Transport in Dilute Media

Abstract

This Chapter is an introduction to the kinetic theory of gases. As part of a book on micro and nanoscale heat transfer, the aims are twofold:

• To introduce the necessary concepts and tools, and in particular, the idea of a distribution function and the Boltzmann equation, to describe heat transfer in dilute gases on short length and time scales.

• To introduce general notions in the theory of transport, based on the kinetic approach, which will prove useful in later Chapters of the book, especially for describing the transport of electrons and phonons in solids.

The Chapter is organised as follows. We begin by introducing the ideas of distribution function, average and flux. We then discuss the particular context of thermodynamic equilibrium and show that, to describe systems that are out of equilibrium, which provide the conditions for macroscopic transfer, one must be able to calculate the distribution function in the most general situation. We introduce the underlying formalism of the Boltzmann equation and a highly simplified model based on the relaxation time. We can then discuss the idea of local thermodynamic equilibrium (LTE), and also situations that are close enough to LTE to be treated by perturbation methods. We shall show in particular how to demonstrate the Fourier law in this regime and obtain an expression for the thermal conductivity of a gas. We then turn to non-LTE regimes and in particular the ballistic transport regime which arises when the characteristic size of the system is smaller than the mean free path (or the observation time is shorter than the average time elapsed between two collisions). We end with a concrete example in which we compare and comment upon the orders of magnitude of exchanged fluxes in different regimes (convection, Fourier-type conduction, ballistic transport).