Dynamics of Stochastic Systems

Aubrey Truman, Ian M Davies

Tetsuya Misawa, Kunio Yasue

A canonical stochastic dynamical system for time-symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations. A certain class of stochastic dynamical systems gives a Hamiltonian formalism of Nelson’s stochastic mechanics. In a manner analogous to classical mechanics, the notions of a stochastic Poisson bracket and canonical transformation are introduced to the stochastic dynamical systems. It is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation.

Yun‐An Yan, Jiushu Shao

Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.

Folkert Kuipers

Khavtgain Namsrai

A. A. Villaeys, Alex Boeglin

In the preceding paper we have developed a formal theoretical approach to describe the dynamics of stochastic systems in nonlinear optics. The main emphasis has been on the description of the equation of motion of the free system, since the optical fields are usually treated by perturbation expansion. Here we explicitly develop the dynamics of complex models, which should be helpful in realistic situations. Two different models are considered. The dynamics of a stochastic three-level system is introduced first. Then the case of two configurations stochastically coupled is explicitly treated. This last model is of interest in order to understand the effects of nonradiative transitions on the nonlinear-optical properties of stochastic systems.