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Appendix 8.1 The *H*-theorem for classical polyatomic molecules

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Appendix 8.1 The *H*-theorem for classical polyatomic molecules

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

**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**

*p***is a vector in 2**

*P**n*− 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

**will now be included). The Boltzmann equation for the distribution function**

*x**f*(

**,**

*x, p**t*) can be written as follows:

*f*

_{*}denote, as usual, that the function

*f*has (besides

**and**

*x**t*) arguments

**’, , and**

*p*

*p*_{*}, respectively, whereas (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

_{t}is the total scattering cross-section (assumed to be finite). The latter may of course depend on

**and**

*p*

*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 , provided that we introduce false collisions in which no change occurs for those values of

**and**

*p*

*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

**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.**

*p*It is important to notice that the transformation of variables from the ** p**’s to the

*p*^{−}’s, with fixed

**, preserves the volume in the space described by the variables**

*x***.**

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

*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

*p, p*_{*}to

*p*^{−}, (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

*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

*f*and integrate with respect to

**, thus obtaining**

*p**f*and

*W*are non-negative and that the following (previously used) elementary inequality holds: