Analytical solution of the elastic transport equation
An example of a random process is the propagation of a particle (or photons, or acoustic wave) in a turbid medium, where particles undergo multiple scattering by randomly distributed scatterers in the medium. The kinetic equation governing particle propagation is the classic Boltzmann transport equation, which is also called the radiative transfer equation in the case of light propagation. The search for an analytical solution of the time-dependent elastic Boltzmann transport equation has lasted for many years. This chapter considers the problem of the classic elastic Boltzmann transport equation based on cumulant expansion. An analytical expression for cumulants of the spatial distribution of particles at any angle and time, exact up to an arbitrarily high order, is derived in an infinite uniform scattering medium. Up to the second order, a Gaussian approximation of the distribution function for the Boltzmann transport equation is obtained, with exact average center and exact half-width with time.
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