The New Statistical Mechanics 



By 



KARL K. DARROW 



THIS is the second article upon statistical mechanics which I have pub- 

 lished 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 statisti- 

 cal 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, 



A definition of statistical mechanics may well be asked for at this point, 

 especially since in the former article I failed to give one. Like many other 

 things either subtle or familiar, statistical mechanics cannot fully be defined 

 till it is fully understood, by which time a definition may seem nugatory. 

 As an attempt at an advance definition, I suggest that statistical mechanics 

 is the theory which, starting frojn the assumption that matter {and, in due course, 

 radiation) is an assemblage of particles, undertakes to explain (1) entropy, (2) 

 temperature, (J) specific heats, and (4) the distribution-in-energy of the particles 

 in thermal equilibrium. The critical reader may justly say that these are 

 four aspects of a single problem, but I think it well to separate them not- 

 withstanding. The word "particle" often has to be construed as standing 

 for an elaborate structure, but in dealing with monatomic gases (and with 

 radiation) we may let it stand for a point endowed with energy and 

 momentum. 



How does the classical statistics succeed in handling these four problems? 

 To take them in reverse order: it does very well with the fourth, for material 

 gases (but not for radiation). It does very well with the third, for mona- 

 tomic gases (but not for polyatomic gases nor for radiation). It produces 

 an adequate theory of temperature for monatomic gases, identifying the 

 temperature with the mean kinetic energy of the atoms multiplied by a 

 certain factor. It has a very strange adventure with entropy, producing 

 a theory which in part is remarkably successful and in part is disconcertingly 



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