RECENT STATISTICAL THEORIES 685 



We proceed to apply this method to the swarm of points in mo- 

 mentum-space representing the assemblage. 



Like the coordinate-space, the momentum-space is to be divided 

 into equal compartments large enough to contain each a multitude 

 of particles. We are to define a distribution by specifying how many 

 particles are in each compartment, and calculate as before the number 

 W which is to measure the "probability" of the distribution. The 

 values for W, for log W and for the variation of log W are obtained 

 just as before. There is however an important novelty. Since the 

 energy of a particle depends on its position in momentum-space, 

 different distributions usually entail different values for the total 

 energy of the assemblage. If we compute the variation of log W 

 due to a slight change in distribution, we shall usually be computing 

 a variation in log W correlated with a certain variation of the total 

 energy U of the assemblage. 



We now take the very great step of identifying the quantity log W 

 with entropy. 



More precisely, we assume that the entropy 5 is proportional to 

 the logarithm of W: 



5 = y^ log W, (13) 



introducing a constant factor k, and relying on subsequent experiments 

 to teach us its numerical value. 



Now when a gas being initially in thermal equilibrium at temperature 

 T receives an infinitesimal amount of energy dE, and regains thermal 

 equilibrium with its augmented energy, its entropy ascends by the 

 amount of dS given by the equation: 



dSjdE = l/T. (14) 



If then the foregoing model of the gas and the foregoing picture of 

 entropy are justified, the variation of log W in passing from the 

 most probable distribution consonant with a total energy E to the 

 most probable distribution consonant with a total energy E + dE 

 (the total number of particles remaining the same) must be equal 

 to {l/kT)dE. 



If we start from the most probable distribution for energy E and 

 make any slight change in it involving an energy-change dE, the 

 new distribution will presumably differ but little from the most 

 probable distribution for E + dE. We therefore say: the most 

 probable distribution for energy E is the one of which the first variation 

 is dE/kT. This expression vanishes, if we are comparing distributions 

 for which E is the same; which is as it should be. 



