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Bell System Technical Journal

The New Statistical Mechanics

Karl K. Darrow

Published Oct 1, 1943• 1 citation
Publisher

Abstract

THIS is the second article upon statistical mechanics which I have published this year in this Journal. The first, which appeared in the January (1943) issue, was devoted to the oldest form of the theory, which is variously known as the old, the classical, or the Boltzmann statistics. The word “ statistics,” I repeat from the former article, is a synonym for statistical mechanics, objectionable but (because of the length of the alternative) hardly to be avoided. The “ new statistics,” frequently divided into “ the Bose-Einstein statistics” and “ the Fermi-Dirac statistics,” emerged in the middle twenties and ever since it has been gradually pushing its ancestor aside. In this article I propose to expound the new statistics, laying especial emphasis on the theory of monatomic gases, to which the former article was strictly limited.

DOI
10.1002/j.1538-7305.1943.tb00447.x

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THIS is the second article upon statistical mechanics which I have published this year in this Journal. The first, which appeared in the January (1943) issue, was devoted to the oldest form of the theory, which is variously known as the old, the classical, or the Boltzmann statistics. The word “ statistics,” I repeat from the former article, is a synonym for statistical mechanics, objectionable but (because of the length of the alternative) hardly to be avoided. The “ new statistics,” frequently divided into “ the Bose-Einstein statistics” and “ the Fermi-Dirac statistics,” emerged in the middle twenties and ever since it has been gradually pushing its ancestor aside. In this article I propose to expound the new statistics, laying especial emphasis on the theory of monatomic gases, to which the former article was strictly limited.
Abstract 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 dQ = TdS. 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 dQ, deduce from dQ 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.
Some fundamental changes have been introduced into statistical theory since the publication of a paper by Bose in 1924. The characteristic features of the new statistics, including that of Bose and Einstein, and the later one of Fermi and Dirac, are pointed out in this lecture and compared with the classical statistics. A number of recent applications of the new ideas are described, including the work of Thomas and Fermi on the distribution of electrons in atoms, and the work of Sommerfeld and others on the electron theory of metals. An account is also given of some recent results obtained by the author and H. J. Woods on the distribution of electrons in a metal.
It is widely accepted that the papers by Bose in 1924 and by Einstein in 1924-1925 definitely opened the path to the quantum concept of ‘indistinguishability’. Since then, distinction in physics between ‘identity’ and ‘indistinguishability’ of particles has come to be required. However, even before the rise of quantum physics, Maxwell, Boltzmann and Gibbs, among others, had to tackle the difficulties and paradoxes that appear when counting identical particles. Later, after Sackur’s and Tetrode’s first quantum derivation of the canonical partition function of an ideal gas, much confusion arose around its dependence on the number of molecules in the system. Many other physicists, such as Planck, Lorentz or Schrodinger took part in those debates. In 1921, Ehrenfest and Trkal devoted a whole paper to this subject. Clearly, in the days before Bose’s seminal work, embroiling and misunderstanding were commonplace. In considering the emergence of the distinction between ‘identical’ and ‘indistinguishable’ particles, we will go through the so-called statistical route to quantum mechanics. The first developments in statistical mechanics became immediately confused and entangled with the irruption of quantum physics. In the first quarter of the twentieth century the mutual influence between both disciplines was profuse. We will pay special attention to the story of the so-called ‘Gibbs Paradox.’ We will argue that what is now often known by this name in some textbooks of statistical mechanics did not exist as such in those foundational days.
We derive Bose-Einstein statistics and Fermi-Dirac statistics by Principle of Maximum Entropy applied to two families of entropy functions different from the Boltzmann-Gibbs-Shannon entropy. These entropy functions are identified with special cases of modified Naudts' ϕϕϕ-entropy.
Quantum statistical mechanics is the arithmetic of indistinguishable particles: bosons that pile into a single state and fermions that refuse to share---statistics whose consequences range from the Bose-Einstein condensate, a macroscopic quantum state, to the Fermi gas's pressure that holds white dwarfs against collapse. This article presents a narrative review of the primary literature of that statistics, from Bose's Planck's law and the light quantum hypothesis and Einstein's quantum theory of the monatomic ideal gas with its 1925 second paper predicting the condensation, through Fermi's quantization of the monatomic perfect gas and Dirac's theory of quantum mechanics, which independently founded the Fermi-Dirac statistics, London's identification of the lambda-phenomenon of liquid helium with the Bose-Einstein degeneracy, the experimental vindication of Anderson, Ensher, Matthews, Wieman, and Cornell's observation of Bose-Einstein condensation in a dilute atomic vapor and Davis and colleagues' sodium condensate, DeMarco and Jin's onset of Fermi degeneracy in a trapped atomic gas, and the syntheses of Leggett's fundamental concepts, Pitaevskii and Stringari's Bose-Einstein condensation, and Bloch, Dalibard, and Zwerger's many-body physics with ultracold gases. The synthesis is organized around three themes: the statistics' invention, in which indistinguishability's consequences were computed before any condensate existed; the condensate's realization, in which the dilute atomic vapors delivered Einstein's sixty-year-old prediction; and the many-body era, in which the ultracold gases became quantum simulators of matter's collective behavior. It is concluded that quantum statistics is the theory whose consequences were verified in reverse---predicted abstractly, found in liquid helium, created in atomic vapors---and that its bosons and fermions remain the universe's two grammars.
The published work for which the honor of the Nobel prize for the year 1954 has been accorded to me does not contain the discovery of a new phenomenon of nature but, rather, the foundations of a new way of thinking about the phenomena of nature. This way of thinking has permeated experimental and theoretical physics to such an extent that it seems scarcely possible to say anything more about it that has not often been said already. Yet there are some special aspects that I should like to discuss.