erfcðX P þUÞ � erfcðX PÞ

1� erfcðX PÞ þ expfðX P þUÞ2 � X 2Pg erfcðX P þU

and

hx2i ¼ x2ðtÞ=CðtÞ

¼ 2Dot1� ð1þ 4X 2

P þ 2X P=U þ 1=U2Þ erfcðX PÞ þ ð2=ffiffiffippÞðX P þ 1=UÞ ex

1� erfcðX PÞ þ expfðX P þUÞ2 � X 2Pg erfcðX P þUÞ

(

þð1=U2 þ 2X 2

PÞ expfðX P þUÞ2 � X 2Pg erfcðX P þUÞ

1� erfcðX PÞ þ expfðX P þUÞ2 � X 2Pg erfcðX P þUÞ

).

In these expressions the reduced distance X P has been considered as positiveloss of generality. The general behavior of diffusion can be described wiexpressions and special characteristic limits are given in Discussion. Grthese functions are shown together with the graph of Pðx; tÞ in Figs. 1(a)–Figs. 1(b)–4(b) corresponding graphs of lattice model can be seen and acomparison thus between the results of the Gaussian model and the discremodel can be done.

3. Formulation of the delta function potential on D-dimensional hypercubic lattice and

h vertexigin and

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-2 -1 0 1 20

1

2

3

2

0

1

-1

-2

X

-2 -1 0 1 20

1

2

1

-1

0

2

-2

2(D

ot)1/

2 P

2(D

ot)1/

2 P

X(a) (b)

Fig. 1. The reduced probability 2ðDotÞ1=2 Pðx; tÞ as a function of the reduced variable X ¼ x=ffiffiffiffiffiffiffiffiffiffi4Dotp

for

the position of the pole X p ¼ xp=ð4DotÞ1=2 ¼ 0:5 and various values of the reduced interaction parameter

U ¼ ut=ffiffiffiffiffiffiffiffiffiffi4Dotp

¼ þ1; 2; 1; 0;�1;�2 as it is shown in both figures from the analytical model (a) and the

computational one (b). Both graphs tend to the limit of a delta function at X p for U !�1.

0 20

1

2

3

4

2

1

0

-1

-2

(π/(

2t))

1/2 <

v>

(π/(

2t))

1/2 <

v>

Xp

00

1

2

3

1

0

-1

2

-2

Xp

21 1

(a) (b)

Fig. 2. The mean numbers of contactsffiffiffiffiffiffiffiffiffiffiffiffiffip=ð2tÞ

pov4 of the walker with the pole as a function of the

reduced position X p ¼ xp=ffiffiffiffiffiffiffiffiffiffi4Dotp

of the pole for various values of the reduced variable U ¼ ut=ffiffiffiffiffiffiffiffiffiffi4Dotp

¼

2; 1; 0;�1;�2 for the analytical (a) and the computational (b) model, respectively.

E. Bakalis et al. / Physica A 360 (2006) 1–168

enumerations for the 1-D and 2-D cases

In the discrete formulation a D-dimensional hypercubic lattice where eacis connected with 2-D neighbor vertices is used. The diffusant starts at the or

in each step it jumps from a lattice vertex to any of its nearest neighbors with equalhat of ahe deltalaced inre Ri aren latticeiffusant

ARTICLE IN PRESS

0

-3

-2

-1

0

1

0

-1

1

-2

2

<x>

/xp

Xp0

-3

-2

-1

0

1

0

1

-1

2

-2

<x>

/xp)

Xp1 2 1 2

(a) (b)

+∞

−∞

Fig. 3. The nonvanishing ratio ox4=xp of the mean displacment of the walker as a function of X p, for

various values of U ¼ �1;�2;�1; 0; 1; 2;1 for the analytical (a) and computational (b) model.

0 30

1

2

1

2

-1

-2

0

<x2 >

/(2D

ot)

Xp

00

1

2

0

-1

1

-2

2

<x2 >

/(2D

ot)

Xp

21 2 1

(a) (b)

−∞

+∞

Fig. 4. The reduced mean end to end square displacmentox24=ð2DotÞ as a function of the position X p of

the pole for various values of the reduced interaction parameter U ¼ �1;�2;�1; 0; 1; 2;1 for the

analytical (a) and the computational (b) model, respectively.

