AND OF THE SECOND LAW 27 



(2) What is Meant by the Probability of a State ? Example 



To come back to the matter in hand we will now show what 

 is here meant by the probability of any state. 



When we speak of the probability W of a particular " elementar- 

 ungeordnete " state, we thereby imply that this state may be 

 variously realized. For every state (which contains many like 

 independent constituents) corresponds to a certain " distribu- 

 tion," namely, a distribution among the gas molecules of the 

 location co-ordinates and of the velocity components. But such 

 a distribution is a permutation problem, is always an assignment 

 of one set of like elements (co-ordinates, velocity components) 

 to a different set of like elements (molecules). So long as only a 

 particular state is kept in view, it is of consequence as to how 

 many elements of the two sets are thus interchangeably assigned 

 to each other and not at all as to which individual elements of 

 the one set are assigned to particular individual elements of the 

 other set. 1 Then a particular state may be realized by a great 

 number of assignments individually differing from one another, 

 but all equally likely to occur. 2 If with PLANCK we call such an 

 assignment a " complexion," 3 we may now say that in general a 

 particular state contains a large number of different, but equally 

 likely, complexions. This number, i.e., the number of the com- 

 plexions included in a given state can now be defined as the proba- 

 bility W of the state* 



Let us present the matter in still another form. BOLTZMANN 

 derives the expression for magnitude of the probability by at 



1 For an example of such permutations see pp. 28 and 61, 62. 



2 LIOUVILLE'S theorem is the criterion for the equal possibility or equal proba- 

 bility of different state distributions. 



3 A happy term, but one not in vogue among English-speaking physicists. 



4 The identity of entropy with the logarithm of this state of probability W 

 is established by showing that both are equal to the same expression. It seems 

 an easy step from this derivation to BOLTZMANN'S definition of entropy as the 

 "measure of the disorder of the motions in a system of mass points." 



