144 MOTION OF SYSTEMS AND ENSEMBLES 



index of probability (77) is a function of the functions men- 

 tioned.* It is therefore a permanent distribution, f and the 

 only permanent distribution consistent with the invariability 

 of the distribution with respect to the functions of phase 

 which are constant in time. 



It would seem, therefore, that we might find a sort of meas- 

 ure of the deviation of an ensemble from statistical equilibrium 

 in the excess of the average index above the minimum which is 

 consistent with the condition of the invariability of the distri- 

 bution with respect to the constant functions of phase. But 

 we have seen that the index of probability is constant in time 

 for each system of the ensemble. The average index is there- 

 fore constant, and we find by this method no approach toward 

 statistical equilibrium in the course of time. 



Yet we must here exercise great caution. One function 

 may approach indefinitely near to another function, while 

 some quantity determined by the first does not approach the 

 corresponding quantity determined by the second. A line 

 joining two points may approach indefinitely near to the 

 straight line joining them, while its length remains constant. 

 We may find a closer analogy with the case under considera- 

 tion in the effect of stirring an incompressible liquid.^ In 

 space of 2 n dimensions the case might be made analyti- 

 cally identical with that of an ensemble of systems of n 

 degrees of freedom, but the analogy is perfect in ordinary- 

 space. Let us suppose the liquid to contain a certain amount 

 of coloring matter which does not affect its hydrodynamic 

 properties. Now the state in which the density of the coloring 

 matter is uniform, i. e., the statt, of perfect mixture, which is 

 a sort of state of equilibrium in this respect that the distribu- 

 tion of the coloring matter in space is not affected by the 

 internal motions of the liquid, is characterized by a minimum 



* See Chapter XI, Theorem IV. 



t See Chapter IV, sub init. 



J By liquid is here meant the continuous body of theoretical hydrody- 

 namics, and not anything of the molecular structure and molecular motions 

 of real liquids. 



