RECENT STATISTICAL THEORIES 679 



by 5' is increasing; in certain simple cases we can evaluate this rate 

 of change of entropy. We know that when a gas is in its stable 

 condition, its entropy is at a maximum; we know how to compute 

 the entropy (except perhaps for an additive constant) of a given 

 quantity of a gas in this condition, as a function of its temperature 

 and others of its measurable properties. And when we have evaluated 

 both the entropy 6" and the energy £ of a gas under any specific 

 conditions, we know that its absolute temperature is determined by 

 the following equation, 



dS/dE = 1/T, (2) 



which is the definition of absolute temperature. 



If we had obtained by some independent way an adequate atomic 

 picture of entropy, so that whenever a distribution-function was 

 suggested we could compute the value of S: then necessarily the 

 stable distribution would be the one for which 5" has the greatest 

 value compatible with the given number of particles and the given 

 amount of energy. We do not have an independent way. But if 

 instead we adopt some tentative atomic picture of entropy, some 

 function S of which we can compute the value for any given distribu- 

 tion: then the test of our picture will be, whether the distribution 

 for which this tentative 5' has its greatest value is verified by experi- 

 ment to be the stable one. It will be found that this distribution 

 "of maximum 6"' involves the derivative dSjdE, and therefore the 

 absolute temperature; so the temperature enters into the postulated 

 distribution-function in the course of its derivation, not by separate 

 assumption or by an afterthought. 



This method is the very notable one invented by Boltzmann, and 

 continued by Planck. One choice of the function 5 which is to be 

 identified with entropy leads to the classical or Maxwell-Boltzmann 

 distribution-law; another leads either to the Bose or to the Fermi 

 distribution, the difference between these two entering in at another 

 point. 



Each of these suggested functions is logarithmic; it is proportional 

 to the logarithm of a function which is called probability. In theo- 

 retical physics it is a fairly general rule, that when a theorist introduces 

 the word probability he is abandoning all hope of explaining by cause- 

 and-effect the phenomena of which he is discoursing. This is the 

 disadvantage of Boltzmann's method. The "distribution of maximum 

 6"' is baptized "the most probable distribution"; there is even a 

 numerical estimate of its "probability," and in general it turns out 

 to have so much greater a probability than all the others put together 



