Equipartition Theorem

a) For a classical system with Hamiltonian

at a temperature τ, show that

b) Using the above, derive the law of Dulong and Petit for the heat capacity of a harmonic crystal.

c) For a more general Hamiltonian,

prove the generalized equipartition theorem:

where x₁ = q₁,…, x_N = q_N, x_N+1 = p₁,…,x_2N = p_N. You will need to use the fact that U is infinite at q_i = ±∞.

d) Consider a system of a large number of classical particles and assume a general dependence of the energy of each particle on the generalized coordinate or momentum component q given by ε (q) where

Show that, in thermal equilibrium, the generalized equipartition theorem holds:

What conditions should be satisfied ε (q)  for to conform to the equipartition theorem?

SOLUTION

a) For both of these averages the method is identical, since the Hamiltonian depends on the same power of either or q. Compose the first average as follows:

(1)

where the energy is broken into the p_i - dependent term and E', the rest of the sum. The second integrals in the numerator and denominator cancel, so the remaining expression may be written

(2)

where, as usual, β ≡ 1/τ. A change of variables produces a piece dependent on β and an integral that is not:

(3)

The (k_i/2) ⟨q²_i⟩ average proceeds in precisely the same way, yielding

(4)

b) The heat capacity, C_V, at constant volume is equal to ∂E/∂τ. From part (a), we have

(5)

where we now sum over the 3-space and momentum degrees of freedom per atom. The heat capacity,

(6)

is the law of Dulong and Petit.

c) Now take the average:

(7)

Integration by parts yields

(8)

where the prime on the product sign in the first term indicates that we integrate over all x_i except i = j. If i ≠ j, then the first term in the numerator equals zero. If x_j is one of the q's, then by the assumption of U infinite, the term still equals zero. Finally, if i = j > N ,then by l’Hôpital’s rule the first term again gives zero. In the second term, ∂x_i/∂x_j = δ_ij, so the expression reduces to

(9)

Finally,

(10)

d) By definition,

(11)

Given a polynomial dependence of the energy on the generalized coordinate:

(12)

(11) yields

(13)

To satisfy the equipartition theorem:

(14)

Thus, we should have n = 2.