114 BELL SYSTEM TECHNICAL JOURNAL 



The Boltzmann Statistics 



Boltzmann invented a way of appraising the probability of any imagined 

 state of a gas, which has the following very remarkable features: 



(a) It gives so sharp a definition to the key-word "probability", that not 

 only can the state of maximum probability be identified, but the ratio of 

 the probabilities of any two states can be computed. 



{b) For the distribution-in-momentum of the molecules in the most 

 probable state, it derives a formula identical with that which springs from 

 the Maxwell statistics. This of course is why the formula is known as the 

 Maxwell-Boltzmann law. 



(c) For the distribution-in-space of the molecules in the most probable 

 state, it derives the uniform distribution. 



All this does not entail that the Boltzmann statistics is necessarily right. 

 It does, however, lead to consequences, which it is the privilege and the 

 affair of experimental physics to verify or to reject. 



I can now write down a phrase into which the Boltzmann statistics, and 

 equally well those which came later, can be fitted: 



The probability of a state is the number of different ways in which the state can 

 be realized. 



This is another of those oracular sayings which acquire a meaning only 

 after some meaning is given to the key-word, which is this case is ways. I 

 could now rewrite the basic theorem without the word "probability", and 

 so can the reader; but the only effect would be to transfer the mystery out of 

 the word "probability" and into the word "way". Boltzmann, however, 

 assigned a meaning to the latter word. It is this meaning which we now 

 must strive to realize. 



For this purpose I propose a game of which the outfit consists of a sack, 

 an enormous number TV of balls, and a smaller number M of baskets. The 

 game is played by reaching into the sack, drawing out the balls one after 

 another, and tossing them into the baskets. All of the balls feel precisely 

 alike to the hand, so that there is never the least inclination to put one aside 

 and pick up another as one's hand gropes around in the sack. Nevertheless 

 when one looks at the balls after they have fallen into the baskets, one sees 

 that they are nicely adorned with the integer numbers running from 1 to N. 

 Incidentally the baskets also are numbered. It is this numbering which 

 gives point to the game. 



Someone or other — someone who might be designated as the caller, after 

 the man who calls the figures of a square-dance — has prescribed a sequence 

 of M numbers Ni and N-i and Nz and so on to N m , all of them positive 

 integers and totalling up to N. A single inning of the game consists in 

 drawing all of the balls out of the sack one after another, and dropping the 

 first Ni which come out into the basket I, the next N2 which emerge into the 



