Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium

The Boltzmann equation is a fundamental equation in kinetic theory that describes the evolution of a gas of particles in the absence of external forces. It is a nonlinear integro-differential equation that is difficult to solve analytically. However, direct methods have been developed that can be used to solve the Boltzmann equation numerically.

Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows (Fluid Mechanics and Its Applications Book 60)

by V.V. Aristov

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Direct methods are based on the idea of representing the solution to the Boltzmann equation as a sum of basis functions. The basis functions are chosen so that they satisfy the boundary conditions of the problem and so that they can be used to approximate the solution to the Boltzmann equation with high accuracy.

Direct methods have been used to solve a wide variety of problems in kinetic theory, including problems in fluid dynamics, plasma physics, and materials science. They have also been used to study nonequilibrium phenomena, such as shock waves and turbulence.

Theoretical Foundations

The theoretical foundations of direct methods for solving the Boltzmann equation are based on the theory of integral equations. The Boltzmann equation can be written as an integral equation of the form

f(x, v, t) = f_0(x, v) + \int_0^t \int_{\mathbb{R}^3}K(x, v, x', v', t - t') f(x', v', t') dx' dv' dt'

where f(x, v, t) is the distribution function of the gas, f_0(x, v) is the initial distribution function, and K(x, v, x', v', t) is the Boltzmann collision kernel.

The Boltzmann collision kernel is a nonlinear operator that describes the interactions between the particles in the gas. It is a complicated operator that depends on the type of gas and the temperature.

Direct methods for solving the Boltzmann equation are based on the idea of representing the solution to the Boltzmann equation as a sum of basis functions. The basis functions are chosen so that they satisfy the boundary conditions of the problem and so that they can be used to approximate the solution to the Boltzmann equation with high accuracy.

The most common type of basis function used in direct methods is the Hermite polynomial. Hermite polynomials are orthogonal polynomials that are defined on the interval (-∞, ∞). They can be used to approximate any smooth function with high accuracy.

Once the basis functions have been chosen, the solution to the Boltzmann equation can be approximated by a linear combination of the basis functions. The coefficients of the linear combination are determined by solving a system of linear equations.

The system of linear equations that is obtained by using direct methods is typically large and sparse. However, there are a number of efficient algorithms that can be used to solve large sparse systems of equations.

Practical Applications

Direct methods have been used to solve a wide variety of problems in kinetic theory, including problems in fluid dynamics, plasma physics, and materials science. They have also been used to study nonequilibrium phenomena, such as shock waves and turbulence.

One of the most important applications of direct methods is in the simulation of rarefied gas flows. Rarefied gas flows are gas flows in which the mean free path of the particles is large compared to the characteristic length of the flow. Direct methods can be used to simulate rarefied gas flows with high accuracy and efficiency.

Direct methods have also been used to study plasma physics. Plasma is a gas that is composed of charged particles. Direct methods can be used to simulate the behavior of plasma in a variety of applications, such as fusion energy and plasma processing.

In materials science, direct methods have been used to study the properties of semiconductors and other materials. Direct methods can be used to calculate the transport coefficients of semiconductors and to study the effects of defects on the properties of materials.

Study of Nonequilibrium

Direct methods have been used to study a wide variety of nonequilibrium phenomena, such as shock waves and turbulence. Shock waves are waves that travel through a gas at supersonic speeds. They are characterized by a sharp increase in pressure, temperature, and density.

Turbulence is a state of

Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows (Fluid Mechanics and Its Applications Book 60)

by V.V. Aristov

4.5 out of 5

Language	:	English
File size	:	13890 KB
Text-to-Speech	:	Enabled
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
X-Ray for textbooks	:	Enabled
Word Wise	:	Enabled
Print length	:	422 pages

FREE