66-69] The H-theorem analysed 59 



Analysis of the H-theorem. 



67. The assumption of molecular chaos (corrected, if necessary, in 

 accordance with 66) will therefore give correct results, provided it is 

 interpreted with reference to the basis of probability supplied by our 

 generalised space, and provided it is understood that it gives probable, 

 and not certain, results. If we wish to obtain strictly accurate results the 

 quantities calculated from it must not be regarded as applying to a single 

 system, but must be supposed to be averaged over all the systems in the 

 generalised space which satisfy certain conditions. For instance, the value 



J}/ 1 )/" 



of ^ given by equation (12) is merely the value of ~ averaged throughout 



all those parts of the space for which the system has a given f. So also we 



O TT 



must interpret the value of -^- given by equation (20) as an average value 



r> ff 



for ^ , taken over all systems in our space which have a given value for H. 



68. We now come to what is, at first sight, a paradox. Let us suppose 

 that f is an even function of u, v, w different from the normal function 

 Ae- hm ^+* 2 +^. Then from Chapter II. ( 23) it follows that dH/dt is 

 negative, dH/dt indicating, as we have just seen, the value of dfl/dt averaged 

 over all the systems in our space which have this given f. Now these 

 systems may be divided up into pairs. Corresponding to any system there 

 will be a second system of which the positional coordinates will be the same 

 as those of the first system and of which the velocity coordinates will be the 

 same in magnitude but opposite in sign. Since f is an even function of the 

 velocity coordinates, the value of/ will be the same for each of these two 

 systems and both systems are equally to be included in the average of dH/dt. 

 But the motion of the first system is exactly the reverse of that of the second 

 system. It would therefore appear as though the values of dH/dt must be 

 equal and opposite for the two systems, so that the average of dH/dt for 

 these two must be zero. Since the whole of the systems corresponding to a 

 given f fall into pairs of this type, it might be inferred that the average 

 value of dH/dt must be zero. 



69. The explanation of the apparent paradox will be found to be 

 contained in the result of 52, if we substitute H for K. By a proof 

 similar to that which led to the result in the case of the function K, we can 

 prove the result for the function H. We therefore see that, of the systems 

 for which H has a value greater than some value H l other than the minimum 

 value for H, all except an infinitesimal fraction have a value for H which 

 only differs infinitesimally from H^. If, therefore, we select at random a 

 point at which the value of H is H ly it is infinitely probable that H will 



