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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.287) Appendix 8.1 The H-theorem for classical polyatomic molecules

(p.287) Appendix 8.1 The H-theorem for classical polyatomic molecules

Source:
Ludwig Boltzmann
Publisher:
Oxford University Press

In this appendix, following ref. [12], we shall be concerned with the Boltzmann equation with particular reference to the H-theorem, which provoked Lorentz’s objections that we mentioned in the main text. First of all, let us remark that there are no formal difficulties in writing a Boltzmann equation for molecules with n > 3 degrees of freedom. Together with the position vector of the centre of mass x and the corresponding velocity ξ, we shall also need other variables; we shall denote by p a vector in a (2n − 3)-dimensional space which includes all the variables except the coordinates of the centre of mass (while the components of ξ are not excluded). Then the equations of the motion between two subsequent collisions are

(A8.1) Appendix 8.1 The H-theorem for classical polyatomic molecules
where P is a vector in 2n − 3 dimensions that describes the partial derivatives (with the appropriate sign) of the Hamiltonian with respect to the coordinates and momenta different from ξ (but x will now be included). The Boltzmann equation for the distribution function f(x, p, t) can be written as follows:
(A8.2) Appendix 8.1 The H-theorem for classical polyatomic molecules
where f’, Appendix 8.1 The H-theorem for classical polyatomic molecules and f * denote, as usual, that the function f has (besides x and t) arguments p’, Appendix 8.1 The H-theorem for classical polyatomic molecules, and p *, respectively, whereas Appendix 8.1 The H-theorem for classical polyatomic molecules (which will include some factors given by the Dirac delta function, in order to ensure conservation of momentum and energy) is essentially the differential scattering cross-section multiplied by the relative speed and in fact we have
(A8.3) Appendix 8.1 The H-theorem for classical polyatomic molecules
where Σt is the total scattering cross-section (assumed to be finite). The latter may of course depend on p and p *. However, we can get rid of this dependence by a trick that makes the proof much simpler. Since the total cross-section is assumed to be finite, there will be a maximum distance r 0 beyond which the molecules do not mutually interact. We can then let Appendix 8.1 The H-theorem for classical polyatomic molecules, provided that we introduce false collisions in which no change occurs for those values of p and p * for which possibly there is no interaction, although r < r 0. The case in which the collision cross-section is infinite can be obtained by first cutting off the interactions occurring for r > r 0 and then letting r 0 → ∞ in the final inequality that we are going to prove.

The microscopic motion equations (A8.1) are of course assumed to be time-reversible; this means that there exists a transformation (x, p, t) → (x, p , −t) (where, typically, a component of p is to be equal to the corresponding one of p if it has the meaning of a coordinate, and opposed to the corresponding one of p if it has the meaning of a momentum canonically conjugated to a coordinate). Incidentally, we remark that one frequently treats the changes in the coordinates as if they could be ignored in the average and just considers the changes in the momenta; this aspect however will not enter into what follows.

It is important to notice that the transformation of variables from the p’s to the p ’s, with fixed x, preserves the volume in the space described by the variables p.

The time reversibility of the microscopic equations even during a collision implies that the following relation, called reciprocity, holds:

(A8.4) Appendix 8.1 The H-theorem for classical polyatomic molecules
(p.288) If the interaction possesses spherical symmetry, as is the case for mass points or perfectly smooth hard spheres, then the following stronger property holds:
(A8.5) Appendix 8.1 The H-theorem for classical polyatomic molecules
which is called detailed balance. This property is more or less explicitly used in the proof of the H-theorem for monatomic gases. Whenever it holds, the proof given in Chapter 5 can be transferred without change to the case of polyatomic gases. Boltzmann’s mistake, pointed out by Lorentz, lies precisely in the implicit assumption that eqn (A8.5) holds for a generic polyatomic gas. If this property fails, one can introduce Boltzmann’s argument on the “closed cycle of collisions”, which, although not so convincing, contains the basic idea of the proof that we shall presently give, following ref. [12], i.e. the fact that one must consider not single collisions but subsets of collisions.

