# (p.271) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution

# (p.271) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution

In this appendix we investigate the existence of positive functions *f* which give a vanishing collision integral:

*H*-theorem, which was the second declared aim of his paper [1]: if

*f*is a non-negative function such that (log

*f*)

*Q(f, f)*is integrable and the assumptions of the previous section hold when ø = log

*f*, then the following

*Boltzmann inequality*holds:

*f*is a collision -invariant, or, equivalently,

*f*is given (almost everywhere) by

*a*,

**,**

*b**c*are constant with respect to ξ. To prove eqn (A5,28) it is enough to use eqn (A5.8) with ø = log

*f*(and the well-known properties of the logarithmic function):

*y*=

*z*; thus the equality sign holds in eqn (A5.28) if and only if

*f*satisfies eqn (A5.9) and its most general expression is thus given by eqn (A5.10). Then

*f*= exp(ø) is given by eqn (A5.29).

We remark that in the same equation *c* must be negative, since *f* ∊ *L* ^{1} (ℜ^{3}). If we let *c* = −β, ** b** = 2β

**(where**

*v***is another constant vector), eqn (A5.29) can be rewritten as follows:**

*v**A*is a positive constant related to

*a, c*, |

**|**

*b*^{2}(β,

**,**

*v**A*constitute a new set of constants). The function appearing in eqn (A5.33) is the so-called

*Maxwell distribution*or

*Maxwellian*, which we have frequently mentioned before. Frequently one considers Maxwellians with

**= 0 (p.272) (non-drifting Maxwellians), which can be obtained from drifting Maxwellians by a change of the origin in velocity space.**

*v*Let us return now to the problem of solving eqn (A5.27). Multiplying both sides by log *f* gives eqn (A5.28) with the equality sign. This implies that *f* is a Maxwellian almost everywhere, by the result which has just been proved. Suppose now that *f* is (almost everywhere) a Maxwellian; then *f* = exp(ø), where ø is a collision-invariant and eqn (A5.32) holds; eqn (A5.27) then also holds. Thus there are functions which satisfy eqn (A5.27) and they are all Maxwellians, given by eqn (A5.33).