Schrödinger: Statistical Thermodynamics
The object of this seminar is to develop briefly one simple, unified standard method, capable of dealing, without changing the fundamental attitude, with all cases (classical, quantum, Bose-Einstein, Fermi-Dirac, etc.) and with every new problem that may arise. The interest is focused on the general procedure, and examples are dealt with as illustrations thereof. It is not a first introduction for newcomers to the subject, but rather a 'repetitorium'. The treatment of those topics which are to be found in every one of a hundred text-books is severely condensed; on the other hand, vital points which are usually passed over in all but the large monographs (such as Fowler's and Tolman's) are dealt with at greater length.
There is, essentially, only one problem in statistical thermodynamics: the distribution of a given amount of energy E over N identical systems. Or perhaps better: to determine the distribution of an assembly of N identical systems over the possible states in which this assembly can find itself, given that the energy of the assembly is a constant E. The idea is that there is weak interaction between them, so weak that the energy of interaction can be disregarded, that one can speak of the 'private' energy of every one of them and that the sum of their 'private' energies has to equal E. The distinguished role of the energy is, therefore, simply that it is a constant of the motion - the one that always exists, and, in general, the only one. The generalization to the case, that there are others besides (momenta, moments of momentum), is obvious; it has occasionally been contemplated, but in terrestrial, as opposed to astrophysical, thermodynamics it has hitherto not acquired any importance.
'To determine the distribution' means in principle to make oneself familiar with any possible distribution-of-the-energy (or state-of-the-assembly), to classify them in a suitable way, i.e. in the way suiting the purpose in question and to count the numbers in the classes, so as to be able to judge of the probability of certain features or characteristics turning up in the assembly. The questions that can arise in this respect axe of the most varied nature, and so the classification really needed in a special problem can be of the most varied nature, especially in relation to the fineness of classification. At one end of the scale we have the general question of finding out those features which are common to almost all possible states of the assembly so that we may safely contend that they 'almost always' obtain. In this case we have well-nigh only one class - actually two, but the second one has a negligibly small content. At the other end of the scale we have such a detailed question as: volume (= number of dates of the assembly) of the 'class' in which one individual member is m a particular one of its states. Maxwell's law of velocity distribution is the best-known example.
This is the mathematical problem - always the same; we shall soon present its general solution, from which in the case of every particular kind of system every particular classification that may be desirable can be found as a special case.
But there are two different attitudes as regards the physical application of the mathematical result. We shall later, for obvious reasons, decidedly favour one of them; for the moment, we must explain them both.
The older and more naive application is to N actually existing physical systems in actual physical interaction with each other, e.g. gas molecules or electrons or Planck oscillators or degrees of freedom ('ether oscillators') of a 'hohlraum'. The N of them together represent the actual physical system under consideration. This original point of view is associated with the names of Maxwell, Boltzmann and others.
But it suffices only for dealing with a very restricted class of physical systems - virtually only with gases. It is not applicable to a system which does not consist of a great number of identical constituents with 'private' energies. In a solid the interaction between neighbouring atoms is so strong that you cannot mentally divide up its total energy into the private energies of its atoms. And even a 'hohlraum' (an 'ether block' considered as the scat of electromagnetic-field events) can only be resolved into oscillators of many - infinitely many - different types, so that it would be necessary at least to deal with an assembly of an infinite number of different assemblies, composed of different constituents.
Hence a second point of view (or, rather, a different application of the same mathematical results), which we owe to Willard Gibbs, has been developed. It has a particular beauty of its own, is applicable quite generally to every physical system, and has some advantages to be mentioned forthwith. Here the N identical systems are mental copies of the one system under consideration - of the one macroscopic device that is actually erected on our laboratory table. Now what on earth could it mean, physically, to distribute a given amount of energy E over these N mental copies? The idea is, in my view, that you can, of course, imagine that you really had N copies of your system, that they really were in 'weak interaction' with each other, but isolated from the rest of the world. Fixing your attention on one of them, you find it in a peculiar kind of 'heat-bath' which consists of the N - 1 others.
Now you have, on the one hand, the experience that in thermodynamical equilibrium the behaviour of a physical system which you place in a heat-bath is always the same whatever be the nature of the heat-bath that keeps it at constant temperature, provided, of course, that the bath is chemically neutral towards your system, i.e. that there is nothing else but heat exchange between them. On the other hand, the statistical calculations do not refer to the mechanism of interaction; they only assume that it is 'purely mechanical', that it does not affect the nature of the single systems (e.g. that it never blows them to pieces), but merely transfers energy from one to the other.
These considerations suggest that we may regard the behaviour of any one of those N systems as describing the one actually existing system when placed in a heat-bath of given temperature. Moreover, since the N systems are alike and under similar conditions, we can then obviously, from their simultaneous statistics, judge of the probability of finding our system, when placed in a heat-bath of given temperature, in one or other of its private states. Hence all questions concerning the system in a heat-bath can be answered.
We adopt this point of view in principle - though all the following considerations may, with due care, also be applied to the other. The advantage consists not only in the general applicability, but also in the following two points:
(i) N can be made arbitrarily large. In fact, in case of doubt, we always mean lim N = ∞ (infinitely large heat-bath). Hence the applicability, for example, of Stirling's formula for N!, or for the factorials of 'occupation numbers' proportional to N (and thus going with N to infinity), need never be questioned.
(ii) No question about the individuality of the members of the assembly can ever arise--as it does, according to the 'new statistics', with particles. Our systems are macroscopic systems, which we could, in principle, furnish with labels. Thus two states of the assembly differing by system No. 6 and system No. 13 having exchanged their roles are, of course, to be counted as different states - while the same may not be true when two similar atoms within system No. 6 have exchanged their roles; but the latter is merely a question of enumerating correctly the states of the single system, of describing correctly its quantum-mechanical nature.
JOC/EFR August 2007
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