( 303 ) 



Physics. — "On the kinetic derivation of the second Laiv of Thermo- 

 dynamics." By Dr. 0. Postma. (Communicated by Prof. H. A. 



Lorentz.) 



§ 1. In a previous paper 1 ) I tried to set forth how an ensemble 

 of molecular systems possessing- only kinetic energy may bethought 

 gradually to pass to a state in which all the combinations of place 

 and also all the combinations of velocity of the molecules occur 

 with the same frequency. In this final condition the molecules of 

 by far the majority of the systems of the ensemble will, as was 

 shown, be distributed about uniformly over the vessel and have 

 Maxwell's distribution of velocities. 



This result, however, requires some amplification. 



As the problem of the distribution of place and that of the distri- 

 bution of velocity were treated quite separately, the above-mentioned 

 result implies only, that in the end the molecules will be distributed 

 uniformly over the vessel for <tll the velocities together, and that they 

 will have Maxwell's distribution of velocities f 'or the vessel considered 

 as a whole. This, however, is not what is generally understood by 

 uniform distribution over the vessel and Maxwell's distribution of 

 velocities ; we mean by this that even for a limited amount of velocities 

 the molecules will be spread about uniformly over the vessel, and 

 that even for a limited portion of the vessel Maxwell's distribution 

 of velocities will hold on the main. So the question remains, how 

 this result may be obtained. 



Let us first observe that in a canonical or microcanonical ensemble 

 the uniform distribution of place and Maxwell's distribution of velo- 

 cities in the latter sense is really obtained. This is very easily seen 

 for the canonical ensemble. It is, however, also the case for a 

 microcanonical ensemble, where the frequency of a certain distribu- 

 tion of place and velocity is proportional to the number of com- 

 binations possible. This number of combinations may be given 



— I 1 f log fdodoi 

 Ce J J 



in the form Ce J J just as it is given in the form 



— \fl°gfdw 

 Ce J when only the distribution of the velocities is con- 



sidered. So in the most frequently occurring system I iflogfdodu) 

 or - II is maximum. With given kinetic energy I his is the case, if 



l) These PrOC X, p. 3yO. 



