102-104] Maxwells Treatment of Equipartition 91 



Hence the solution becomes 



(221), 



where E,E'... are the energies of the separate molecules, and W is the 

 potential energy of the intermolecular forces acting between molecules, this 

 not now being included in E and E '. 



This solution agrees with that of 85. 



v 104. We have therefore seen that Maxwell's treatment of equipartition 

 can be made to lead to the right result if we make the assumption of 

 molecular chaos, or, more accurately, if we assume that the number of 

 collisions of a given kind is that given by expression (208). It is infinitely 

 probable, but not certain, that this expression will be accurate, so that it is 

 infinitely probable, but not certain, that (221) will be the solution in a steady 

 state. This is exactly the result arrived at before. 



It is of interest to notice that it could have been predicted a priori, that/ 

 it would be necessary to supplement Maxwell's treatment by some assumption 

 of this kind. 



This, as we shall now see, follows from the fact that the problem is 

 a "statistical" problem, and not a dynamical problem of the ordinary type. 

 A dynamical problem may, in accordance with accepted usage, be said to be 

 one of statistical mechanics when the data and objects of inquiry are not the 

 actual values of the various coordinates, but the law of distribution of these 

 coordinates. Since the data of a problem in statistical mechanics do not 

 completely specify the dynamical coordinates of the system, we are, in 

 a problem of statistical mechanics, discussing an infinite number of different 

 systems at once, and without differentiation inter se. The motion of these 

 systems will naturally diverge in the course of time. It may be that after 

 the motion a single statistical specification can be given which covers all 

 except an infinitesimal fraction of the systems. In this case a solution may 

 be said to have been found to the problem. It cannot be that a solution can 

 be obtained which covers all the systems, the reason for this being that, even 

 after the initial system has been fully specified statistically, there are still an 

 infinite number of undetermined variables; and, by giving suitable values to 

 these, we can obtain any chosen infinite number of relations between the 

 coordinates of the final result, and can therefore cause this final result to 

 disagree with any single statistical specification. It is therefore clear that 

 a statistical problem must always have an element of uncertainty in its final -' 

 solution, although in virtue of the infinite number of the variables, this 



