Statistical Mechanics · Problem 6.10

Ideal Bose Systems — R. K. Pathria

Physics — Statistical Mechanics · relativistic-boltzmannian-gas-momentum-distribution

Problem

(a) Show that the momentum distribution of particles in a relativistic Boltzmannian gas, with

\varepsilon = c(p^2 + m_0^2 c^2)^{1/2}

, is given by

f(p)\,dp = Ce^{-\beta c(p^2+m_0^2c^2)^{1/2}}p^2\,dp,

with the normalization constant

C = \frac{\beta}{m_0^2 c K_2(\beta m_0 c^2)},

K_v(z)

being a modified Bessel function. (b) Check that in the nonrelativistic limit

(kT \ll m_0 c^2)

we recover the Maxwellian distribution,

f(p)\,dp = \left(\frac{\beta}{2\pi m_0}\right)^{3/2} e^{-\beta p^2/2m_0}(4\pi p^2\,dp),

while in the extreme relativistic limit

(kT \gg m_0 c^2)

we obtain

f(p)\,dp = \frac{(\beta c)^3}{8\pi}e^{-\beta pc}(4\pi p^2\,dp).

\langle pu \rangle = 3kT.