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Category: PHY 831

Description

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Possibly useful results

Z ∞

−∞

dxe−ax2+bx =

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)

1. (a) (2 points) Prove the relationship

∂S

∂V

T,N

=

∂P

∂T

V,N

(b) (4 points) Using only the properties P = NT/V and CV = T

∂S

∂T

N,V

=

3

2N for an ideal gas, find the adiabatic sound speed squared

c

2

s = −

V

2

Nm

∂P

∂V

S,N

in terms of T, N, and V using partial derivative relations. Here, m is the

mass of the particles in the gas. The result of part (a) should be useful.

2. Consider a classical, non-interacting gas of N indistinguishable particles that

can only move in one-dimension and obey the single-particle dispersion relation e = |~p|c (i.e. the Hamiltonian is given by H = ∑

N

i=1

|~pi

|c) and are confined

to a length L.

(a) (8 points) Find the canonical partition function when N = 1.

(b) (2 points) Find the canonical partition function for arbitrary N.

(c) (2 points) Calculate the entropy of the N particle system.

3. (8 points) Consider a systems with an equation of state given by

P =

E

V

V

V0

λ

E

E0

λ

and a temperature given by

T = E

E

E0

λ

,

where E0, V0, and λ are constants. Given the entropy at E0 and V0 is S(E0, V0) =

0, find the entropy at arbitrary energy E and volume V.

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