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 tr ansport equation in an infinite uniform isotropic medium with an arbitrary phase function. We calculate (1) the exact distribution in angle, (2) the s patial cumulants at any angle, exact up to an arbitrary high order n. At th e second order, n = 2, an analytical, hence extremely useful combined distr ibution in position and angle, is obtained as a function of time. This dist ribution 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 comple te distribution approaches this Gaussian distribution as the number of coll isions (or time) increases. The center of this distribution advances in tim e, and an ellipsoidal contour that grows and changes shape provides a clear picture of the time evolution of the particle migration from near ballisti c, through snake-like, and into the final diffusive regime. This second-ord er cumulant approximation also provides the correct ballistic limit. Algebr aic expressions for the nth order cumulants are provided. The number of ter ms grows rapidly with n, but our expressives are recursive and easily autom ated.