Department of Physics, University of Athens, GR-15771 Athens, Greece.
Chaos. 2012 Jun;22(2):026120. doi: 10.1063/1.3697399.
The standard description of Fermi acceleration, developing in a class of time-dependent billiards, is given in terms of a diffusion process taking place in momentum space. Within this framework, the evolution of the probability density function (PDF) of the magnitude of particle velocities as a function of the number of collisions n is determined by the Fokker-Planck equation (FPE). In the literature, the FPE is constructed by identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle velocity and its square in the course of one collision. Although this treatment leads to the correct solution after a sufficiently large number of collisions have been reached, the transient part of the evolution of the PDF is not described. Moreover, in the case of the Fermi-Ulam model (FUM), if a standard simplification is employed, the solution of the FPE is even inconsistent with the values of the transport coefficients used for its derivation. The goal of our work is to provide a self-consistent methodology for the treatment of Fermi acceleration in time-dependent billiards. The proposed approach obviates any assumptions for the continuity of the random process and the existence of the limits formally defining the transport coefficients of the FPE. Specifically, we suggest, instead of the calculation of ensemble averages, the derivation of the one-step transition probability function and the use of the Chapman-Kolmogorov forward equation. This approach is generic and can be applied to any time-dependent billiard for the treatment of Fermi-acceleration. As a first step, we apply this methodology to the FUM, being the archetype of time-dependent billiards to exhibit Fermi acceleration.
在一类时变碰撞系统中，费米加速的标准描述是用在动量空间中发生的扩散过程来表示的。在这个框架内，作为碰撞次数 n 的函数，粒子速度大小的概率密度函数（PDF）的演化由福克-普朗克方程（FPE）决定。在文献中，通过将输运系数与在一次碰撞过程中粒子速度大小及其平方变化的系综平均值相关联，来构建 FPE。尽管这种处理在经过足够多次的碰撞后会得到正确的解，但 PDF 的演化的瞬态部分并没有被描述。此外，在费米-乌拉姆模型（FUM）的情况下，如果采用标准的简化方法，则 FPE 的解甚至与用于推导它的输运系数的值不一致。我们工作的目标是为时变碰撞中的费米加速提供一种自洽的处理方法。所提出的方法避免了对随机过程的连续性和正式定义 FPE 的输运系数的极限的任何假设。具体来说，我们建议代替计算系综平均值，推导单步转移概率函数并使用 Chapman-Kolmogorov 正向方程。这种方法是通用的，可以应用于任何时变碰撞来处理费米加速。作为第一步，我们将这种方法应用于 FUM，它是表现出费米加速的时变碰撞的原型。
