( 306 ) 



regions extend over the total amount of' possible combinations of 



place, but the approach to uniformity is also found over finite small 

 portions of this amount, (the regions b , What happens now if the 

 two actions take place simultaneously? In the collision the represent- 

 ing points shift in horizontal direction, which modifies] the distribution 

 in the regions a, and the first action, which would make the distribution 

 over the regions a uniform, is counteracted. This continues to be the 

 case, as we think the regions a infinitely narrow ; if however, we 

 consider an element from the figure, the horizontal dimension of 

 which is indicated by a, and the vertical dimension by b (the rect- 

 angle A), then the distribution in the elements .1 lying one above 

 the other will also approach to uniformity by the first action whereas 

 the disturbing influence with which the second action counteracts the 

 first, will continually decrease and approach to zero. So it seems to 

 me that we may assume that the mean density in the elements A 

 lying above each other becomes the same in course of time; this 

 reasoning will, however, also hold for the elements lying side by 

 side in horizontal direction (if we take now the second action as the 

 principal, one, and the first as the disturbing action), so that we get 

 a "rough" microcanonical ensemble in the end 1 ). 



If the above reasoning i^ correct, we have obtained the result that 

 every arbitrary ensemble of molecule systems with purely kinetic 

 energy proceeds towards a state where uniform distribution of place 

 and Maxwell's distribution of velocities i> most frequent. In the 

 meantime we must assume that every system in itself has a reversible 

 motion and so after some time it will get again very near to its 

 initial state, and will do so repeatedly. Whether Boi.tzmannV 



// — j \ f log f do dm will decrease for the majority of the cases 



depends on the initial state of the ensemble. It is conceivable that 

 this state is such that the majority of the systems are nearer to the 

 state occurring finally maximum than is the case for a micro-cano- 

 nical ensemble; then the // would increase instead of decrease for 

 those systems. It is, however, evident,, that this will not be the case 

 for an ensemble that represents a system in which recently some 

 disturbance of' equilibrium has taken place. For such a system // 

 will most probably decrease. 



l ) It would be doing Gibbs an injustice if we did not admit, that in his Statistical 

 Mechanics he already pointed to this remaining constant öf the entropie fine, in 

 opposition to the decrease of the entropie grossière when he says treating the 

 analogy of the coloured liquid: "If treating the elements of volume as constant .. . 



etc." p. 145. 



