Path integral formulation of general diffusion processes

Horacio S. Wio

Yan Wang

Stochastic differential equation (SDE) and Fokker-Planck equation (FPE) are two general approaches to describe the stochastic drift-diffusion processes. Solving SDEs relies on the Monte Carlo samplings of individual system trajectory, whereas FPEs describe the time evolution of overall distributions via path integral alike methods. The large state space and required small step size are the major challenges of computational efficiency in solving FPE numerically. In this paper, a generic continuous-time quantum walk formulation is developed to simulate stochastic diffusion processes. Stochastic diffusion in one-dimensional state space is modeled as the dynamics of an imaginary-time quantum system. The proposed quantum computational approach also drastically accelerates the path integrals with large step sizes. The new approach is compared with the traditional path integral method and the Monte Carlo trajectory sampling.

Xing Min Wang

In this work, we advance the development of the Probability Bracket Notation (PBN), a formalism inspired by Dirac's notation in quantum mechanics, to provide a unified framework for probability modeling. We demonstrate that under a Special Wick Rotation (SWR), an imaginary-time map, the Schrödinger equation, the transition amplitude, and its associated path integral in Dirac notation transform into the master equation, the transition probability, and its Euclidean path integral in the PBN, from which we can reproduce the master equation, representing induced micro-diffusion processes. By extending this approach through a General Wick Rotation (GWR) and employing an anti-Hermitian wave-number operator, we perform parallel derivations of path integrals in both the Dirac and PBN frameworks. This leads to the formulation of the Euclidean Lagrangian for induced diffusions and the strong-damping harmonic oscillator (described by the Smoluchowski diffusion equation). Our findings highlight the versatility of the PBN in bridging quantum mechanics and stochastic processes, offering a coherent notation system for analyzing time evolution and path integrals across these domains.

Pontificia Universidad Católica de Chile. Instituto de Física, T. B. Lantaño

Ph. Blanchard, Weian Zheng

Walter T. Strunz

The general theory of path integral propagators for the solution of linear quantum state diffusion (LQSD) stochastic Schr\"odinger equations describing open quantum systems is developed. Both Hamiltonian and, where possible, Lagrangian path integrals are derived and their connection established. The Hamiltonian version turns out to be more suitable. The results also show how the stochastic terms in the LQSD equation introduce a weight functional under the path integral, thus restricting the set of contributing paths. The center of this weight functional is determined by the stochastic processes governing the LQSD equation. In general, this picture holds in a semiclassical limit only. Some peculiarities of stochastic path integrals are pointed out. We evaluate the stochastic path integral in closed form for soluble models, gaining further insight into the behavior of the solutions of the LQSD equation. \textcopyright{} 1996 The American Physical Society.