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Ludwig BoltzmannThe Man Who Trusted Atoms$

Carlo Cercignani

Print publication date: 2006

Print ISBN-13: 9780198570646

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198570646.001.0001

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

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

Source:
Ludwig Boltzmann
Publisher:
Oxford University Press

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

(A5.27) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
As we said in the main text, this problem is one of the motivations laid down by Boltzmann in the introduction to his basic paper [1]. In order to solve eqn (A5.27), we prove a preliminary result which plays an important role in the theory of the Boltzmann equation and in the proof of the 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:
(A5.28) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
further, the equality sign applies if and only if log f is a collision -invariant, or, equivalently, f is given (almost everywhere) by
(A5.29) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
where 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):
(A5.30) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
and eqn (A5.28) follows thanks to the elementary inequality (please note that the two factors are positive, negative, or zero at the same time):
(A5.31) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
Equation (A5.31) becomes an equality if and only if y = z; thus the equality sign holds in eqn (A5.28) if and only if
(A5.32) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
applies almost everywhere. But, taking the logarithms of both sides of eqn (A5.32), we find that ø = log 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 fL 1 (ℜ3). If we let c = −β, b = 2βv (where v is another constant vector), eqn (A5.29) can be rewritten as follows:

(A5.33) Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
where 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 v = 0 (p.272) (non-drifting Maxwellians), which can be obtained from drifting Maxwellians by a change of the origin in velocity space.

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).