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Possibly useful information:
Z ∞
−∞
dxe−ax2+bxdx =
r
π
a
e
b
2
4a (for a > 0)
Γ(n) = (n − 1)! =
Z ∞
0
dxxn−1
e
−x
ln N! ≈ N ln N − N (for N 1)
ζ(m) =
∞
∑
n=1
n
−m =
1
Γ(m)
Z ∞
0
dx x
m−1
e
x − 1
ζ(1) = ∞ ζ(2) = π
2
6
ζ(3) = π
4
90
Z π
0
dθ sin θ = 2, R π
0
dθ sin θ cos θ = 0, Z π
0
dθ sin θ cos2
θ =
2
3
1. (4 points) Approximately draw P vs. v for a van der Waals gas for a single
temperature below the critical temperature. Label the unstable region and approximately draw the Maxwell construction for this isotherm.
2. (4 points) Consider the mean-field approximation to the Ising model
HMF = −
q J
2
hσi
N
∑
i=1
σi − h
N
∑
i=1
σi
where q is a factor of order unity and h is the scaled external magnetic field.
This system has the partition function
Z = ∑
σ1
…∑
σN
e
−βHMF({σi}) = 2
NcoshN
(β
q J
2
hσi + βh)
Find the self-consistency equation for hσi.
3. (8 points) Assuming a three-dimensional system and given the single-particle
distribution function
f1(~x,~p, t0) = exp
−β
~p
2
2m
+ βµ
(1 + a cos θ), (1)
where the angle θ is defined by ~p = p(sin θ cos φ, sin θ sin φ, cos θ), a is a constant with a magnitude less than one, µ is the chemical potential, calculate the
energy flux in the z-direction.
4. Assume that you have a one-dimensional gas that can be modeled using Euler’s equations of hydrodynamics. This gas has a temperature that is constant
in time and space, i.e. T(x, t) = T0, a velocity that is constant in space but
changes linearly with time, i.e. u(x, t) = αt, and an initial density ρ(x, t =
0) = ρ0 exp(−λx). Here λ, α, ρ0, and T0 are constants. Assume the gas is ideal
so that the pressure is give by P = ρT/m and the internal energy per mass
is given by e = 3
2m
T (m is the mass of a particle). In one dimension, Euler’s
equations (assuming there are no external forces) are
∂ρ
∂t
+ u
∂ρ
∂x
= −ρ
∂u
∂x
∂u
∂t
+ u
∂u
∂x
= −
1
ρ
∂P
∂x
∂e
∂t
+ u
∂e
∂x
= −
P
ρ
∂u
∂x
(2)
(a) (2 points) Show that the energy evolution equation is satisfied by the assumed temperature and velocity profiles.
(b) (2 points) Find the value of λ that is required for the fluid velocity evolution equation to be satisfied at time zero.
(c) (4 points) Use the density evolution equation and the results from part
(b) to find the density at all times and positions by assuming it has the
form ρ(x, t) = ρ(x, t = 0)f(t). Briefly describe what the solution means
physically.
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