Density matrix formulation of dynamical systems
Density matrix formulation of dynamical systems
复制标题
动力系统的密度矩阵公式
DOI：
10.1103/physreve.106.054135
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发表时间：
2022
影响因子：
2.4
通讯作者：
Green, Jason R.
中科院分区：
文献类型：
--
作者：
Das, Swetamber;Green, Jason R.
耗散、混合和发展湍流的物理系统也不可逆地传递统计密度。然而，对于非稳定、开放和驱动的非平衡过程，从原子和分子尺度动力学预测密度的演变是具有挑战性的。在这里，我们建立了一个理论来解决经典动力系统的这一挑战，类似于量子力学的密度矩阵公式。根据非哈密顿系统的Liouville定理和Liouville方程的推广，我们证明了经典密度矩阵类似于相空间度量，并随时间演化。在没有耗散或驱动的情况下，通过施加痕迹保存或考虑哈密顿动力学，恢复了传统的刘维利形式。动态不稳定性和混沌的局部测度被嵌入到经典的对易子和反对易子中，并且当动力学是哈密顿时直接与泊松括号相关。因为经典密度矩阵是建立在李雅普诺夫向量基础上的，它为非平衡过程的统计力学提供了一个可选的计算处理基础，适用于驱动、瞬态、耗散、规则和混沌的系统。
Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equation for non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.