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Suppose we have l degrees of freedom (generalized coordinates) q1, q2,..., ql and corresponding momenta p1, p2,..., pl, and all q's are close to their equilibrium values (take them to be zero, and qi will be deviations from equilibrium positions): . We want to calculate thermodynamics of such a system.
Potential energy: let's expand
The first term does not matter--we'll drop it! The second term is zero (we are at equilibrium!)
In harmonic approximation both potential and kinetic energy are quadratic forms of coordinates and momenta.
Expanding U, we recover quadratic form (1)
Classical partition function:
The integral does not depend on temperature!
with X not dependent on T. Entropy, energy and heat capacity:
We have l degrees of freedom, and energy is lkBT: one kBT for each degree of freedom!
What happens, if for some qi potential energy is zero (rotational and translational degrees of freedom)? Then it contributes kBT/2.
In harmonic approximation for classical systems each degree of freedom contributes kBT to energy and kB to heat capacity unless potential energy is identically zero. In the latter case it contributes kBT/2 and kB/2 correspondingly.
Equipartition theorem is fine--except sometimes it does not work!
© 1997 Boris Veytsman and Michael Kotelyanskii