Book contents
- Frontmatter
- Dedication
- Contents
- List of Figures
- List of Tables
- Acknowledgements
- Nomenclature
- 1 Introduction
- 2 The Boltzmann Equation 1: Fundamentals
- 3 The Boltzmann Equation 2: Fluid Dynamics
- 4 Transport in Dilute Gas Mixtures
- 5 The Dilute Lorentz Gas
- 6 Basic Tools of Nonequilibrium Statistical Mechanics
- 7 Enskog Theory: Dense Hard-Sphere Systems
- 8 The Boltzmann–Langevin Equation
- 9 Granular Gases
- 10 Quantum Gases
- 11 Cluster Expansions
- 12 Divergences, Resummations, and Logarithms
- 13 Long-Time Tails
- 14 Transport in Nonequilibrium Steady States
- 15 What’s Next
- Bibliography
- Index
8 - The Boltzmann–Langevin Equation
Published online by Cambridge University Press: 18 June 2021
- Frontmatter
- Dedication
- Contents
- List of Figures
- List of Tables
- Acknowledgements
- Nomenclature
- 1 Introduction
- 2 The Boltzmann Equation 1: Fundamentals
- 3 The Boltzmann Equation 2: Fluid Dynamics
- 4 Transport in Dilute Gas Mixtures
- 5 The Dilute Lorentz Gas
- 6 Basic Tools of Nonequilibrium Statistical Mechanics
- 7 Enskog Theory: Dense Hard-Sphere Systems
- 8 The Boltzmann–Langevin Equation
- 9 Granular Gases
- 10 Quantum Gases
- 11 Cluster Expansions
- 12 Divergences, Resummations, and Logarithms
- 13 Long-Time Tails
- 14 Transport in Nonequilibrium Steady States
- 15 What’s Next
- Bibliography
- Index
Summary
The linearized Boltzmann equation is generalized to include a fluctuating source term to account for fluctuations in the distribution function about the average behavior given by the Boltzmann equation. The result is the Boltzmann-Langevin equation with fluctuations taken to be Gaussian and correlated by space and time delta functions. Linearized Navier-Stokes equations are derived with fluctuation terms identical to those obtained by hydrodynamical arguments by Landau and Lifshitz, when applied to dilute gases. The Boltzmann-Langevin equation is used to obtain the average correlations in density fluctuations needed for the spectrum of long wavelength light scattered by a dilute gas in equilibrium. The Rayleigh-Brillouin spectrum is obtained. The method is then extended, with appropriate modifications, to obtain equations that describe the scattering of light by a fluid that is maintained in 9 a non-equilibrium stationary state, with a fixed temperature gradient. The light scattering spectrum is dramatically different from the equilibrium case.
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- Contemporary Kinetic Theory of Matter , pp. 317 - 350Publisher: Cambridge University PressPrint publication year: 2021