PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #2
(1) A system consists of N ’molecules’. Each molecule consists of four ’spins’: σ, µ1 , µ2 ,
and µ3 , where each spin polarization can takes values ±1. The molecular Hamiltonian is
ĥ = Jσ(µ1 + µ2 + µ3 ) − H 3σ − µ1 − µ2 − µ3 .
(a) Enumerate all the molecular energy states along with their degeneracies.
(b) Find the molecular partition function ζ(T, H).
(c) Compute the magnetic susceptibility χ(T, H = 0).
(2) In §4.9.4 of the lecture notes, we considered a simple model for the elasticity of wool in
which each of N monomers was in one of two states A or B, with energies εA,B and lengths
ℓA,B . Consider now the case where the A state is doubly degenerate due to a magnetic
degree of freedom which does not affect the energy or the length of the A± monomers.
(a) Generalize the results from Eqs. 4.221 and 4.222 of the lecture notes and show that
you can write the Hamiltonian Ĥ and chain length L̂ in terms of spin variables Sj ∈
{−1, 0, 1}, where Sj = ±1 if monomer j is in state A± , and Sj = 0 if it is in state B.
Construct the appropriate generalization of Eqn. 4.223.
(b) Find the equilibrium length L(T, τ, N ) as a function of the temperature, tension, and
number of monomers.
(c) Now suppose an external magnetic field is present, so the energies of the A± states
are split, with εA± = εA ∓ µ0 H. Find an expression for L(T, τ, H, N ).
(3) Consider a system of identical but distinguishable particles, each of which has a nondegenerate ground state with ε0 = 0, and a g−fold degenerate excited state with energy
ε > 0. Study carefully problems #1 and #2 in the example problems for chapter 4 of the
lecture notes, where this system is treated in the microcanonical and ordinary canonical
ensembles. Here you are invited to work out the results for the grand canonical ensemble.
(a) Find the grand partition function Ξ(T, z) and the grand potential Ω(T, z). Express
your answers in terms of the temperature T and the fugacity z = eµ/kB T .
(b) Find the entropy S(T, z).
(c) Find the number of particles, N (T, z).
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(d) Show how, in the thermodynamic limit, the entropy agrees with the results from the
microcanonical and ordinary canonical ensembles.
(4) The grand partition function for a system is given by the expression
Ξ = (1 + z)V /v0 1 + z αV /v0 ,
where α > 0. In this problem, you are to work in the thermodynamic limit. You will also
need to be careful to distinguish the cases |z| < 1 and |z| > 1.
(a) Find an expression for the pressure p(T, z).
(b) Find an expression for the number density n(T, z).
(c) Plot v(p, T ) as a function of p for different temperatures and show there is a first
order phase transition, i.e. a discontinuity in v(p), which occurs for |z| = 1. What is
the change in volume at the transition? .
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