CCP - lect.10 - Review of Statistical Mechanics
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Computational Physics I Luigi Scorzato Lecture 10: Review of Classical Statistical Mechanics
Transcript of CCP - lect.10 - Review of Statistical Mechanics
Computational Physics ILuigi Scorzato
Lecture 10: Review of Classical Statistical Mechanics
• Equivalence (under conditions) of the average over time with the average over phase space. Ergodic principle, ist abuse and other arguments.
• Ensembles and associated thermodynamic functions
• Examples of statistical models:
1. Classical Molecules with Lennard-Jones potentials
• Pair interaction approximation
• Lennard-Jones potentials
• (besides thermodyn quantities as before) Pressure via Virial theorem
• static pair correlation function g(r) and multipoint correlators
2. Lattice Models (Ising)
• correlators of fields defined on lattice points.
Summary:
On the equivalence of the time average and the ensemble average
A := limT→∞
1T
T
0A(t)dt
A :=
dpdq ρ(p, q) A(p, q)=
A fundamental assumption in Statistical Mechanics is that the (physical) time average and the ensemble average coincide. This goes under the name of Ergodic Principle (or theorem in those cases in which it can be proved).
• Physically, the 1st one is more relevant.
• For Analytic computation the 2nd one can be more easily dealt with.
• For Numerical Simulations, the 1st one leads to the idea of Molecular Dynamic (MD) approach, in which one follows the Newton Eq. of motion (or their QM correspondents) and will occupy us in Lecture 13. The 2nd one leads to the idea of Monte Carlo (MC) integration, that we will study in Lectures 11, 12, 14.
Is the Ergodic Principle true?!? In some cases it can be proved, but this is physically not so relevant, because the argument typically implies a time scale ~ TPoincaré, which is typipcally larger than the age of the universe and hence does not explain why we see the equivalence.
What matters is the equality in that portion of the phase space near the maximum of ρ, since it is known that ρ(p,q) ~0 almost everywhere. This is hard to prove, but believed to be true in most cases, except when...It is certainly not true for very good physical reasons: Symmetry Breaking...In MC or MD simulations, it is usually important to prove that the algorithm is ergodic, but it is also important to use the right algorithm and the right set-up when simulating phases with broken symmetries.
Classical Ensembles(quick review of formulae that you should know)
We assume that an Hamiltonian H(p,q) is given. I will use x= Γ=(p,q) to denote a point in phase space.
Microcanonical (NVE ensemble)
Number of states:
Expectation value of an observable:
Fundamental thermodynamical quantity: Entropy S(N,V, E) = kB log Ω(N,V, E)
Derived thermodynamical quantities: T =
∂S
∂E
−1
N,V
µ = −T
∂S
∂N
E,V
P = T
∂S
∂V
E,N
Ω(N,V, E) =1
N !
x
δ(H(x)− E)
A =1
ΩN !
x
A(x)δ(H(x) − E)
Canonical (NVT ensemble)
Partition function:
Expectation value of an observable:
Fundamental thermodynamical quantity: Free Energy
Derived thermodynamical quantities:
F (N,V, T ) = −kBT log Z(N,V, T )
S = −
∂F
∂T
N,V
µ =
∂F
∂N
T,V
P = −
∂F
∂V
T,N
Z(N,V, T ) =1
N !
x
e−βH(x) β :=1
kBT
A =1
ZN !
x
A(x)e−βH(x)
Isobaric-Isotermal (NPT ensemble)
Partition function:
Expectation value of an observable:
Fundamental thermodynamical quantity: Gibbs Potential
Derived thermodynamical quantities:
G(N,P, T ) = −kBT log Q(N,P, T )
S = −
∂G
∂T
N,P
µ =
∂G
∂N
T,P
V =
∂G
∂P
T,N
Q(N,P, T ) =1
N !
dV
x
e−βPV−βH(x) =
dV Z(N,V, T )
A =1
QN !
dV
x
A(x)e−βPV−βH(x)
Gran Canonical (μVT ensemble)
Partition function:
Expectation value of an observable:
Fundamental thermodynamical quantity: Gran Canonical Potential
Derived thermodynamical quantities:
ΩG(µ, V, T ) = −kBT logZ(µ, V, T )
S = −
∂ΩG
∂T
µ,V
N = −
∂ΩG
∂µ
T,V
P = −
∂ΩG
∂V
T,µ
Z(µ, V, T ) =
N
1N !
x
eβµN−βH(x) =
N
eβµNZ(N,V, T )
A =
N
x
1ZN !
A(x)eβµN−βH(x)
Other useful relations
(h,M)Other important conjugate variables: applied magnetic field and magnetization:
U = Hensemble
CV =
∂U
∂T
N,V
CP =
∂U
∂T
N,P
F = U − TS
G = F + PV
H = U + PV
Later we will introduce Monte Carlo methods. They provide a method to compute integrals of the form: <A> . The energy U is such an example. CV and CP or the Magnetization M are other examples.
