﻿834 Messrs. G. G. Darwin and R. H. Fowler on 

 § 6. " The increasing property" of Entropy. 



We have now obtained a quantity S st ., the statistical 

 entropy, which is evidently related to the entropy of thermo- 

 dynamics Sth.j hut we must examine what right we have to 

 make the identification complete. By its definition (5*1), 

 S st . has the additive property for separation, and we can 

 easily show that for junction it has the property Sj •+■ S 2 < S 12 ,, 

 which may be called the increasing property. 



Consider the special case of two assemblies, and suppose 

 that in their junction only changes of temperature are con- 

 cerned — not of volume or any other parameter. We shall 

 also simplify by supposing that in each assembly there is 

 only one type of system, different for the two. Ks we do- 

 not intend to base our final result on the present paragraph, 

 this will be general enough. By definition, for the first 

 assembly before junction the entropy is given by 



Srt.7* = M' \og m f(s')-w log y. 



Now when the energy E' is given, the temperature 3' is 

 determined by (2*4), and this is equivalent exactly to the 

 condition that S s t/ should be a minimum for given E'. So,. 

 if 3 has any value different from 3', 



Set//* < M' log/ (3) -E' log $. 



Similarly, 



J S st / 7&<M'Mog /''(>) -E'' logS, 



if 3 is different from y, the temperature of the second, 

 assembly. It follows that unless 3' =.&" = S we have 



(S st .' + S st .")/£ < M' log/" (a) + M" log/"(S) -E log d, 



where E' + E" = E, the energy of the joint assembly. Now 

 with a suitable choice of 3 this is S s t./&, where S s t. is the- 

 entropy of the joint assembly after combination ; so we 

 have proved that (when no volume or other such changes 

 take place) 



S st .' + S st ."<S st ., (E'+E"=E), 



unless the initial temperatures are equal, in which case 



Sst/ + Sst. == &st.- 



Thus statistical entropy has the increasing property. 



It is often taken for granted that if we can find a function 

 of the state which has the increasing property, then that 

 f unction must be the entropy: this assertion tacitly underlies 

 Boltzmann's hypothesis. But the identification of S s t, with, 

 Sth., the entropy of thermodynamics, cannot be established in 



