By C. Cercignani
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Additional resources for Mathematical Methods in Kinetic Theory
1) we have The quantity pv; which appears in Eq. 3) can be interpreted as the momentum density or, alternatively, as the mass flow (in the i direction). Other quantities which will be needed in the following are the momentum flow, the energy density, and the energy flow. 6) Sect. 3] 47 Microscopic Description and Macroscopic Gas Dynamics eJ is obviously the momentum density in phase space. 6) shows that the momentum flow is described by a symmetric tensor of second order. It is to be expected that in a macroscopic description only part of the microscopically evaluated momentum flow will be identified as such, because the integral in Eq.
The Problem of Nonequilibrium States. The Boltzmann Equation In Section 5 we saw that the problem of describing the state of thermal equilibrium of a monatomic ideal gas can be nicely solved; in particular, for the one-particle distribution function P<;,l we found a very simple formula which has a large variety of applications in the statistical description of matter in the gaseous state. , flowing in a channel or surrounding a body in flight in the atmosphere, something very different from being enclosed in a box and kept at uniform temperature and pressure.
Actually, one has to be a little bit more careful, since if one extracts molecules from the gas, Eq. 9) cannot be applied, since it refers to a gas of exactly N molecules. , we expect that if N is very large, Eq. 9) is not essentially altered if N is changed by a small fraction. It is therefore customary to consider the limit of Eq. 9) as N-+ oo, which is quite justified by the huge numbers of molecules which are usually considered. 12) Ifwe put Eq. 11) into Eq. 10), the latter reduces to the Maxwell formula for the velocity distribution which is frequently derived in elementary treatments of kinetic theory by a different kind of argument.