Worrakitpoonpon Tirawut
Institute of Science, <a href="https://ror.org/05sgb8g78">Suranaree University of Technology</a>, Nakhon Ratchasima 30000, Thailand.
Phys Rev E. 2024 May;109(5-1):054118. doi: 10.1103/PhysRevE.109.054118.
We investigate the effect of a finite particle number N on the violent relaxation leading to the quasistationary state (QSS) in a one-dimensional self-gravitating system. From the theoretical point of view, we demonstrate that the local Poissonian fluctuations embedded in the initial state give rise to an additional term proportional to 1/N in the Vlasov equation. This term designates the strength of the local mean-field variations by fluctuations. Because it is of the mean-field origin, we interpret it differently from the known collision term in the way that it effects the violent relaxation stage. Its role is to deviate the distribution function from the Vlasov limit, in the collisionless manner, at a rate proportional to 1/N while the violent relaxation is progressing. This hypothesis is tested by inspecting the QSSs in simulations of various N. We observe that the core phase-space density can exceed the limiting density deduced from the Vlasov equation and its deviation degree is in accordance with the 1/N estimate. This indicates the deviation from the standard mean-field approximation of the violent relaxation process by that 1/N term. In conclusion, the finite-N effect has a significant contribution to the QSS apart from that it plays a role in the collisional stage that takes place long after. The conventional collisionless Vlasov equation might not be able to describe the violent relaxation of a system of particles properly without the correction term of the local finite-N fluctuations.
我们研究了一维自引力系统中有限粒子数N对导致准稳态（QSS）的剧烈弛豫的影响。从理论角度来看，我们证明了初始状态中嵌入的局部泊松涨落会在弗拉索夫方程中产生一个与1/N成正比的附加项。该项表示涨落引起的局部平均场变化的强度。由于它起源于平均场，我们对它的解释与已知的碰撞项不同，即它对剧烈弛豫阶段的影响方式不同。它的作用是在剧烈弛豫进行时，以与1/N成正比的速率使分布函数以无碰撞的方式偏离弗拉索夫极限。通过检查不同N值模拟中的QSS来检验这一假设。我们观察到核心相空间密度可以超过从弗拉索夫方程推导得出的极限密度，并且其偏离程度与1/N的估计值一致。这表明该1/N项导致剧烈弛豫过程偏离了标准平均场近似。总之，有限N效应除了在很久之后发生的碰撞阶段起作用外，对QSS也有显著贡献。如果没有局部有限N涨落的修正项，传统的无碰撞弗拉索夫方程可能无法恰当地描述粒子系统的剧烈弛豫。
