MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � � � � 8.334: Statistical Mechanics II Problem Set # 1 Due: 2/20/08 Phase transitions 1. Critical behavior of a gas: The pressure P of a gas is related to its density n = N/V , and temperature T by the truncated expansion b 2 c 3P = kBT n − n + n ,2 6 where b and c are assumed to be positive, temperature independent constants. (a) Locate the critical temperature Tc below which this equation must be invalid, and the correspo nding density nc and pressure Pc of the critical point. Hence find the ratio kBTcnc/Pc. (b) Calculate the isothermal compressibility κT = −1 ∂V � , and sketch its behavior as a V ∂P Tfunction of T for n = nc. (c) On the crit ical isot herm give an expression for (P − Pc) as a function of (n − nc). (d) The instability in the isotherms for T < Tc is avoided by phase separation into a liquid of density n+ and gas of density n−. For temperatures close to Tc, these densities behave as n± ≈ nc (1 ± δ). Using a Maxwell construction, or otherwise, find an implicit eq uat ion for δ(T ), and indicate its behavior for (Tc − T ) → 0. (Hint: Along an isotherm, variations of chemical potential obey dµ = dP/n.) (e) Now consider a gas obeying Dieterici’s equation of state: a P (v − b) = kBT exp − ,kBT v where v = V/N. Find the ratio P v/kBT at its crit ical point. (f) Calculate the isothermal compressibility κT for v = vc as a function of T − Tc for the Dieterici gas. (g) On the Dieterici critical isotherm expand the pressure to the lowest non-zero order in (v − vc). ******** 2. Magnetic thin films: A crystalline film (simple cubic) is obtained by depo siting a finite number of layers n. Each atom has a three component (Heisenberg) spin, and they interact through the Hamilt onian n n−1 −βH = JH~s α · ~sjα + JV ~s α · ~s α+1 .i ii α=1 <i,j> α=1 i� � �� �� � � �� �� (The unit vector ~siα indicates the spin at site i in the αth layer. The subscript < i, j > indicates that the spin at i interacts with its 4 nearest-neighbors, indexed by j on the square lattice on the same layer.) A mean–field approximation is obtained from the variatio nal density ρ0 ∝ exp (−βH0), with the trial Hamiltonian n −βH0 = ~h α · ~siα . α=1 i (Note that the most general single–site Hamil tonian may include the higher order terms Lαc1,···,cp sαc1 · · · sαc1 , where sc indicates component c of t he vector ~s.) (a) Calculate the partition function Z0 ~h α , and βF0 = − ln Z0. (b) Obtain the magnetizations mα = �h~siαi0 �, and hβH0i0, in terms of the Langevin func-tion L(h) = coth(h) − 1/h. (c) Calculate hβHi0, with the (reasonable) assumption that all the variational fields ~h α are parallel. (d) The exact free energy, βF = − ln Z, satisfies the Gibbs inequality (see below), βF ≤ βF0 + hβH − βH0i0. Give the self-consistent equations for the magnetizations {mα} that optimize βH0. How would you solve these equat ions numerically? (e) Find the critical temperature, and the behavior of the magnetization in the bulk by considering the limit n → ∞. (Note that limm→0 L−1(m) = 3m + 9m3/5 + O(m5).) (f) By l inearizing the self-consistent equations, show that the critical temperature of film depends on the number of layers n, as kTc(n ≫ 1) ≈ kTc(∞) − JV π2/(3n2). (g) Derive a continuum form of the self-consistent equations, and keep terms to cubic order in m. Show that the resulting non-linear differenti al equation has a solution of the form m(x) = mbulk tanh(kx). What circumstances are described by t his solution? (h) How can the above solution be modified to describe a semi–infinite system? Obtain the critical behavi ors of the healing length λ ∼ 1/k. (i) Show that the magnetization of the surface layer vanishes as |T − Tc|. The result in (f) illustrates a quite general trend that the transition temperature of a finite system of size L, approaches its asymptotic (infinite–size) limit from below, as Tc(L) = TC(∞) − A/L1/ν, where ν is the exponent controlling the divergence of the correlation length. However, some liquid crystal films appeared to violate this behavior. In fact, in these films the couplings are stronger on the surface layers, which thus order� � � � � � � � � � � � before the bulk. For a discussion of the dependence of Tc on the number o f layers in this case, see H. Li, M. Pa czuski, M. Kardar, and K. Huang, Phys. Rev. B 44, 8274 (19 91). • Proof of the Gibbs inequality: To approximate the partition function Z = tr e−βH of a difficult problem, we start we a simpler Hamiltonian H0 whose properties are easier to calculate. The Hamiltonian H (λ) = H0 + (H − H0) interpolates between the two as λ changes from zero to one. The corresponding partit ion function Z(λ) = tr{exp [−βH0 − λβ (H − H0)]}, must satisfy the convexity condition d2 ln Z(λ)/dλ2 = β2 (H − H0)2 ≥ 0, and hence 0c d ln Z � ln Z(λ) ≥ ln Z(0) + λ . dλ λ=0 But it is easy to show that d ln Z/dλ|λ=0 = β hH0 − Hi0, where the subscript indicates expectation values with respect to H0. Defining free energies via βF = − ln Z, we thus arrive at the inequality βF ≤ βF0 + hβH − βH0i0 . ******** 3. Superfluid He4–He3 mixtures: The superfluid He4 order parameter is a complex number ψ(x) . In the presence of a concentra tion c(x) of He3 impurities, the system has the following Landau–Ginzburg energy βH[ψ, c] = dd x K |∇ψ|2 + t |ψ|2 + u |ψ|4 + v |ψ|6 + c(x)2 − γc(x)|ψ|2 ,2 2 2σ2 with positive K, u and v. (a) Integrate out the He3 concentrations to find the