700 BELL SYSTEM TECHNICAL JOURNAL 



number of particles N by the already so familiar way of integrating 

 F{es) over the entire energy-range from to <» and setting the integral 

 equal to N. 



Einstein proposed this as an alternative to the Maxwell-Boltzmann 

 law derived from the classical statistics. It is not easy to decide 

 which of the two is supported by experiment, as with increasing 

 temperature the formula (57) becomes more and more nearly like the 

 classical one, and it turns out that throughout the convenient ranges 

 of temperature and pressure the two are indistinguishable. It would 

 be very valuable to determine between the two, as then we should 

 know which of the two ways of defining a distribution and estimating 

 the probability thereof, which of the two pictures of entropy, is the 

 proper one for a material gas. The reader may have remarked that 

 if one were to apply Bose's method to the problem of determining the 

 most probable distribution of particles in ordinary space, one would 

 reach a result at variance with that of the classical statistics, and 

 therefore at variance with intuition. One must deal altogether with 

 the six dimensional phase-space, to be perfectly consistent. This is 

 to be regretted. 



The Fermi Statistics 



The statistics invented by Fermi, and later independently by 

 Dirac, involves the same fundamental assumptions as that of Bose — 

 the same manner of counting the ways in which a distribution may 

 be realized, of defining its probability, of picturing its entropy. But 

 there is an additional assumption, of the nature of a limitation: it is 

 postulated, that a compartment may contain not more than some 

 specific maximum number of particles. In particular for a gas to 

 which no external field is applied, it is postulated that each compart- 

 ment must either be empty, or else contain one particle only. 



The "exclusion-principle of Pauli" gave the hint from which the 

 Fermi-Dirac theory sprang. This principle may be paraphrased as 

 follows. In Bohr's "atomic theory of the atom" the electrons be- 

 longing to an atom are forbidden to revolve in any except certain 

 specific orbits, set apart from the rest as the "permitted" orbits, 

 and labelled by specific "quantum-numbers." In later versions of 

 the theory the "permitted orbits" are less conspicuous, the "permitted 

 quantum-numbers" more so; but the picture is acceptable at all 

 events as a beginning. Upon this prohibition, then, Pauli superposed 

 another; not more than one electron is allowed in each orbit or to 

 each set of quantum-numbers. Perhaps it would be better to say 

 "not more than some definite small number of electrons ..." 



