Problem 16 We showed that the entropy of a system could be written in the form S = kB H where H is the Eta function of Boltzmann given by
H=−
X
Pi ln Pi
i
and Pi is the probability of occupation of quantum state i. The purpose of this problem is to explore some properties of this function. (a) For a system having a finite number of states M, show that this function achieves its maximum value when all of these probabilities are equal. Determine from your result the maximum value of the entropy for such a system. Hint: Use a Lagrange multiplier to handle the constraint among the Pi . (b) Now introduce the explicit form Pi = exp(−βEi )/Z(β) given by the canonical ensemble for a system at temperature T = 1/(kB β) and fixed volume V. Show by straightforward differentiation that
∂S ∂T
= V
P
kB T
j
Pj (εj − U )2 (kB T )2
where U is the internal energy at temperature T. (c) From the result of part (b), what can you deduce about the behavior of S with increasing temperature and the sign of the heat capacity CV of the system? For a system having a finite number of states M, what would be the slope of a graph of ln CV versus ln T at extremely high temperatures? Problem 17 The Helmholtz free energy of an an ultrarelativistic ideal gas having N particles is V BT F = −N kT [ln(8π N ) + 3 ln( khc ) + 1]
where c is the speed of light. (a) What is the canonical partition function Z(β) for this gas? Hint: Use Stirlings approximation in reverse to relate the terms in N to N!. (b) Determine the density of states D(E) for this gas by inverting the Laplace transform of the partition function Z(β). You will need to calculate the contour integral
1 g(E) = 2πi
Z
eβE Z(β)dβ,
Br
where the Bromwich contour Br starts at −i∞, ends at i∞ and lies to the right of all singularities of Z(β). (c) According to the microcanonical ensemble, we showed in problem 9 that Ω for such a gas has the functional form ˜˜ )V N E 3N . Ω = Ω(N 1
˜˜ ). Use the result of part (b) to determine the unknown function Ω(N Problem 18 This is a revised version of Pathria Problem 3.26. The objective of this problem is to compare the partition function of N identical but distinguishable s-dimensional harmonic oscillators and their thermodynamic properties with Ns identical but distinguishable one-dimensional harmonic oscillators. In addition, the chemical potentials should be compared to deduce that µs = sµ1 . (a) First, assume that each dimension is governed by an independent quantum number ni = 0, 1, 2 so that the energy states of a single s-dimensional oscillator are given by
ε(n1 ···ns ) =
s X (ni + 1/2)¯hω i=1
Solve the problem by calculating the partition function for each direction and using the factorization theorem to deduce the partition function of the whole system. (b)Show that the energy levels of the system can be written in the form εj = (j + s/2)¯hω where j = 0, 1, 2 · ·· and that these energy levels have degeneracy (j + s - 1)!/[(j - 1)!s!]. Now obtain the total partition function for the whole system by doing the partition sum of exp(βεj ) over j, including the degeneracy factor. Hint: Since you already know the answer, evaluate the sum by showing that both expressions have the same power series. (c) Calculate the chemical potentials µs and µ1 and compare your results. Problem 19 This is a version of Pathria Problem 3.24 page 86. The Hamiltonian of a relativistic particle of rest mass m can be expressed in terms of its canonical momentum p by
H = mc2 (1 +
p2 1/2 ) − mc2 m2 c2
where c is the speed of light and p2 = p2x + p2y + p2z . Here, energy is measured relative to the rest energy, mc2 , so H becomes p2 /(2m) in the non-relativistic limit and therefore plays the role of the kinetic energy. The momentum is related to the velocity v by
p=
mv . (1 − v 2 /c2 )1/2
(a) Show that these relations are consistent with Hamiltons equations, namely that
∂H = x˙ α ∂pα where a is a cartesian component and the dot denotes a time derivative. (b) Use the results for classical averaging in the canonical ensemble (Sekerka 18.5) to show that equipartition takes the form 2
1 m 2
v2 (1 − v 2 /c2 )1/2
=
3 kB T. 2
(c) Use the corresponding expression in terms of p2 to evaluate approximately the thermal average < H > of the kinetic energy in the non relativistic and the extreme relativistic limits and compute the ratio of you results. Problem 20 This problem is adapted from Pathria 3.36 page 88, for which the given answer is wrong because of a typographical error. The potential energy of two dipoles having dipole moments µ and µ0 separated by a vector R of magnitude R can be written in the form
U=
1 0 ˆ ˆ [µ · µ0 − 3(µ · R)(µ · R)] R3
ˆ = R/R. Pathria’s expression for U is obtained by choosing R ˆ to be along the z-axis where the unit vector R ˆ ˆ of a spherical coordinate system, in which case R = k and ˆ µ = µ sin θ cos φˆi + µ sin θ sin φˆj + µ cos θk and similarly for µ. This gives
U =−
µµ0 µµ0 0 0 0 [2 cos θ cos θ − sin θ sin θ cos(φ − φ )] =: − G(θ, φ, θ0 , φ0 ). R3 R3
The kinetic energy of each dipole can be approximated as if each were a diatomic molecule. In a reference frame where each molecule is rotating but not translating, the kinetic energy takes the form
K(pθ , pφ , θ, pθ0 , pφ0 , θ0 ) :=
1 2 1 (p + p2φ / sin2 θ) + 0 (p2θ0 + p2φ0 / sin2 θ0 ) 2I θ 2I
where I and I’ are moments of inertia and pθ , pφ , pθ0 , pφ0 are conjugate momenta to the coordinates θ, φ, θ0 , φ0 . The Hamiltonian is therefore
H(pθ , pφ , θ, φ, pθ0 , pφ0 , θ0 , φ0 ) = K −
µµ0 G. R3
(a) Returning to the general expression for U, show that the instantaneous force exerted on a dipole at R by a dipole located at the origin is given by
F =
1 3 0 ˆ ˆ ˆ ˆ 0 + 3 (µ0 · R)µ ˆ [3µ · µ0 − 15(µ · R)(µ · R)]R + 4 (µ · R)µ 4 R R R4
(b)The average force is given by 3
R exp(−βH)Fdθdφdθ0 dφ0 dpθ dpφ dpθ0 dpφ0 . hFi = R exp(−βH)dθdφdθ0 dφ0 dpθ dpφ dpθ0 dpφ0 Show that integration over the canonical momenta leads to factors in both integrals that are essential to integration over the coordinates. Cancel any common factors and rewrite the resulting expression for hFi in terms of integrals over the four angles. ˆ give components of force that are not (c) Note that the last two terms in the general expression for F ˆ necessarily along R. Show that the average values of these terms give rise to forces that are only along ˆ Hint: G depends only on φ and φ0 only through the variable Φ = φ − φ0 . ˆ = k. R (d) Now expand the exponentials in powers of a 1 and do the remaining integrals only to the extent necessary to obtain a non-vanishing answer to lowest order in a. (e) Compare your result with that of Pathria and give two reasons why his result cannot possibly be correct.
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