CHAPTER X. 



ON A DISTRIBUTION IN PHASE CALLED MICROCANONI- 

 CAL IN WHICH ALL THE SYSTEMS HAVE 

 THE SAME ENERGY. 



AN important case of statistical equilibrium is that in which 

 all systems of the ensemble have the same energy. We may 

 arrive at the notion of a distribution which will satisfy the 

 necessary conditions by the following process. We may 

 suppose that an ensemble is distributed with a uniform den- 

 sity-in-phase between two limiting values of the energy, e' and 

 e", and with density zero outside of those limits. Such an 

 ensemble is evidently in statistical equilibrium according to 

 the criterion in Chapter IV, since the density-in-phase may be 

 regarded as a function of the energy. By diminishing the 

 difference of e' and e", we may diminish the differences of 

 energy in the ensemble. The limit of this process gives us 

 a permanent distribution in which the energy is constant. 



We should arrive at the same result, if we should make the 

 density any function of the energy between the limits e' and 

 e", and zero outside of those limits. Thus, the limiting distri- 

 bution obtained from the part of a canonical ensemble 

 between two limits of energy, when the difference of the 

 limiting energies is indefinitely diminished, is independent of 

 the modulus, being determined entirely by the energy, and 

 is identical with the limiting distribution obtained from a 

 uniform density between limits of energy approaching the 

 same value. 



We shall call the limiting distribution at which we arrive 

 by this process microcanonical. 



We shall find however, in certain cases, that for certain 

 values of the energy, viz., for those for which e* is infinite, 



