标题： 基于阴影的可靠性退化在软化n体模拟中的影响 显示英文标题

标题： Shadowing-based reliability decay in softened n-body simulations

摘要： 混沌系统数值解的影子轨道是一条长时间内始终接近该数值解的动力学方程精确解。 在一个无碰撞的n体系统中，我们知道粒子的运动由全局势能决定，而非粒子间的相互作用。 因此，系统中每个单独粒子的轨迹都可以独立地找到影子轨道。 所以测量随着时间变化具有可影子轨迹的粒子数是有意义的。 我们发现具有可影子轨迹的粒子数随时间呈指数衰减，其形式为exp(-μt)，并且当软化参数ε在[~0.2,1]范围内（以局部平均粒子间距$\bar n$为单位）时，衰减常数μ与跳跃积分器的时间步长h、软化参数ε以及模拟中的粒子数N之间存在明确的关系。 因此，给定N和ε后，可以预先计算出达到特定模拟时间长度后所需的时间步长h，以获得期望比例的可影子粒子。 我们证明即使粒子每一步移动的距离达到软化长度的约1/3，超过100个穿越时间尺度内仍然有大量粒子具有可影子轨迹。 然而，如果时间步长增加，使得粒子在一时间步长内移动的距离超过软化长度的1/3，或者如果软化参数ε降低到约0.2$\bar n$以下，则可影子粒子的数量会急剧下降。 显示英文摘要

摘要： A shadow of a numerical solution to a chaotic system is an_exact_ solution to the equations of motion that remains close to the numerical solution for a long time. In a collisionless n-body system, we know that particle motion is governed by the global potential rather than by inter-particle interactions. As a result, the trajectory of each individual particle in the system is independently shadowable. It is thus meaningful to measure the number of particles that have shadowable trajectories as a function of time. We find that the number of shadowable particles decays exponentially with time as exp(-mu t), and that for eps in [~0.2,1] (in units of the local mean inter-particle separation $\bar n$), there is an explicit relationship between the decay constant mu, the timestep h of the leapfrog integrator, the softening eps, and the number of particles N in the simulation. Thus, given N and eps, it is possible to pre-compute the timestep h necessary to acheive a desired fraction of shadowable particles after a given length of simulation time. We demonstrate that a large fraction of particles remain shadowable over ~100 crossing times even if particles travel up to about 1/3 of the softening length per timestep. However, a sharp decrease in the number of shadowable particles occurs if the timestep increases to allow particles to travel further than 1/3 the softening length in one timestep, or if the softening is decreased below ~0.2$\bar n$.