Equilibrium Solutions of Boltzmans Equation

By definition, the equilibrium distribution /o is one which does not depend on time. For simplicity assume further that the system is uniform in space so that /o is not a function of position, and set all external forces F = 0. The LHS of equation 9.32 

A sufficient condition for fo to solve equation 9.33 is that the term in the curly brackets itself vanishes  

Taking the logarithm of both sides of the expression on the RHS of this equation, we have 

A gas is not in equilibrium when its distribution function differs from the Maxwell-Boltzman distribution. On the other hand, it can also be shown that if a system possesses a slight spatial nonuniformity and is not in equilibrium, then the distribution function will monotonically relax in velocity space to a local Maxwell-Boltzman distribution, or to a distribution where p = N/V, v and temperature T all show a spatial dependence [bal75]. 

Boltzman s H-Theorem Let us consider a binary elastic collision of two hard-spheres in more detail. Using the same notation as above, so that v, V2 represent the velocities of the incoming spheres and v, V2 represent the velocities of the outgoing spheres, we have from momentum and energy conservation that 

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