Derivation of the Boltzmann equation with no "molecular chaos"-type approximation

Abstract

The paper resolves the problem of the derivation of a completely closed evolution equation for $s$-particle distribution function $F_s(t)$ ($s \le N$) from the Liouville equation for $N \gg 1$-particle distribution function $F_N(t)$ with arbitrary initial condition $F_N(0)$ and without any use of the "molecular chaos" type approximation. The initial correlations are accounted for in this equation in the kernel governing the evolution of $F_s(t)$ via the special projection operator which exactly transforms the inhomogeneous Nakajima-Zwanzig Generalized Master Equation (GME) with an irrelevant initial condition term into the homogenous one. This equation is further simplified by presenting its kernel in the linear in the particles' density $n$ approximation. In this approximation the equations for one-particle $F_1(t)$ and two-particle $F_2(t)$ distribution functions are derived. It is shown that the terms describing the influence of initial correlations in the equation for $F_1(t)$ disappear at the large timescale $t \sim t_{\text{rel}} \gg t_{\text{cor}}$ ($t_{\text{cor}}$ is a short correlation time as compared to a relaxation time $t_{\text{rel}}$ of $F_1(t)$) resulting in the linear Boltzmann equation. This equation can be presented as the nonlinear Boltzmann equation in the time interval $t_{\text{cor}} \ll t \ll t_{\text{rel}}$. At $t_{\text{rel}} \to \infty$ (mean free path $l \to \infty$) the Boltzmann equation holds for all finite times $t \gg t_{\text{cor}}$.