MEMORIAL TO CLASSICAL STATISTICS 113 



which serves just as well for all of its purposes and even better for some. Let 

 the velocity-components v^ , Vy , and Vz be laid out along the axes of a Cartesian 

 coordinate-frame, and the vector for any molecule be drawn from the origin: 

 the point at its tip is the point in question. Point and coordinate-frame are 

 said to be "in velocity-space". Statistical mechanics prefers as a rule to 

 deal with the momenta of the molecules rather than their velocities. This 

 is for valid and powerful reasons, one of which is that the transition to the 

 case of photons becomes much easier.^ In the case of material gases it 

 makes no practical difference, since the momentum of a molecule is its 

 velocity-vector multiplied by the mass of the molecule which is practically a 

 constant, and every statement about the distribution-in-velocity can with 

 the utmost ease be translated into a statement about the distribution-in- 

 momentum and vice versa. The momentum-vector may be replaced by the 

 point at its tip, having coordinates p^c , p,j and pz in a coordinate-frame in 

 "momentum-space". If we consider together with these the three co- 

 ordinates X, y, z of the molecule in ordinary space, we may say that we are 

 locating the molecule in six-dimensional space. I have yet to meet someone 

 who claims that he can visualize a six-dimensional space, and yet there is no 

 doubt that the phrase fulfills a psychological need and has a practical value. 

 The six-dimensional space of these particular six variables is called "the 

 /Li-space". 



It seems odd to bring in the /x-space before considering by itself the three- 

 dimensional "ordinary" or "coordinate-space" in which the gas is located. 

 Is there nothing to be said about the most probable distribution of the 

 molecules in the coordinate-space? Well, "every schoolboy knows" that 

 the state to which a gas tends and in which it remains is a state of uniform 

 density. Maxwell, I think, accepted this as one of the facts behind which 

 one cannot, or does not, go. For a complete statement of the Maxwell 

 statistics I therefore offer the following: 



A gas is very much more likely to be in its "most probable state" than in any 

 other. The most probable state is that in which isotropy and independence pre- 

 vail among the momentum-vectors, ivhile the distribution in coordinate-space is 

 uniform. 



So in the Maxwell statistics the distribution-in-momentum of the mole- 

 cules is derived from assumptions ostensibly more basic, while the distri- 

 bution-in-ordinary-space is simply affirmed. If a theory could be devised in 

 which both were derived from assumptions apparently more basic, one would 

 be likely to feel that something had been gained. Now this is a char- 

 acteristic, and one of the principal virtues, of Boltzmann's theory known as 

 the "Boltzmann Statistics" or as the "Classical Statistics". 



1 Another reason has to do with "Liouville's theorem," for which unfortunately I 

 cannot make room without overloading this article. 



