Question

3) Relativistic Particles Consider a gas of N relativistic particles with Hamiltonian rn n= where c is the speed of light, m is the rest mass, and pn is the momentum, with pn (Pnz, Pny, pns) and p; = pn . pn-Ping +pny +Pie. Show that p2e2 = 3kgT.

Answer 1

The partition function Z = 1/h^3 N integral d^3 P_1 Ellipsis d^3 P_N d^3 n_1 Ellipsis d^3 n_N e^-beta H since H is independent of n, integral d^3 n_1 Ellipsis d^3 n_N = v^N, v = volume of space Therefore Z = v^N/h^3 N integral d^3 P_1 Ellipsis d^3 P_N - e^-beta Sum_n = 1^N squareroot m^2 c^4 + P_n^2 c^2 = v^N/h^3 N pi_n = 1^N integral d^3 P_n - e^-beta squareroot m^2 c^4 + P_n^2 c^2 = v^N/h^3N (integral d^3 P e^-beta squareroot m^2 c^4 + P^2 c^2)^N but d^3 P = 4 pi P^2 dP Therefore Z = v^N/h^3 N (4 pi)^N integral_0^infinity dP P^2 e^-beta c squareroot P^2 + m^2 c^2 \r\n Z = (4 pi v/h^2)^N (integral_0^infinity dP P^2 e^-beta c squareroot P^2 + m^2 c^2)^N but notified Bessel function k_r (x) = integral_0^infinity e^-cos ht cos hrt dt Put P = nc cos hx (cos h^2 x - sin h^2 x = 1) dP = nc cos hx dx