Question

Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and the heat capacity C1 per particle from Z(T, L. [12 points] (c) Calculate the Boltzmann entropy S(E,N) of all N particles. Consider them as indistinguishable. [16 points] Hint: Use Stirling's formula for large N : N! /2nNNe-N.

Answer 1

1
Energy F ith particle is e_1 = c1 Vertical-bar i Vertical-bar in natural units put c = 1 e_i = 1 Vertical-bar i Vertical-bar p_i values from -infinity to infinity partition function for single particle z_i = integral^L_x = 0 integral^infinity_p_i = -infinity e^-beta Vertical-bar i Vertical-bar infinity times dt_i = 2 integral^L_x = 0 integral^infinity_k = 0 e^-beta p_i d times p_i z_i = 2L integral^infinity_k = 0 e^-beta p_i dp_i Z_i = 2L/beta partition function for N particles is z = integral^L_x_i = 0 integral^L_x_2 = 0 ..... integral^L_x_n = 0 integral^infinity_p_i = infinity e^- beta sigma^N_i = 1 Vertical-bar p_i Vertical-bar dx_1,....dx_N dp_1,...dp_N value of all p_1, p_2,...p_N vary from - infinity to infinity \r\n z = integral^L_x_1 = 0 integral^L_X_2 = 0 .... E^-beta( Vertical-bar p_1 Vertical-bar + .... + Vertical-bar p_N Vertical-bar ) z = (integral^L_x_1 = 0 integral^v_si = -infinity e^-p Vertical-bar p Vertical-bar dx dp_x)^N z = z^N_1 so average energy for system is < E > = -partial differential/partial differential beta ln z < E. = -1/z partial differential/partial differential beta z^N < E > = -partial differential/partial differential beta ln z^N_1 = -N/Z_1 partial differential z_1/ partial differential beta
2

34

5