The chaotic diffusion for particles moving in a time dependent potential well is described by using two different procedures: (i) via direct evolution of the mapping describing the dynamics and ; (ii) by the solution of the diffusion equation. The dynamic of the diffusing particles is made by the use of a two dimensional, nonlinear area preserving map for the variables energy and time. The phase space of the system is mixed containing both chaos, periodic regions and invariant spanning curves limiting the diffusion of the chaotic particles. The chaotic evolution for an ensemble of particles is treated as random particles motion and hence described by the diffusion equation. The boundary conditions impose that the particles can not cross the invariant spanning curves, serving as upper boundary for the diffusion, nor the lowest energy domain that is the energy the particles escape from the time moving potential well. The diffusion coefficient is determined via the equation of the mapping while the analytical solution of the diffusion equation gives the probability to find a given particle with a certain energy at a specific time. The momenta of the probability describe qualitatively the behavior of the average energy obtained by numerical simulation, which is investigated either as a function of the time as well as some of the control parameters of the problem.

中文翻译：

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粒子在时间相关势阱中运动的混沌扩散

通过使用两种不同的程序来描述在时间相关势阱中运动的粒子的混沌扩散：（i）通过描述动力学的映射的直接演化和；(ii) 由扩散方程的解。扩散粒子的动态是通过使用二维、非线性区域保持图来表示变量能量和时间。系统的相空间混合了混沌、周期性区域和限制混沌粒子扩散的不变跨越曲线。粒子系综的混沌演化被视为随机粒子运动，因此由扩散方程描述。边界条件规定粒子不能穿过不变的跨越曲线，作为扩散的上边界，也不是最低能量域，即粒子从时间移动势阱中逸出的能量。扩散系数是通过映射方程确定的，而扩散方程的解析解给出了在特定时间找到具有特定能量的给定粒子的概率。概率动量定性地描述了通过数值模拟获得的平均能量的行为，它被研究为时间的函数以及问题的一些控制参数。