Analytical solution of the elastic Boltzmann transport equation in an infinite uniform medium using cumulant expansion

We study the analytical solution of the time-dependent elastic Boltzmann transport equation in an infinite uniform isotropic medium with an arbitrary phase function. We calculate (1) the exact distribution in angle, (2) the spatial cumulants at any angle, exact up to an arbitrary high order n. At the second order, n = 2, an analytical, hence extremely useful combined distribution in position and angle, is obtained as a function of time. This distribution is Gaussian in position, but not in angle. The average center and spread of the half-width are exact. By the central limit theorem the complete distribution approaches this Gaussian distribution as the number of collisions (or time) increases. The center of this distribution advances in time, and an ellipsoidal contour that grows and changes shape provides a clear picture of the time evolution of the particle migration from near ballistic, through snake-like, and into the final diffusive regime. This second-order cumulant approximation also provides the correct ballistic limit. Algebraic expressions for the nth order cumulants are provided. The number of terms grows rapidly with n, but our expressives are recursive and easily automated.