Problem Sheet
Advanced Statistical Physics
Sheet 5
Prof. Dr. U. Schollwöck, SS 2015
Problem 12: Order-disorder transition and phase separation
In the binary alloy of Cu and Zn (brass), the atoms occupy the sites of a body-centered cubic (bcc)
lattice. The distribution of atoms on these sites is disordered above Tc ' 740 K. Below Tc there is
ordering with atoms of each kind preferentially distributed on one of the two simple cubic sublattices
of the bcc lattice. We want to obtain a mean-field description of this transition.
Define NAA , NBB , NAB to be the number of nearest-neighbor pairs of Cu-Cu, Zn-Zn and Cu-Zn type,
respectively. The energy of a configuration of atoms then is
E = NAA eAA + NAB eAB + NBB eBB ,
where eAA , eAB and eBB are, respectively, the energies of an AA, AB, and BB bond. Let N be the
number of lattice sites and NA = cA N and NB = cB N the number of Cu and Zn atoms, respectively.
The concentrations of both species add to one cA + cB = 1. NA1 and NB1 are the number of atoms of
each type on sublattice 1, NA2 and NB2 the number of atoms of each type on sublattice 2. We have
NA1 + NA2 = NA
N
NA1 + NB1 =
2
NB1 + NB2 = NB
N
NA2 + NB2 = .
2
An atom on sublattice 1 has q = 8 nearest neighbors, which are all on sublattice 2.
(a) Give expressions for NAA , NBB , NAB assuming statistical independence of site occupations
(Bragg-Williams mean-field theory).
(b) Define an appropriate order parameter m to describe the order-disorder transition (m = 0 in the
disordered phase, −1 < m < 1).
(c) Express E and S in terms of this order parameter to compute the free energy F = E − T S. To
compute S, only consider the leading terms in Stirling’s Formula. Plot (F (m) − F (0))/(qN ε)
versus m, −0.3 < m < 0.3, for cA = 0.5 and different τ = kBqεT , 0.49 < τ < 0.51, where
ε = 12 (eAA + eBB ) − eAB .
(d) Derive the following equation for the position of the minima of F
0 = −2qcA εm + 21 kB T ln
(1 + m)(cB + cA m)
,
(1 − m)(cB − cA m)
(1)
Analytically determine the transition temperature Tc at which, coming from high temperatures,
the disordered state becomes unstable if ε > 0. In the light of the experimental evidence, why
is ε > 0 a meaningful assumption? For which concentration cA is Tc maximal, and for which
physical reason? Do you find a similarity to a known model for a certain value of cA ?
(e) Numerically solve the equation of state (1) and plot m versus τ =
Plot the phase diagramm in the cA –τ plane.
kB T
qε
for cA ∈ {0.2, 0.3, 0.4, 0.5}.
(f) How can you recycle the preceding calculation to describe phase separation of a homogeneous
mixture below some critical temperature?
Problem 13: Potts model
An example that leads to a Landau expansion with a cubic term, which breaks inversion symmetry
and gives rise to a a first order transition, is the Potts model. Consider a system of N spins, each of
which can be in any of p states. Each spin only interacts with the q nearest neighbor spins of the same
type as itself:
N
X
H = −J
δSi ,Sj , J > 0.
hi,ji
For p = 2 this is just the Ising model. We consider the case p = 3 and label the states A, B, C. Let
nA = NA /N , nB = NB /N and nC = NC /N .
(a) Show that the free energy in the Bragg-Williams mean-field theory becomes:
F =−
qN J 2
(nA + n2B + n2C ) + N kB T (nA ln nA + nB ln nB + nC ln nC )
2
(b) In the disordered high-temperature phase nA = nB = nC = 1/3. In general, these concentrations
are subject to the constraint nA + nB + nC = 1. A possible parametrization that anticipates a
meaningful definition of an order parameter is
√
√
nA = 13 (1 + 2y),
nB = 31 (1 + 3x − y),
nC = 13 (1 − 3x − y)
Graphically represent the allowed values of x and y.
(c) The possible ordered phases have preferential occupation of either the A, B or C state. Because
of symmetry we can restrict ourselves to the A state (x = 0) and choose as order parameter
m = y, −1/2 ≤ m ≤ 1. Show that around Tc , the free energy can be expanded as
F/N = −qJ/6 − kB T ln(3) − (qJ/3 − kB T )m2 −
kB T 3 kB T 4
m +
m + ...
3
2
(d) Plot the exact (no expansion) free energy F/(N qJ) versus m for different temperatures around
kB Tc /(qJ) = 4 ln1 2 ' 0.3607. Determine the transition temperature Tc and the value of m at Tc
numerically. Using the numeric value of m, you can confirm the exact result for Tc analytically.
(e) Since the transition is first order, it is accompanied by latent heat L. Show, via computing the
entropy per spin in both phases, that L = qJ/12.
Problem 14 (Bonus): Spruce Budworm model
A population of spruce budworms reproduces exponentially and infests trees up to a maximum number
of q budworms per tree. Budworms are prey to birds. If there are few budworms, birds mainly eat
other insects. If there are more budworms, birds can minmize the effort for finding food by specializing
on budworms: birds increase their daily budworm ration up to a maximum value.
(a) Explain the terms in the following time evolution equation for the mean number n of budworms
per tree
dn
n2
A
= n(1 − n/q) − a
, A, a constants.
(2)
dt
1 + n2
Analogously to Fick’s law of diffusion, one postulates that the rate of approach to thermal equilibrium should be characterized by the deviation of an appropriate “free energy” F from its
equilibrium value: dn
∝ − ∂F
.
dt
∂n
(b) Discuss the phase transition w.r.t. variation of a that occurs in the steady states of (2). It suffices
to argue with plots of F and the√two equivalent equations
of state: n vs. a and 1/n vs. 1/a.
√
Hint: Inspect the case q = qtc = 3 3, a ∼ atc = 8/(3 3), and the case q > qtc .