( 310 ) 



was shown by Lorkntz V this quantity lias the property thai in 



dQ 



reversible changes of the system the differential is equal to — , in 



which T, the modulus of the ensemble, has the properties of the 

 temperature. This entropy was only defined by Gibbs for the state 

 of equilibrium. When, however, we represent a gas which is not in 

 equilibrium by a non-canonical ensemble, and detine the entropy in 

 the same way, also the second property will hold for this entropy ; 



the quantity I P log Pdx will namely gradually decrease if the elements 



dx are not taken infinitely small, because each portion of the ensemble 

 with given energy approaches to a rough micro-canonical one. In 

 the special case considered in thermodynamics that the parts of the 

 system are in equilibrium this will also be the case '). 

 Calculating this — i} for a perfect gas, we find, as 



e — xp 



-•'= f 



whereas 



s —— NT and = -A' log (InmT) 4- N log v, 



2 7 2 



3 3 



— tj = - N -f - A' log (InmT) -f N log v . 



When comparing this value with Boltzmann's entropy we must 

 bear in mind that this 7' does not agree perfectly with the i> of 



BOLTZMANN : viz. : 



mean kin. energy per mol. , mean kin. energy per system 



ih — ' ' and T = - - ' • . 



u 3 



A r 

 2 



■j 

 from this follows T= — fi X mean a taken over the ensemble). 



So for comparison we must take: 



') See Abhandlung XI. 



-i The objection advanced by Lorentz to this way of defining the entropy, that 

 it would namely be difficult to understand how a non-canonical ensemble should 

 be determined by a system that is not in equilibrium, does not seem to be con- 

 clusive to me. It is true, that the entropy and the ensemble are not determined 

 in the same way as for a stationary system, but as we know more about the 

 place and the velocities of the molecules or the way in which they have assumed 

 their places and velocities, the ensemble is determined more accurately. If we e.g. 

 know that everywhere a certain pressure and temperature prevails, we consider 

 the ensemble as a sum of canonical ensembles, etc. 



