# (1)

```PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #3
P
(1) Show that the Boltzmann entropy S = −kB n Pn ln Pn agrees with the statistical entropy S(E) = kB ln D(E, V, N ) in the thermodynamic limit.
(2) [Kardar 4.12] Consider rod-shaped molecules with moment of inertia I, and a dipole
moment &micro;. The contribution of the rotational degrees of freedom to the Hamiltonian is
2
Ĥrot =
pφ
p2θ
− &micro;E cos θ ,
+
2I
2I sin2 θ
where E is the external electric field, and (θ, φ) are polar and azimuthal angles describing
the molecular orientation.
(a) Calculate the contribution of the rotational degrees of freedom of each dipole to the
classical partition function.
(b) Obtain the mean polarization P = h&micro; cos θi of each dipole.
∂P
.
(c) Find the zero-field isothermal polarizability, χ(T ) = ∂E
E=0
(d) Calculate the rotational energy per particle at finite field E, and comment on its high
and low-temperature limits.
(e) Sketch the rotational heat capacity per dipole as a function of temperature.
(3) A surface consisting of Ns adsorption sites is in thermal and particle equilibrium with
an ideal monatomic gas. Each adsorption site can accommodate either zero particles (energy 0), one particle (two states, each with energy ε), or two particles (energy 2ε + U ).
(a) Find the grant partition function of the surface, Ξsurf (T, Ns , &micro;). and the surface grand
potential Ωsurf (T, Ns , &micro;).
(b) Find the fraction of adsorption sites with are empty, singly occupied, and double
occupied. Express your answer in terms of the temperature, the density of the gas,
and other constants.
(4) Consider the Hamiltonian below with N kinetic DOF, N2 quadratic potential DOF, and
N4 quartic DOF, with N ≥ N2 + N4 .
Ĥ =
1
2
N
X
i,j
m−1
ij pi pj
+
N2
X
i=1
N2 +N4
2
1
2 Ki qi
+
X
4
1
4 Ai qi
.
i=N2 +1
Find the free energy F and the internal energy E in terms of T , V , N , N2 , and N4 . Assume
the matrix mij is nondegenerate.
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```