By Amedeo Odoni, Richard de Neufville

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**Example text**

T/2 at+~. 22) 0 where the arguments off' and f{ are related to~. ~ 1 , E, and() by Eqs. 17); the integration with respect to ~ 1 extends to the whole threedimensional velocity space E, the integration with respect to E goes, as obvious, from 0 to 2n, while the angle () varies from 0 (head-on collisions, r = 0) to n/2 (grazing collisions, r = o"). It is seen that all the complicated details of the two-body interactions are summarized by the quantity B((), V), which gives essentially the (unnormalized) probability density of a relative deflection equal to 1/J = n - 2() for a pair of molecules having relative speed V.

2) has been used in the above. The result we have found means that although a random molecule is likely to be at any point of the region occupied by the gas (P~> does not depend upon xd, the distribution of velocities is by no means uniform. If we open a little hole at any point of the boundary and let a very few molecules (few with respect to the total number) escape, when we measure their velocities and plot the distribution ofthe molecules according to their velocities, in the limit of infinitely repeated sampling, we find that the occupation density of the interval (; 1 , ~ 1 + d~d does depend upon ~ 1 and is given by Eq.

22) of Chapter I have been omitted here for simplicity in notation. It is clear that although the free-streaming operator has its own interest and presents nontrivial problems when realistic boundary conditions are prescribed, the collision term is what characterizes the Boltzmann equation because of its unusual form. It therefore seems appropriate to study some properties which make the manipulation of Q possible in many problems of basic character, in spite of its complicated form. 2) Sect. 1) Elementary Manipulations of the Collision Operator It is clear that when g = f Eq.