THROUGH LONG PERIODS OF TIME. 143 



Let us next consider whether an ensemble of isolated 

 systems has any tendency in the course of time toward a 

 state of statistical equilibrium. 



There are certain functions of phase which are constant in 

 time. The distribution of the ensemble with respect to the 

 values of these functions is necessarily invariable, that is, 

 the number of systems within any limits which can be 

 specified in terms of these functions cannot vary in the course 

 of time. The distribution in phase which without violating 

 this condition gives the least value of the average index of 

 probability of phase (77) is unique, and is that in which the 



in various ways describe an ensemble in statistical equilibrium, while the 

 same language applied to the critical value of the energy would fail to do 

 so. Thus, if we should say that the ensemble is so distributed that the 

 probability that a system is in any given part of the circle is proportioned 

 to the time which a single system spends in that part, motion in either direc- 

 tion being equally probable, we should perfectly define a distribution in sta- 

 tistical equilibrium for any value of the energy except the critical value 

 mentioned above, but for this value of the energy all the probabilities in 

 question would vanish unless the highest point is included in the part of the 

 circle considered, in which case the probability is unity, or forms one of its 

 limits, in which case the probability is indeterminate. Compare the foot-note 

 on page 118. 



A still more simple example is afforded by the uniform motion of a 

 material point in a straight line. Here the impossibility of statistical equi- 

 librium is not limited to any particular energy, and the canonical distribu- 

 tion as well as the microcanonical is impossible. 



These examples are mentioned here in order to show the necessity of 

 caution in the application of the above principle, with respect to the question 

 whether we have to do with a true case of statistical equilibrium. 



Another point in respect to which caution must be exercised is that the 

 part of an ensemble of which the theorem of the return of systems is asserted 

 should be entirely denned by limits within which it is contained, and not by 

 any such condition as that a certain function of phase shall have a given 

 value. This is necessary in order that the part of the ensemble which is 

 considered should be any assignable fraction of the whole. Thus, if we have 

 a canonical ensemble consisting of material points in vertical circles, the 

 theorem of the return of systems may be applied to a part of the ensemble 

 defined as cqntained in a given part of the circle. But it may not be applied 

 in all cases to a part of the ensemble defined as contained in a given part 

 of the circle and having a given energy. It would, in fact, express the exact 

 opposite of the truth when the given energy is the critical value mentioned 

 above. 



