Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 407

Solution:
(a) The assumptions made are the following.

1. A collision can be considered to take place at a point as the collision
   time is generally much shorter than the average time interval between two
   collisions.

2) The particle number density f(r,v, t) is the same throughout the

interval d3rd3v.

3) Particles of different velocities are completely independent, i.e., the

distribution for particles of different velocities v1,vz can be expressed as

4) Only central forces and two-body elastic collisions need to be con-

sidered.

(b) By assumption l), we need consider only the collision rate

(af1/&),,,, of particles with v1 in the interval d3rd3vl.

By assumption 2), the probability for a particle with v1 is to be scat-

tered by a particle with v2 into solid angle R in time interval dt can be

written as

/vl - v21u(R)f(r,v2, t)dnd3v2dt.

By assumption 3) we can write the number density of particles with

v1 scattered into solid angle R as

Iv1 - v21u(R)f(r,v2,t)f(r,v1, t)dRd3v2d3vldt.

Similarly, the increase in the number density of particles with v1 in the

space d3rd3vl after the collision (vi,vh) --+ (v1,vz) is given by

Ivi - v~lu(n)r(r,v~,t)f(r,v:, t)dRd3v$d3vidt.

Assumption 4) gives d3vld3v2 = d3vid3v', and

Hence