Gaveau Bernard, Moreau Michel
Faculty of Physics and Engineering, Sorbonne Université, LPTMC, Case 121, 4, Place Jussieu, 75231 Paris Cedex 05, France.
Chaos. 2020 Aug;30(8):083104. doi: 10.1063/5.0001435.
We study the coarse-grained distribution of a Hamiltonian system on the space partition determined by the initial measurement inaccuracies. Using methods of coding theory, introduced by Shannon and further researchers, Kolmogorov treated the stationary case for a discretized time, when the microscopic system is initially uniformly distributed. Following his work, we consider the non-stationary mesoscopic process induced by the Hamiltonian evolution from an inhomogeneous initial distribution. In general, this process has an infinite memory, but we show that its memory fades out with time: with any finite accuracy a, it can be approximated by a process with a memory limited to the n past events, n depending only on a. As a result, under suitable hypotheses, the mesoscopic process obeys an approximate Markov equation on groups of n successive states. More roughly, one obtains an ordinary Markov system by introducing a time coarse-graining on n successive elementary time steps. So, in a generic case, the system eventually tends to equilibrium for any initial mesoscopic distribution.
我们研究了哈密顿系统在由初始测量误差确定的空间划分上的粗粒化分布。利用香农及后续研究者引入的编码理论方法，柯尔莫哥洛夫处理了离散时间下的平稳情形，此时微观系统最初是均匀分布的。继他的工作之后，我们考虑由哈密顿演化从非均匀初始分布诱导出的非平稳介观过程。一般来说，这个过程具有无限记忆，但我们表明其记忆会随时间消退：对于任何有限精度(a)，它可以由一个记忆限于过去(n)个事件的过程来近似，(n)仅取决于(a)。结果，在适当的假设下，介观过程在(n)个连续状态的组上服从一个近似的马尔可夫方程。更粗略地说，通过在(n)个连续的基本时间步上引入时间粗粒化，可得到一个普通的马尔可夫系统。所以，在一般情况下，对于任何初始介观分布，系统最终都会趋向于平衡。
