An Introduction to the Statistical Theory of Classical by G.H. A. Cole

By G.H. A. Cole

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It is this minimum value that is located at a distance from the centre of the particle which is associated with the particle diameter of the simple kinetic theory of gases. At distances beyond this minimum the potential remains negative but approaching zero, and with its gradient always remaining positive. e. for particle spherical symmetry, ip is essentially vanishing at a distance of three or four particle diameters. It is not easy to translate this qualitative data into a quantitative form for specific cases, especially if the form is to have some mathematical simplicity.

DG . 10). Let us now average this expression over some time interval r, taking the limit when x becomes indefinitely great. 14) Qàt\. 15) we obtain the general relation, This relation is known as the general virial theorem. The force F· does not need to be conservative or derivable from a potential function. 8a). g. due to the boundary walls). 16) depends upon the exact form of G. The best known is that given by Clausius and follows by setting G = £(prr3). 16) gives - Σ <*·**> = Z S · (2 19) · The left hand side is the virial of Clausius and is the average of the product of minus a force times a distance: the right hand side is the average of twice the total particle kinetic energy.

Suppose Q(vN, rN) is some phase function for a system of N interacting particles which does not explicitly involve the time and which always 37 MICROSCOPIC REPRESENTATION remains finite throughout the motion. 29). 4. The left hand side of this expression vanishes because Q is not explicitly a function of time and all phase is to be encompassed in the motion. >· for ally in the range 1 to N. 16). 32) which result from specific choices of Q can be of physical importance. For instance if ß = Z(Pr'i).

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