18-22] The Statistical Method 19 



u, v', w and over all elements of spherical surface da> for which a collision is 

 possible. Obviously the quantity obtained in this way will represent the 

 total number of molecules of class A which enter into collision in the interval 

 dt*. So also expression (10) integrated through the same range of values 

 gives the total number of molecules of class A which emerge from collision 

 during the same interval. 



The net gain to class A in the interval dt is therefore the difference of 

 these two integrals, and this is 



ifdudvdwdt I I (ff'-ff'} V<r*cos0du'dv'dw'da> ......... (11), 



J J J J J __ 



in which f,f',f,f' are written for f(u, v, w), f(u', v', w'}, f(u, v, w) and 

 f(u', v', w') respectively. 



21. The number of molecules which belong to class A at the beginning 

 of the interval dt is known to have been vf dudvdw per unit volume, whilst 

 the number at the end of this interval may be supposed to be 



v (f+ 



dt dudvdw. 



The gain to class A is therefore 



?if 



v ~- dudvdw dt. 

 ot 



Equating this quantity to that given by expression (11), we obtain the 

 equation 



a 2 cos0dudv'dw'da> .. ....... (12). 



The condition for a steady state is that df/dt shall vanish for all values of 

 u, v and w. No progress can however be made by equating the right-hand 

 side of equation (12) to zero : the problem of determining the steady state 

 has to be attacked in a different manner. 



The H-theorem. 

 22. Consider the quantity H denned by 



H= Iff flog f dudvdw (13), 



in which the integration is to extend over all possible values of u, v, w, so that 

 77 is a pure quantity and not a function of u, v, w. This quantity depends 

 solely upon the law of distribution of velocities and therefore remains un- 



* Not the total number of collisions in which a molecule initially of class A is involved, since 

 collisions for which both molecules are of class A are counted twice. 



22 



