## Extract

It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation *d*Q = T*d*S. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining *ab initio* a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of *d*Q, deduce from *d*Q the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.

## Footnotes

This text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR.

- Received November 3, 1926.

- Scanned images copyright © 2017, Royal Society