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1 results

  • 8 - The Boltzmann–Langevin Equation

  • J. R. Dorfman, University of Maryland, College Park, Henk van Beijeren, Universiteit Utrecht, The Netherlands, T. R. Kirkpatrick, University of Maryland, College Park
  • Book:
    Contemporary Kinetic Theory of Matter
    Published online:
    18 June 2021
    Print publication:
    24 June 2021, pp 317-350

  • 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.