Mathematical Methods in Kinetic Theory by C. Cercignani

By C. Cercignani

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We have assumed that the interaction force is zero for intermolecular distances larger than (J; it is known, however, that the interaction force between a pair of molecules is different from zero at any distance, being, in general, strongly repulsive at short distances and weakly attractive for large separations. We have neglected this weakly attractive part, as is justified for ideal gases; accordingly, we could define the molecular diameter (J as the distance at which repulsion changes into attraction.

II We have thus obtained four different expressions for the same quantity: Eqs. 13). We can now obtain more expressions by taking linear combinations of the four basic ones; we are particularly interested in the combination which is obtained by adding the above four expressions and dividing by four. 15) E x (cp cp~)B(e, V) d~ d~l d£ de This equation expresses a basic property of the collision term, which will be frequently used in the following. In the particular case of g = 1, Eq. 16) We now observe that the integral appearing in Eq.

How will this second part manifest itself in a macroscopic description? If we take a fixed region of the gas and observe the change of momentum inside it, we shall find that (in the absence of external body forces) the changes can only in part be attributed to the matter which enters and leaves the region, leaving a second part which has no macroscopic explanation unless we admit that the contiguous regions of the gas exert a force on the region of interest, which is thus subject to stresses. 8) (a complete identification is correctly justified by the fact that, as we shall see later, Pij plays, in the macroscopic equations to be derived from the Boltzmann equation, the same role as the stress tensor in the conservation equations derived from macroscopic considerations).

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