( 307 ) 



§ 2. Now (lie question may be raised what place we have to 



assign to Boltzmann's proof that H= \ If log/ do dm would decrease 



for an "ungeordnetes" system with regard to the above reasoning 

 and the one preceding it. li seems to have to decrease for such a 

 system, which as ZERMELO and others pointed out can hardly be 

 always the case for one definite system, and can only be assumed 

 as occurring in general for an ensemble. 



In the first place we must observe with regard to this that by no 

 means certainly prevails that a system which is in an "ungeordnet" 

 state at a certain moment, will continue to be in such a state. 



In the second place the properties of such a system, as BoLTZMANN 

 applies them in the derivation of the variation of the function H in 

 consequence of the collisions (see form. 17 and 1 05;, can occur for 

 one definite system with a sufficient degree of accuracy only when 

 the elements (/a>, <// etc. occurring- in these formulae, are taken rather 

 large. There are, however, also objections to this (as that it is assumed 

 that in a collision of two molecules the velocity-points always get 

 outside the elements dm and <ho x ). 



Independent of the size of the elements the property of being "unge- 

 ordnet" cannot occur for one definite system, it can, however, for 

 the average of a whole ensemble (or in course of lime if we think 

 the systems taken at random from a certain ensemble). 



This ensemble is formed by the whole of the possible systems 

 obtained if we think the places and velocities of the n molecules 

 assigned to them by chance, so that every time the chance to a 

 certain combination of place and velocity is represented by a constant 

 function ƒ of the coordinates and velocities. If n is large, the majority 

 of these systems have a distribution of place and velocity the course 

 of which is mainly indicated by the function f. It does not hold 

 exactly for any definite system that in the neighbourhood of every 

 molecule the number of molecules of a certain kind are determined 

 by the size of the spacial element considered and the / holding 

 there, but on an average it does hold for the whole ensemble. So 

 we may say that this ensemble represents Boltzmann's "ungeordnetes" 



system. On an average would, therefore, be negative for this 



dt 



ensemble. 



<)n further comparison of Boltzmann's way of treatment and the 

 results of § J we meet with an important point of difference. On 

 the whole 11 will decrease for the majority of the systems for an 

 arbitrary ensemble on account of the tendency towards uniformity 



