Question

1. (10 pts) Consider a system of N classical, independent harmonic oscil- lators. In the microcanonical ensemble, calculate Ω(E) and Ω(B) exactly. From them, calculate the entropy S(E, N) and temperature T in the large N limit. 2. (10 points) Consider the same system as in problem 1. Calculate the average energy and entropy starting from the canonical ensemble.

Answer 1

Hamiltonian is: H = 1/2m sigma^N_i = 1 p_i^2 + 1/2 k sigma^N_i = 1 x_i^2, w = squareroot k/m equation (1) We first calculate ohm (E) = 1/h^3 N integral_H < E (d^3 p_1 ... d^3 P_N) (d^3 q_1 ... d^3 q_N) equation (2) h = dimension of momentum times distance, such that ohm(E) dimensionless Let R_1 = squareroot 2mE and R_2 = squareroot 2E/k ohm(E) = 1/h^3 N gamma_3N(R_1) gamma_3N(R_2) equation (3) gamma_3N(R_1) rightarrow volume of an n-sphere of radius R_1 gamma_n(R_1) = integral_p_1^2 + p_2^2 + ... + p_n^2 < R^2 (dp_1 dp_2 ... dp_n) clearly, gamma_n (R_1) = C_n R_1^n equation (3, a) where C_n is a constant. To find C_n, consider the identity integral^+ infinity_-infinity dp_1 ... integral^+ infinity_-infinity dp_n e^- (p_1^2 + p_2^2 + ... + p_n^2) = (integral^+ infinity_-infinity dp e^-p^2)^n = pi^n/2 equation (4) \r\n Let s_n (R_1) identicalto d ohm_n (R_1)/dR be the surface area of an n-sphere of radius R, Then