Functions of the state distribution

Functions of the state distribution. If we displace the phase point G continuously every time the G-point passes from one Z-star to another, the set of integers a,- changes in a discontinuous way. Hence every function of the numbers o,—or function F(Z) of the distribution of state—if considered as a function of the phase (q, p), will be a discontinuous phase function. 

These functions are characteristic tools of the kinetic theory. 

All the F(Z.ys which one uses in the kinetic theory of gases to approximate the exact values of continuous phase functions are also in this category. Let us denote by i the kinetic energy of a molecule when its phase point is exactly at the center of cell in the p-space. Since very small a cells are chosen, the total kinetic energy of all a molecules whose phase point lies in the cell au can be approximated very well by 

Hence the exact value of the kinetic energy of the whole gas model, L(q, p)—a continuous phase function—can be approximated by the F(Z) s  

One can then easily verify the following statement, which will be used in Section 13  

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