130 BELL SYSTEM TECHNICAL JOURNAL 



equal average energy makes the distributions of the two gases just the 

 same. 



We have already seen that kB is the reciprocal of the temperature: for 

 it is the reciprocal of {dU/dS)v in our statistical picture, and the definition 

 of absolute temperature T is precisely that T is this derivative. The state- 

 ment to which we have come is, that the most probable state of the mixture is 

 the one in which T is the same for both components. It is often expressed in 

 this way : classical statistics shows that for two (or more) gases in equilibrium 

 with each other, the temperature must be the same. It is indeed a fact of 

 experience, and a most important one, that when two systems (be they gases 

 or be they not) are in thermal equilibrium, their temperatures are the same. 

 This has not hitherto been mentioned, and yet we seem to have derived it. 

 Quite a rabbit for the magician of the classical statistics to have pulled out 

 of the hat! 



However, skeptical people who see a rabbit pulled out of a hat are inclined 

 suspect that either the rabbit was in the hat beforehand, or else there is no 

 rabbit. Let us inquire into the contents of the hat and see whether we can 

 find the rabbit there. 



The first (and the last) question to be asked is: what is the difference be- 

 tween "different" kinds of gas in the statistical picture? 



To the physicist or the chemist, different kinds of gas will be (for example) 

 mercury and helium. These differ in their spectra, boiling-points, chemical 

 properties, and quantities of other features. None of these features however 

 appears in the theory, and therefore none of them can contribute to the 

 result. The atoms also differ in mass, and for a moment this seems to be a 

 difference of which the statistical picture takes account, since the letter m 

 appears in some of our equations. However, it appears only in the ultimate 

 equations, those such as (29) in which the distribution-in-momentum is 

 expressed. It does not appear in the original form of the Maxwell-Boltz- 

 mann distribution-in-energy, the form shown in equation (8). It appears in 

 particular in the last term of equation (31), but not elsewhere. Apart from 

 this it may be said that in the classical statistics, all gases are the same gas. 



This is a paradox, but only one of two. The other paradox is, that in 

 the classical statistics two parts of the same gas are different gases. This second 

 paradox arises from the numbering of the molecules, which is an essential 

 feature of the classical statistics. 



Therefore in the statistical picture a mixture of N' atoms of mercury and 

 N" atoms of helium is distinguished by the fact that the mercur}^ atoms bear 

 one set of integer numbers (say those from 1 to N') and the helium atoms 

 another set (say those from iV' + 1 to TV' + N"). But if the atoms were all 

 helium atoms or all mercury atoms, they would also be divisible in many 



