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# A3.1 Kinetic theory: transport and fluctuations

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A3.1 Kinetic theory: transport and fluctuations book

# A3.1 Kinetic theory: transport and fluctuations

DOI link for A3.1 Kinetic theory: transport and fluctuations

A3.1 Kinetic theory: transport and fluctuations book

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## ABSTRACT

The kinetic theory of gases has a long history, extending over a period of a century and a half, and is responsible for many central insights into, and results for, the properties of gases, both in and out of thermodynamic equilibrium [1]. Strictly speaking, there are two familiar versions of kinetic theory, an informal version and a formal version. The informal version is based upon very elementary considerations of the collisions suffered by molecules in a gas, and upon elementary probabilistic notions regarding the velocity and free path distributions of the molecules. In the hands of Maxwell, Boltzmann and others, the informal version of kinetic theory led to such important predictions as the independence of the viscosity of a gas on its density at low densities, and to qualitative results for the equilibrium thermodynamic properties, the transport coefficients, and the structure of microscopic boundary layers in a dilute gas. The more formal theory is also due to Maxwell and Boltzmann, and may be said to have had its beginning with the development of the Boltzmann transport equation in 1872 [2]. At that time Boltzmann obtained, by heuristic arguments, an equation for the time dependence of the spatial and velocity distribution function for particles in the gas. This equation provided a formal foundation for the informal methods of kinetic theory. It leads directly to the Maxwell-Boltzmann velocity distribution for the gas in equilibrium. For non-equilibrium systems, the Boltzmann equation leads to a version of the second law of thermodynamics (the Boltzmann H -theorem), as well as to the Navier-Stokes equations of fluid dynamics, with explicit expressions for the transport coefficients in terms of the intermolecular potentials governing the interactions between the particles in the gas [3]. It is not an exaggeration to state that the kinetic theory of gases was one of the great successes of nineteenth century physics. Even now, the Boltzmann equation remains one of the main cornerstones of our understanding of non-equilibrium processes in fluid as well as solid systems, both classical and quantum mechanical. It continues to be a subject of investigation in both the mathematical and physical literature and its predictions often serve as a way of distinguishing different molecular models employed to calculate gas properties. Kinetic theory is typically used to describe the non-equilibrium properties of dilute to moderately dense gases composed of atoms, or diatomic or polyatomic molecules. Such properties include the coefficients of shear and bulk viscosity, thermal conductivity, diffusion, as well as gas phase chemical reaction rates, and other, similar properties.