AI 中文总结
研究从\(N\)粒子刘维尔方程推导\(s\)粒子分布函数封闭演化方程，不采用“分子混沌”近似。利用特殊投影算子考虑初始关联，经密度线性近似简化方程，推导单双粒子分布函数方程，得出不同时间尺度下的玻尔兹曼方程。
AI 中文摘要
本文解决了从具有任意初始条件\(F_N(0)\)的\(N\gg1\)粒子分布函数\(F_N(t)\)的刘维尔方程出发，推导\(s\)粒子分布函数\(F_s(t)\)（\(s\leq N\)）的完全封闭演化方程的问题，且不使用任何“分子混沌”型近似。通过特殊投影算子在控制\(F_s(t)\)演化的核中考虑初始关联，将具有无关初始条件项的非齐次中岛 - 兹万齐格广义主方程转化为齐次方程。通过在粒子密度\(n\)的线性近似下呈现其核来进一步简化该方程，推导了单粒子\(F_1(t)\)和双粒子\(F_2(t)\)分布函数的方程。结果表明，在大时间尺度\(t\sim t_{\text{rel}}\gg t_{\text{cor}}\)（\(t_{\text{cor}}\)是与\(F_1(t)\)的弛豫时间\(t_{\text{rel}}\)相比的短关联时间）下，\(F_1(t)\)方程中描述初始关联影响的项消失，得到线性玻尔兹曼方程。在时间区间\(t_{\text{cor}}\ll t\ll t_{\text{rel}}\)内，该方程可表示为非线性玻尔兹曼方程。当\(t_{\text{rel}}\to\infty\)（平均自由程\(l\to\infty\)）时，玻尔兹曼方程对所有\(t\gg t_{\text{cor}}\)的有限时间都成立。
英文摘要
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 $t \gg t_{\text{cor}}$.