E. Bakalis et al. / Physica A 360 (2006) 1–16 9

probability. When interactive poles are absent, the diffusive motion is trandom walker while presence of poles gives rise to extra interactions. Tfunction potential ujd½RðsÞ � rj� of the continuous Gaussian model are repthe case of the lattice model with Kronecker delta type potential ujdRi ;rj

, whepositions of the diffusant and rj the positions of the poles, fixed at certaivertices. This potential obtains non-zero values only when the traveling d

visits vertices with poles. It describes repulsions and attractions between poles andas in thels ud toial basictime thisal walksonly inefore inssage ofthe 1-Dpossible.ten withoffs arehat uj inunits of

umericale on d-order toation R

d by thetion 2 it

ð3:1Þ

he walkifferenceof theidentitythe last

dR;Rn.

elation

(3.2)

j is they of the

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E. Bakalis et al. / Physica A 360 (2006) 1–1610

diffusant when uj parameters obtain positive or negative values respectively,continuous Gaussian model. Employment of the delta function potentiarepresent interactions with poles gives rise to the simplest possible initproblem of non-ideal diffusion of the particle for both models. It is the firstsimple one-body potential is included in the study of diffusion with non-ideon lattices. It is a soft potential permitting touch of diffusant and pole andextreme limit of u!þ1 behaves like a potential of hard-core type used bproblems of random walks [15]. With this soft potential the touch and the pathe diffusant through position of pole is permitted even in the case ofproblem where extension of motion to the other side of pole is alsoPotentials coming from complicated sets of scatterers are easier to be writthis simple potential and the interest comes from the fact that no cut-necessary for the determination of properties in any dimension d. Notice tthis discrete model are dimensionless while uj in the continuous model bearvolume/time, becoming length/time for the 1-D case.

For computational simplicity an iteration relation amenable to ntreatment is given. We consider in this discrete model the general casdimensional regular lattice a walk of n steps, which starts at the origin. Infind the un-normalized probability PuðR; nÞ that the particle will be at locafter n steps, all different paths must be taken into account each one weighteenergy of interaction with poles. Similar to the path integral formalism of Seccan be written as

PuðR; nÞ ¼X1

X2

� � �X

n

exp �X

i

Xj

V ðRi � rjÞ

( )dR;Rn

,

V ðRi � rjÞ ¼ ujdRirj,

where the n summations are done over all possible different n steps of tproducing thus all walks starting at the origin and ending at Rn ¼ R. A dbetween the discrete and the continuous model comes from vanishingKronecker dRi ;rj

in the discrete model except at Ri ¼ rj, which leads to theexpf�ujdRi ;rj

g ¼ 1þ kjdRi ;rjwith kj ¼ �1þ expð�ujÞ. Using this identity in

step of the walk, the probability (3.1) takes the form:

PuðR; nÞ ¼X1

X2

� � �Xn�1

Xn

exp �X

i

Xj

ujdRi ;rj

" #1þ

Xj

kjdRn;rj

( )

Upon summing over the positions of the last Rn step we end up with the r

PuðR; nÞ ¼XDR

PuðR� DR; n� 1Þ 1þX

j

kjdR;rj

( ),

whereP

DR is the sum over all neighbor vertices of the vertex at R andP

summation over all positions of centers. Eq. (3.2) connects the probabilit

walker to be at position R in its nth step with probabilities of walks of previousR. Thetions by3.2). Itses whichlengths.2 while

both for

babilitypole at

(3.3)

the first. Finallynpxpn

of view,ting theo loopspable of00 timember oflly from

ation ofis equale shownmodel,

properthe idealith the=t is theð2tÞ can

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E. Bakalis et al. / Physica A 360 (2006) 1–16 11

generation with n� 1-steps to end at the neighboring positions R� Ddependence of Pu’s of all generations starting with n ¼ 1; 2; . . . on interacmeans of kj parameters comes through the iterative application of Eq. (inherent inclusion of the smallest length of a step makes it free of singularitithe continuous counterpart contains in the limit of vanishing trajectorySingularities do not permit exact solutions of the continuous problem at dX

the application of (3.2) of the lattice model can be used at all dimension d

determination of generating functions and numerical studies.For the 1-D problem walker starts at position x ¼ 0 of the axis with a pro

defined equal to Puð0; 0Þ ¼ 1. For the one-dimensional problem and for onexP we take from Eq. (3.2) that

Puðx; nÞ ¼ fPuðx� 1; n� 1Þ þ Puðxþ 1; n� 1Þgf1þ kdx;xPg .

The computer program, constructed from this iteration relation, starts withprobability Puð0; 0Þ ¼ 1 and determines the random walk history iterativelyall the values of the un-normalized probability Puðx; nÞ in the whole range �are recorded. The advantage of the method from a computational pointcompared to the classical exact enumeration methods, is that by calculaprobabilities of the walk using the iteration relation, we avoid the n-dnecessary for the description of an interactive n-step walk and we are thus caexecuting the program for very large number of steps. Typically the 100 0steps on a Pentium 2.4GHz machine needs 1 h CPU time. The total nucontacts with the pole and the mean displacements are obtained numericathe following expressions:

hvi ¼ �1

C

qC

qu,

hxi ¼1

C

Xn

x¼�n

xPuðx; nÞ

and

hx2i ¼1

C

Xn

x¼�n

x2Puðx; nÞ ,

where the normalization function C is equal toPn

x¼�n Puðx; nÞ. If t is the dura step taken equal to 1 unit of time for simplicity, the total time of the walkto t ¼ nt converting thus the number n of steps into time. These functions arin Figs. 1(b)–4(b) and behave exactly like those of the continuous Gaussianproving the capability of two models to describe walks and diffusion.

