Question

2. The one-particle distribution function of the velocity of a particle obeys Maxwell Boltzmann statistics: where 2 2mexp Use direct integration of the probability density function to answer the following: (a) Show that the probability that the particle has any velocity must be unity (b) Show that the probability that all three components of the velocity are negative is 1/8. (c) Using the full probability density function fe, show that the average value of is lurl is 2kBT 1/2

Answer 1

f_v^vector dv^vector = g(v_x) g(v_y) g(v_z) dv_x dv_y dv_z where, g(v_x) = (beta m/2 pi)^1/2 e^-beta m/2 v_x^2 g(v_y) = (beta m/2 pi)^1/2 e^-beta m/2 v_y^2: beta = 1/k_B T g(v_z) = (beta m/2 pi)^1/2 e^-beta m/2 v_z^2 (a) p(v) = integral_All space f_v^vector dv^vector = integral^infinity_-infinity integral^infinity_-infinity integral^infinity_-infinity g(v_x) g(v_y) g(v_z) dv_x dv_y dv_z Now, integral^infinity_-infinity g(v_x) dv_x = (beta m/2 pi)^1/2 integral^infinity_-infinity e^-beta m/2 v_x^2 dv_x = (beta m/2)^1/2 middot 2/squareroot pi integral^infinity_0 v_x^0 e^-beta m/2 v-x^2 dv_x = (beta m/2 pi)^1/2 middot 2/squareroot pi middot 1/2 middot (2/beta m)^1/2 middot squareroot 1/2 {Using integral^infinity_0 x^n e^-alpha x^2 dx = 1/2 middot 1/alpha^n + 1/2 middot squareroot n + 1/2 = 1/squareroot pi times squareroot pi = 1 (Because squareroot 1/2 = squareroot pi) Similarly, integral^infinity_-infinity g(v_y) dv_y = 1 = integral^infinity_-infinity g(v_z) dv_z Therefore p(v) = 1 middot 1 middot 1 = 1 Ans proved. Implies Probability of any velocity is unity.