Solutions of Boltzmann Equation and Transport Processes

Abstract

An integral approximation (IA) method is proposed for the solution of certain integro‐differential equations of which the linearized Boltzmann equation is one example. The lowest‐order solution in this method consists of replacing the integral operator of the equation by a known function such that the solution has the correct initial value, correct initial slope in time, and correct behavior at large times. The deviation of the integral operator from the function is treated as a perturbation in higher orders. The method is applied as an example to the calculation of time correlation functions and thermal transport coefficients. Deviations from the exponential behavior of the correlation functions are explicitly evaluated. Another method of solution which involves a cumulant expansion (CU) is also used for the evaluation of these quantities. Both methods are then compared with the Chapman–Enskog (CE) method. The IA method provides a better physical approximation and better numerical estimates for the thermal transport coefficients than the CU or CE methods.