On the contrary, quantities like the Free Energy F are easily expressed as derivatives of Z (which is very useful analytically) but not in the form F=<A>. Such quantities are more difficult numerically. We will see some tricks later.
Classical MoleculesWe consider first (and only) the simple situation with N spherically symmetric identical classical particles interacting
via a potential given by the sum of pair interactions. Moreover, the potential only depends on the distance
V2(r) = VLJ(r) = 4
σ
r
12−
σ
r
6
0.5 1 1.5 2 2.5 31
0.5
0
0.5
1
1.5
2
-ε21/6σ
The typical choice for the potential is the one of Lennard-Jones:
Repulsive core
Short range attraction
Note that you can remove σ and ε by a redefinition of the Energy and Length scales. What really matters is the ratio w.r.t. other quantities entering the game (mass, Temperature).
H(p, q) =N
i=1
p2i
2m+ VN (r1, . . . , rN ) =
N
i=1
p2i
2m+
12
i =j
V2(|ri − rj |)
(p, q) = (p1, . . . , pN ; r1, . . . , rN )
Besides the internal Energy U (both kinetic and potential) one can compute the Pressure P via the virial theorem. I give no proof, but remember that the virial theorem says that foreach degree of freedom we have
V P =N
β− 1
3
i<j
ri,j · ∇i,jVLJ(ri,j)
x∂H
∂x =
1β
virial th. (3N d.o.f.)external forces internal forces
Pressure
Pair Correlation Function
n 2 (r, rʼ) = probability that one particle is in r and a second one is in rʼ ≠ r
all N(N-1) terms are equal, becauseI can rename the ri .
n 2 (r, rʼ) actually depends only on r =|r- rʼ| and it is simply denoted by n 2 (r). Proof: shift both r and rʼ by the same amount
It is convenient to normalize ... w.r.t. pair correlation in an ideal gas where no correlation is present.The normalized correlation is called g2(r)
=
N
i=1
δ(ri − r)
N
j=1
δ(rj − r)
ensamble
=N(N − 1)
ZN
dr3 drNe−βVN (r,r,r3,...,rN )
n2(r)(ideal) = n2 n =N
Vg2(r) = n2(r)/n2
Expected behaviours: • Low density (gas)• Medium densities (liquid)• high densities (solids)
Structure Functions: (accessible experimentally via photon scattering of frequency q.
S(q) =
N
i=1
e−iq·ri
= (...) = 1 + n
dre−iq·rg2(r)
Lattice Models: e.g. Ising ModelIt is the simplest lattice model to formulate. Nevertheless it is extremely rich of nontrivial features.
It represents a very much simplified ferromagnet. Every site hosts an electron whose only degree of freedom is its spin, which can be either up or down.
A single configuration is defined by N=Ld variables σi i=1...N that can be either 1 (up) or -1 (down).The Hamiltonian is:
H = −J
i,j
σiσj − h
i
σi
means that i and j are nearest neighbors i.e.: i =(x,y), j =(x+/-1,y) or =(x,y+/-1)
Z =
σ
e−βH[σ]
F = − 1β
log Z
We use the canonical Ensemble to definePartition Function Z and Free Energy F:
Physically interesting quantities are:
internal energy
heat capacity
magnetization
magnetic suscettivity
U = H
Ch =
∂U
∂T
N,h
M =
∂F
∂h
N,T
χ =1V
∂M
∂h
N,T
Another crucial quantity is the pair correlation function g (r) and the correlation length ξ:
Similarly to the percolation case, also here we have a critical point Tc . It is the point where the Magnetization (when h=0) starts to be non-zero in the infinite volume limit.
Also here we observe empirical scaling lawsnear the critical point.In fact the singular parts behave like:
The definition above is only useful for the time average. Moreover it does not tell what is its sign? The sign of the time average depends on the initial state of the system.
The ensamble average has a different problem: it is symmetric by exchange of the sign of all spins. How can it be non-zero? In fact it is always =0, as long as h=0. The precise definition, valid also for the ensamble average is:
limL→∞M = 0 (T > Tc)
limL→∞M = 0 (T < Tc)
limh→0
limL→∞M(L, h) = 0 (T > Tc)
limh→0
limL→∞M(L, h) = 0 (T < Tc)
g(i, j) = g(r = |i− j|) = σiσj − σ2 ∼r→∞ e−r/ξ
Ising Model. Exact solutions
The Ising model is easily solved analytically for d=1(see e.g. Huangʼs textbook “Statistical Mechanics”). Most one dimensional statistical system can be solved exactly.
Unfortunately, the result is not so interesting because it displays no phase transition.
The Ising model is one of the very few models that can also be solved analytically for d=2.
The solution (due to Onsager in 1944) is very complicated (~20 pages in Huangʼs textbook) and not so instructive, since the same method could be used to solve any other systems.
However, Onsagerʼs solution in d=2 displays a phase transition, whose properties, in particular the exponents of the scaling laws, can all be computed analytically and provides a crucial benchmark for the numerical methods (See K.Huangʼs textbook for more details).
