CHAPTER XII. 



ON THE MOTION OF SYSTEMS AND ENSEMBLES OF SYS- 

 TEMS THROUGH LONG PERIODS OF TIME. 



AN important question which suggests itself in regard to any 

 case of dynamical motion is whether the system considered 

 will return in the course of time to its initial phase, or, if it 

 will not return exactly to that phase, whether it will do so to 

 any required degree of approximation in the course of a suffi- 

 ciently long time. To be able to give even a partial answer 

 to such questions, we must know something in regard to the 

 dynamical nature of the system. In the following theorem, 

 the only assumption in this respect is such as we have found 

 necessary for the existence of the canonical distribution. 



If we imagine an ensemble of identical systems to be 

 distributed with a uniform density throughout any finite 

 extension-in-phase, the number of the systems which leave 

 the extension-in-phase and will not return to it in the course 

 of time is less than any assignable fraction of the whole 

 number; provided, that the total extension-in-phase for the 

 systems considered between two limiting values of the energy 

 is finite, these limiting values being less and greater respec- 

 tively than any of the energies of the first-mentioned exten- 

 sion-in-phase. 



To prove this, we observe that at the moment which we 

 call initial the systems occupy the given extension-in-phase. 

 It is evident that some systems must leave the extension 

 immediately, unless all remain in it forever. Those systems 

 which leave the extension at the first instant, we shall call 

 the front of the ensemble. It will be convenient to speak of 

 this front as generating the extension-in-phase through which it 

 passes in the course of time, as in geometry a surface is said to 



