12 Dr. W. F. G. Swann on Equipartition of 



energy. Even though the theorem itself is admitted, how- 

 ever, considerable care is necessary in its application, and in 

 what follows, without questioning the truth of the theorem, 

 even for ordinary matter, we shall inquire whether some of 

 the difficulties associated with its apparent consequences are 

 not due to its erroneous application. 



Equipartition of energy betiveen electrons inside and 

 outside the atoms. 



It is frequently stated that the theorem leads to the con- 

 clusion that there should be equipartition of energy between 

 the electrons inside the atoms, and the free electrons outside. 

 Admitting for the moment that the ordinary coordinates and 

 momenta of the electrons are suitable generalized coordinates 

 in terms of which to express the system, it is to be noted that 

 the theorem tells us that it is infinitely probable that if we 

 take the average energy of any p coordinates at any instant, 

 the result will be the same as that obtained by taking the 

 average energy of any other group containing say q co- 

 ordinates, p and q being large numbers. Now it would 

 obviously not be fair, if, before picking out our p co- 

 ordinates, we asked to be told the energy of each coordinate 

 at the instant and then put only the coordinates of high 

 energy in the p group and only the coordinates of low 

 energy in the q group ; yet, when we discuss the relation 

 between the average energy of the electrons inside, and the 

 average energy of the electrons outside, are we not implicitly 

 grouping the coordinates in a manner very largely deter- 

 mined by their magnitudes, or at any rate by relations in 

 which these magnitudes play a fundamental part ? Are we 

 not, in fact, grouping them in a manner of which the above 

 example is a crude illustration? It is true that the theorem 

 of equipartition of energy places no restriction on the method 

 of grouping the coordinates, but in whatever way we. group 

 the electrons, if for example we place in one group only the 

 electrons which at the particular instant concerned are to be 

 found inside atoms and in the other group only electrons 

 which are to be found outside atoms, we must not be content 

 to merely compare the average energies of the electrons at 

 the instant when we grouped them. We must fix our minds 

 on the identical electrons which we have chosen, and follow 

 them over a very long period of time. During this period, the 

 various electrons which were all inside atoms originally will 

 have opportunities of coming out and re-entering atoms several 

 times, so that for the greater portion of the time the two groups 



