Numerical investigations of low-density nozzle flow by solving the Boltzmann equation

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

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Approach to Equilibrium, the H-Theorem and Irreversibility

10.1093/oso/9780199592357.003.0003 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Boltzmann Equation ◽

Detailed Knowledge ◽

Approach To Equilibrium ◽

The Boltzmann Equation ◽

H Theorem ◽

Irreversible Evolution

Like most of the greatest equations in science, the Boltzmann equation is not only beautiful but also generous. Indeed, it delivers a great deal of information without imposing a detailed knowledge of its solutions. In fact, Boltzmann himself derived most if not all of his main results without ever showing that his equation did admit rigorous solutions. This Chapter illustrates one of the most profound contributions of Boltzmann, namely the famous H-theorem, providing the first quantitative bridge between the irreversible evolution of the macroscopic world and the reversible laws of the underlying microdynamics.

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Boltzmann’s Kinetic Theory

10.1093/oso/9780199592357.003.0002 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Boltzmann Equation ◽

Kinetic Theory ◽

Statistical Physics ◽

Neutron Transport ◽

Condensed Matter ◽

Relativistic Fluids ◽

Classical Statistical Mechanics ◽

Dilute Gas ◽

Equilibrium Processes ◽

The Boltzmann Equation

Kinetic theory is the branch of statistical physics dealing with the dynamics of non-equilibrium processes and their relaxation to thermodynamic equilibrium. Established by Ludwig Boltzmann (1844–1906) in 1872, his eponymous equation stands as its mathematical cornerstone. Originally developed in the framework of dilute gas systems, the Boltzmann equation has spread its wings across many areas of modern statistical physics, including electron transport in semiconductors, neutron transport, quantum-relativistic fluids in condensed matter and even subnuclear plasmas. In this Chapter, a basic introduction to the Boltzmann equation in the context of classical statistical mechanics shall be provided.

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