680 BELL SYSTEM TECHNICAL JOURNAL 



that one accepts without demur the conclusions that in practice it 

 will be stable. But there is no proof that the "most probable" 

 distribution is always or even usually followed by another exactly 

 like it, nor that an "improbable" distribution is always or even 

 usually followed by another of greater probability; there is no study 

 of the way in which one distribution is transformed into another, 

 there are no assumptions about the collisions or encounters which 

 presumably offer to the particles their means of interchanging speed 

 and energy, and to the assemblage its means of approaching the 

 stable distribution. There are other statistical methods in which 

 account is taken of these things, and we shall have a glimpse of one 

 of them in the last section of this paper; but the notion of causality 

 is absent from the method which will be followed in deriving the 

 distribution-laws of Maxwell and Boltzmann, of Bose, and of Fermi 

 and Dirac. 



These three distribution-laws will be applied to freely-flying particles 

 in regions which are either field-free, or else pervaded by a field 

 (electrostatic or gravitational) derivable from a potential. It may 

 surprise the reader to hear so little about oscillators, considering that 

 the statistics which Planck applied to these objects was the first of 

 all the modifications of the classical statistics, was the source of the 

 entire quantum-theory, and therefore the most important advance 

 of the physics of the last quarter-century. The history of this period 

 is very curious; but I cannot mention more than a couple of the 

 salient points. 



The Planckian oscillators served two purposes: they enabled Planck 

 to derive the law of distribution of radiant energy at uniform temper- 

 ature in a cavity, by supposing the radiation to be entirely wavelike 

 and to be in equilibrium with myriads of oscillators in the walls of 

 the cavity; and they enabled various savants to develop, step by 

 step, a progressively improving theory of the specific heat of solids. 

 The Bose statistics made them quite superfluous for the first purpose: 

 by applying this statistics to the radiation supposed to consist of 

 corpuscles, we can derive the same law of distribution without invoking 

 the oscillators at all. As for the second: as early as 1912 (which 

 seems remarkable, now) Debye had replaced the concept of a solid 

 as a latticework of vibrating atoms by the concept of a solid as a 

 system of stationary waves agitating a continuum. I do not mean 

 to imply, of course, that the existence of the atoms was denied; I 

 mean no more than to say, that in these statistical reasonings the 

 individual vibrating atom was replaced by an individual pattern of 

 stationary waves. Today we are becoming familiar with the idea 