In order to have the best possible quantitative approach of the two, achoice of microscopic parameters must be done. Comparing, for example,second moment hx2io ¼ 2Dot of the one-dimensional analytical model wcorresponding expression hx2io ¼ nl2 of the computational one where n ¼ t

number of steps and l the length of each step we see that the choice Do ¼ l2=

be done for a coincidence of the two properties and a better quantitative agreement.ke hx2ios far asthat hviand thisthe twosteps ofo matterand theit of fullmber ofnot truepable ofgiven init.kness ofe of thee modelhas four

1Þ

ð3:4Þ

of steps,ally andsurfaceout thealues oflem too.e finitert-rangecumventto the

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E. Bakalis et al. / Physica A 360 (2006) 1–1612

The values t ¼ 1; l ¼ 1;Do ¼ 1=2 in proper units ensure this choice and mathe same in both models. These are the values that we use for the graphs. Athe equivalence of the U parameters of the two models are concerned we seeobtains a bit smaller values at large negative values of U in the lattice modelobservation touches the limitation of the quantitative comparison betweenmodels. This difference in the limit of u!�1 is due to the discreetness ofthe lattice model. It is built on a lattice and has a number of steps n which nhow large, is always finite. The number of contacts between the polediffusant can be smaller tending to become equal to this number in the limattraction. After the enlargement of attraction beyond a limit this nucontacts remains the same and equal to the maximum value n. The same isfor the continuous model which contains time t as a continuous variable cagiving an infinite number of small steps. The number of contacts in this caseEq. (2.8a) increases with large �u as hvi ¼ �ut=ð2DoÞ without having a lim

Having shown the qualitative equivalence of the two models and the weacontinuous Gaussian model to be extended to dimensions 2 and 3 becausabsence of a small but finite size of the pole, we extend the counterpart latticto higher dimension. For the 2-D case and the pole at RPðxP; yPÞ each vertexneighbor vertexes and Eq. (3.2) becomes

Puðx; y; nÞ ¼ fPuðx� 1; y; n� 1Þ þ Puðxþ 1; y; n� 1Þ þ Puðx; y� 1; n�

þ Puðx; yþ 1; n� 1Þgð1þ kdR;RPÞ; R ¼ ðx; yÞ .

We use a program similar to that of the 1-D case and reach large numbern ¼ 100, sufficient to see pole effects. All properties are determined numericas an example we give plots of Puðx; y; nÞ for RP ¼ xP ¼ 2 in Fig. 5. ThePðx; y; nÞ ¼ Puðx; y; nÞ=CðtÞ of the normalized probability is symmetric abðP;xÞ plane, with cuts at the various x planes, lines of the Gaussian form. VPðR; tÞ and influence of the pole on diffusion are substantial in this 2-D probImportant effects of the presence of the pole are observed for thcomputational time though the soft pseudopotential ud used is of the shotype. Because of the many paths along which the diffusant can travel and cirthe pole the values of PðR; tÞ are smaller, though finite, comparedcorresponding values of the 1-D case.

4. Discussion

u whichinfinitethe 1-Dpsing ofon limitbability

Of interest is the analysis in the whole range of the interaction parametervaries between two extremes: that of infinite repulsions u!þ1, where thebarrier prevents trajectories from passing through the position of the pole atcase and that of infinite attraction with u!�1 where an attractive collathe diffusant and binding on pole takes place. Starting from the preventiu!þ1 we see from Eq. (2.6) that the corresponding un-normalized pro

takes at this limit the familiar expression

(4.1)

nding tonite, forthe roleanied by; tÞ at xP

t showsfrom its!�1.negativeproperlyis zero, to theen used!�1

jx� yjg

can bee limits,

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-20 -10 0 10 20x

-20

-10

0

10

20

y

-20 -10 0 10 20x x

-20

-10

0

10

20

y

10

10

5

5

0

0

-5

-5-10

-10

yx

Fig. 5. The probability Pðx; y; nÞ for the 2D case with the pole at Rp ¼ xp ¼ 2 and for n ¼ 100. The ideal

probability Po with u ¼ 0 is the surface at the center, P for attractions ðu ¼ �1Þ with a maximum at the

position x ¼ xp ¼ 2 is the right surface, while for the case with repulsions ðu ¼ 1Þ we take the third surface

in the left with a maximum at x ¼ �5, pushed away from the pole to the other side of the origin.