The key point is eqn (A8.3), whose right-hand side is clearly invariant under time reversal, and the reciprocity property, expressed by eqn (A8.4). Let us integrate both sides of the latter relation with respect to p and p *. We obtain

(A8.6) Appendix 8.1 The H-theorem for classical polyatomic molecules
where, in the last step, we have changed the integration variables from p, p * to p , Appendix 8.1 The H-theorem for classical polyatomic molecules (using the aforementioned invariance of the volume element) and subsequently abolished the superscript “—”, which is no longer required. But the last integral is that which appears on the left-hand side of (A8.3), apart from the presence of the superscript “—” in the second pair of variables; but because of the invariance of the right-hand side of the same equation with respect to the transformation from variables with superscript “—” to variables without the same superscript, we can suppress the latter in the last integral of (A8.6) and get
(A8.7) Appendix 8.1 The H-theorem for classical polyatomic molecules
This is the new relation that we shall use to prove the H-theorem for polyatomic molecules. The importance of a relation of this kind was underlined by Waldmann [9], who, guided by an analogy with the quantum case, wrote it without proof in the particular case of dumb-bell shaped classical molecules, remarking that “one must get the (purely mechanical) normalization property” expressed by (A8.7). As a possible proof, the same author [9] seems to hint at a complete calculation with the simplifying assumption of “averaging over all possible phase angles […] before and after collision”. This average, albeit useful in some cases as indicated above, to simplify the relations, is not required in the proof which we have just discussed of ref. [12], according to which (A8.7) is a general property which follows from the time reversibility of the microscopic equations of the molecular motion.

Having shown that (A8.7) holds, it is now a relatively simple matter to prove the H-theorem, or, it is better to say, the Boltzmann lemma (whence the H-theorem follows). According to this lemma, if we let

(A8.8) Appendix 8.1 The H-theorem for classical polyatomic molecules
(p.289) we get
(A8.9) Appendix 8.1 The H-theorem for classical polyatomic molecules
To prove this result, let us multiply (A8.8) by log f and integrate with respect to p, thus obtaining
(A8.10) Appendix 8.1 The H-theorem for classical polyatomic molecules
This relation can be easily obtained by the same manipulations as discussed in the Appendices to Chapter 5, i.e. with suitable changes of variables and indices, without using any property of Appendix 8.1 The H-theorem for classical polyatomic molecules. Equation (A8.10) however does not permit the argument used in Chapter 5 to be used; to this end, in fact, we should sum (A8.10) and the relation that can be obtained from it by exchanging the primed and unprimed variables and use the property of detailed balance, (A8.5), which however does not generally hold. At this point we must make use of a trick, apparently first used by Pauli [10] in quantum statistical mechanics. Together with the identity expressed by (A8.10), let us also consider the following one:
(A8.11) Appendix 8.1 The H-theorem for classical polyatomic molecules
This can be obtained by an exchange of name of the variables and expresses the conservation of the number of particles in a collision. We can now make use of (A8.7) on the left-hand side of (A8.11) and rewrite it as follows:
(A8.12) Appendix 8.1 The H-theorem for classical polyatomic molecules
or, equivalently:
(A8.13) Appendix 8.1 The H-theorem for classical polyatomic molecules
We can then subtract the integral appearing on the left-hand side of (A8.13) from the right-hand side of (A8.10), without changing anything. We then have:
(A8.14) Appendix 8.1 The H-theorem for classical polyatomic molecules
Let us now make use of the fact that both f and W are non-negative and that the following (previously used) elementary inequality holds:
(A8.15) Appendix 8.1 The H-theorem for classical polyatomic molecules
(p.290) We can then conclude that (A8.9) has been proved and that the equality sign holds if and only if (almost everywhere)
(A8.16) Appendix 8.1 The H-theorem for classical polyatomic molecules
i.e. if the distribution function describes an equilibrium state.