Classical gas with general dispersion relation
For gas of noninteracting particles in d dimensional box with kinetic energy of the
form (p)=C|p|s prove the following results:
Define =1+(s/d).
In an adiabatic process PV=const.
The energy is E=NkBT/(1)
kB N
The entropy is S =
ln(PV) + f(N) .
1
The heat capacity ratio is Cp/Cv = .
What is the most general form of the function f(N)?
Density of a gas inside a rotating box
A cylinder of radius R and height H rotates about its axis with a constant angular
velocity It contains an ideal classical gas at temperature T. Find the density
distribution as function of the distance r from the axis. Note that the Hamiltonian in
the rotating frame is H'(p,q; ) = H(p,q) L(p,q) where L(p,q) is the angular
momentum. It is conceptually useful to realize that it is formally the Hamiltonian of a
charged particle in a magnetic field (=“Coriolis force”) plus centrifugal potential V(r).
Explain how this formal equivalence can be used in order to make a “shortcut” in the
above calculation.
Virial coefficients
Find the second virial coefficient B of an ideal Bose gas by evaluating the canonical
partition function of two particles. Repeat for an ideal fermion gas.
Find the second virial coefficient for:
Ideal Bose gas.
Ideal Fermi gas.
Classical hard sphere gas.
Hard Sphere gas
Consider a one-dimensional classical gas of N particles in a length L at temperature
T. The particles have mass m and interact via a 2-body "hard sphere" interaction (xi
is the position of the i-th particle): V(xixj) = ∞ if |xixj|<a, and zero otherwise.
Evaluate the exact free energy F(T,L,N). Find the equation of state and identify the
first virial coefficient; compare with its direct definition. Show that the energy is
E=NkBT/2. Why is there no effect of the interactions on E ?
In three dimensions V(|rirj|) is defined as above with r the position vector.
Comment on the form of the free energy.
2D coulomb gas
N ions of positive charge q and N with negative charge q are constrained to move
in a two dimensional square of side L. The interaction energy of charge qi at
position ri with another charge qj at rj is qiqj ln|rirj| where qi,qj=±q.
a) By rescaling space variables to ri'=Cri, where C is an arbitrary constant, show
that the partition function Z(L) satisfies: Z(L)=CN(q 4)Z(CL) .
b) Deduce that for low temperatures -1<q2/4, The function Z(CL) for the infinite
system (Cinfty) does not exist. What is the origin of this instability?
2
Bose gas for general dispersion relation
Consider an ideal Bose gas in d dimensions whose single particle spectrum is given
by =C|p|s. Find the condition on s, d for the existence of Bose-Einstein condensation.
In particular show that for nonrelativistic particles in two dimensions (s=d=2) the
system does not exhibit Bose-Einstein condensation.
s E
d
Show that P= d V
and
CV(T∞) = s NkB
Heat capacity of Bose gas
Consider an ideal Bose gas and show that the ratio
CP/CV = 3 g1/2(z) g5/2(z) / 2g3/22(z) where z is the fugacity.
Why is CP∞ in the condensed phase?
Find  in the adiabatic equation of state. Note that in general ≠CP/CV .
Heat capacity of He4
The specific heat of He4 at low temperatures has the form
Cv = AT3 + B(T)exp(-kT)
What can you deduce about the excitations of the system?
(assume that the density of excitation modes has the form g()~p as ).
What would be the form of Cv for a similar system in a two dimensional world?
Bose gas in a uniform gravitational field
Consider an ideal Bose gas of particles with mass m in a uniform gravitational field of
accelaration g. Show that the critical temperature for the Bose-Einstein condensation
8 1
is Tc = Tc0[1+9
( mgL ) 1/2 ]
(3/2) kBTc0
where L is the height of the container, mgL<<kBTc0 and Tc0 =Tc(g=0).
[hint: g3/2(z)=g3/2(1)2   ln( z ) + O(ln(z)).]
Show that the condensation is accompanied by a discontinuity in the specific heat at
9
mgL 1/2
Tc, with the result CV = 
(3/2)NkB(
) .
16
kBTc0
[Hint: CV is due to discontinuity in (∂z/∂T)N,V]
Quantum Fermi gas in gravitation field of a star
Consider a neutron star as non-relativistic gas of non-interacting neutrons of mass m
in a spherical symmetric equilibrium configuration. The neutrons are held together by
a gravitational potential mMG/r of a heavy object of mass M and radius r0 at the
center of the star (G is the gravity constant and r is the distance from the center).
Assume that the neutrons are classical (Boltzmann) particles at temperature T and find
their density n(r) at r>r0. Is the potential confining? [By definition, for a confining
potential there a solution with n(r)0 at r∞] .
Next, consider the neutrons as fermions at T=0, and find n(r). Is the potential
confining? Extend your result to the case T≠0, and discuss the connection with the
Boltzmann limit.