scholarly journals From the Boltzmann equation to fluid mechanics on a manifold

2011 ◽  
Vol 369 (1944) ◽  
pp. 2362-2370 ◽  
Author(s):  
Peter J. Love ◽  
Donato Cianci
Keyword(s):  
Boltzmann Equation ◽  
Fluid Mechanics ◽  
Hydrodynamic Equations ◽  
The Boltzmann Equation

We apply the Chapman–Enskog procedure to derive hydrodynamic equations on an arbitrary surface from the Boltzmann equation on the surface.

Nonlinear transport coefficients and plane Couette flow of a viscous, heat-conducting gas between two plates at different temperatures

1987 ◽  
Vol 65 (9) ◽  
pp. 1090-1103 ◽  
Author(s):  
Byung Chan Eu ◽  
Roger E. Khayat ◽  
Gert D. Billing ◽  
Carl Nyeland
Keyword(s):  
Boltzmann Equation ◽  
Couette Flow ◽  
Transport Coefficients ◽  
Heat Conducting ◽  
Hydrodynamic Equations ◽  
Nonlinear Transport ◽  
Plane Couette Flow ◽  
The Boltzmann Equation ◽  
Viscous Heat ◽  
Different Temperatures

By using the example of plane Couette flow between two plates maintained at different temperatures, we present a method of calculating flow profiles for rarefied gases. In the method, generalized hydrodynamic equations are derived from the Boltzmann equation. They are then solved with boundary conditions calculated by taking into consideration the interfacial interaction between the surface and the gas molecule. The nonlinear transport coefficients employed in the generalized hydrodynamic equations are obtained from the Boltzmann equation by means of the modified-moment method. The profiles calculated are in agreement with the Liu–Lees theory as long as the boundary values are in agreement. It is found that the viscous-heating effect has a significant influence on the temperature and velocity profiles. The nonlinearity of transport coefficients also has significant effects on the profiles as the Knudsen and Mach numbers increase.

The Boltzmann Equation and the H Theorem (1872–1875)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Transport Phenomena ◽  
Dilute Gas ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

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.

Approach to Equilibrium, the H-Theorem and Irreversibility

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.

Boltzmann’s Kinetic Theory

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.

A SOLVABLE MODEL OF INTERACTING MANY BODY SYSTEMS EXHIBITING A BREAKDOWN OF THE BOLTZMANN EQUATION

2014 ◽  
Vol 28 (03) ◽  
pp. 1450046
Author(s):  
B. H. J. McKELLAR
Keyword(s):  
Boltzmann Equation ◽  
Time Dependence ◽  
Single Particle ◽  
Density Operator ◽  
Particle Density ◽  
Solvable Model ◽  
Many Body ◽  
Single Particle Density ◽  
The Boltzmann Equation ◽  
Many Body Systems

In a particular exactly solvable model of an interacting system, the Boltzmann equation predicts a constant single particle density operator, whereas the exact solution gives a single particle density operator with a nontrivial time dependence. All of the time dependence of the single particle density operator is generated by the correlations.

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