E. Bakalis et al. / Physica A 360 (2006) 1–16 13

PuðX ; tÞ ¼ ½1=ð4pDotÞ1=2�fexpf�X 2g � expf�ðjX Pj þ jX � X PjÞ2gg .

It is equal to the difference of contributions from two image points correspothe absorbing boundary condition [22]. While for XoX P;Puðx; tÞ is fiX4X P, the probability vanishes, proving that in this case the impurity playsof an impenetrable obstacle. The vanishing of Puðx; tÞ at the pole is accompmaximum of gradient dPuðx; tÞ=dx at pole. Decreasing u the probability Puðx

increases towards its maximum value at the limit u!�1, while its gradienthe opposite behavior. The slope of the graph at the pole decreasesmaximum value towards its minimum (maximum negative) at the limit u

Varying u from �1 to þ1, dPuðx; tÞ=dx passes through a zero for somevalue of u where Puðx; tÞ obtains a flat minimum. All situations aredescribed starting from the case of full repulsions where the probabilityand identical to that of absorbing boundary conditions used in the pastfull attractions not so known in the field of diffusion though it has bein quantum mechanics [23]. By means of Eq. (2.6) we see that at u

the normalized reduced probability becomesffiffiffiffiffiffiffiffiffiffiffiffiffi4pDotp

Pðx; tÞ ¼ �u expf�u

[24] and behaves like the delta function 2dðx� yÞ. The rest propertiesstudied at this collapsing limit U !�1 by means of Eqs. (2.8). Th

hxi ¼ hxPi; hx2i ¼ x2P and hvi ¼ �ut=2Do for U !�1 are found and the

n of thepole. Atsant andhe usualanges atthe fullisotropye case ofnetrablelosest isoundaryian forms also antical to

that ford of thethe poleusant tocase ofhe one-ny pathspole atlimiting

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coincidence of both moments with the corresponding values of the positiopole proves, in an analytic way, the collapse of the diffusant and binding onthis limit of u!�1, the average number hvi of contacts between the diffuthe scatterer increases enormously in accordance with the collapse picture. Tsquare root dependence hvi�t1=2 found for the ideal behavior of walks, chthis limit to the stronger power-law dependence hvi�t given also before foradsorption of walks [15]. Notice also that hxia0, which is a result of the anof space caused by the presence of the pole. An interesting point is that in thzero slope with dPu=dx ¼ 0 at x ¼ xP, Puðx; tÞ of Eq. (2.6) is given in this pecase in terms of the error function for some negative value of u. Although cnot identical to what the impenetrable case yields for the reflecting bcondition which is the sum of the two exponential image terms of the Gauss[22]. This impenetrable Puðx; tÞ is defined only on the semi-axis xoxP, it havanishing derivative and a minimum at x ¼ xP, but though close is not idethe first one of the penetrable problem.

For 2-D case, from numerical results given in Fig. 5 we understandattractive interactions a maximum of Pðx; y; nÞ occurs in the neighborhoopole at xP ¼ 2. Pðx; y; nÞ obtains substantial values not only at the point ofbut in an area spread around the pole because of the capability of the diffcircumvent the pole. The same spread of effects of pole is true for therepulsions also, Fig. 5, where the diffusant cannot be confined as in tdimensional case but can circumvent the repulsive pole and travel along maeven for large repulsions. These observations of substantial effects of thed ¼ 2 are not in disagreement with the results of Ref. [16] where thebehaviour at infinite time is considered.

5. Conclusion

rpart onpresenceivalence.h of thediscretes of thewhile iner hand,cause ofe latticelatter isbstantialses withructures

We have compared the continuous Gaussian model and its discrete countea lattice for the description of the 1-D diffusion of a probe molecule in theof a repulsive or an attractive pole. The two models show a qualitative equThe structure of their properties is the same but for the quantitative approactwo the right choice of the microscopic parameters has to be done. Thenature of the lattice model does not permit the average number of visitdiffusant to the pole to increase beyond the number of steps of the walk,the Gaussian continuous model this increase can be unlimited. On the oththe weakness of the Gaussian model to be extended to higher dimension bethe non-existence of the non-ideal terms at the small-scale limit is saved in thdiscrete model with the inherent minimum length of the lattice site. Theextended to the dimension d ¼ 2 where the effects of pole are found to be sufor finite times. Extension of the application of the present lattice model to cainteracting sets of points for description of diffusion in organized microstmet in nanotechnology and biology is now possible.

Acknowledgements

rationalnistry ofn Social

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E. Bakalis et al. / Physica A 360 (2006) 1–16 15

This research was funded by the program ‘‘Heraklitos’’ of the OpeProgram for Education and Initial Vocational Training of the Hellenic MiEducation under the 3rd Community Support Framework and the EuropeaFund.

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