Statistical physics and thermodynamics: Alternative problems I.
1. Determine the density of states of free nonrelativistic particle with mass m in 1D.
2. Show that from a statistical definition of free energy F = −kT lnZ follows for entropy S = −k∑n wn logwn.
3. From a partition function
Z =
1
N!
mkT
2π¯h3
3/2N
VN
determine cp.
4. Using
Ω = −
∞
0
E
0 ρ(E′)dE′ dE
e
E−µ
kT −1
determine the Landau potential of ultrarelativistic bosons in 3D.
5. Show that the fermionic function
Fn(y) =
1
Γ(n)
∞
0
xn−1
ex−y +1
dx
can be written in terms of series as
Fn(y) =
∞
∑
j=1
(−1)j+1 ejy
jn
.
Using this result, determine an approximate equation of state for nonrelativistic fermionic gas in 3D using
first two members of series.
6. The Landau potential of bosonic gas at temperatures lower than the critical temperature Tc is
Ω = −NkT
T
Tc
3/2
ζ(5/2)
ζ(3/2)
.
Determine entropy and heat capacity CV,N.
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