150 MOTION OF SYSTEMS AND ENSEMBLES 



systems in JDF", for example, by the total number of systems, 

 and also by the extension-in-phase of the element, and take 

 the logarithm of the quotient, we would get a number which 

 would be less than the average value of rj for the systems 

 within D V" based on the distribution in phase at the time t 1 .* 

 Hence the average value of 77 for the whole ensemble of 

 systems based on the distribution at t" will be less than the 

 average value based on the distribution at t'. 



We must not forget that there are exceptions to this gen- 

 eral rule. These exceptions are in cases in which the laws 

 of motion are such that systems having small differences 

 of phase will continue always to have small differences of 

 phase. 



It is to be observed that if the average index of probability in 

 an ensemble may be said in some sense to have a less value at 

 one tune than at another, it is not necessarily priority in tune 

 which determines the greater average index. If a distribution, 

 which is not one of statistical equilibrium, should be given 

 for a time ', and the distribution at an earlier time t" should 

 be defined as that given by the corresponding phases, if we 

 increase the interval leaving t' fixed and taking t tf at an earlier 

 and earlier date, the distribution at t" will in general approach 

 a limiting distribution which is in statistical equilibrium. The 

 determining difference in such cases is that between a definite 

 distribution at a definite time and the limit of a varying dis- 

 tribution when the moment considered is carried either forward 

 or backward indefinitely, f 



But while the distinction of prior and subsequent events 

 may be immaterial with respect to mathematical fictions, it is 

 quite otherwise with respect to the events of the real world. 

 It should not be forgotten, when our ensembles are chosen to 

 illustrate the probabilities of events in the real world, that 



* See Chapter XI, Theorem IX. 



t One may compare the kinematical truism that when two points are 

 moving with uniform velocities, (with the single exception of the case where 

 the relative motion is zero,) their mutual distance at any definite time is less 

 than f or t = <x> , or t = oo . 



