Abstract
We provide numerical solutions based on the path integral representation of stochastic processes for non-gradient drift Langevin forces in the presence of noise, to follow the temporal evolution of the probability density function and to compute exit times even for arbitrary noise. We compare the results with theoretical calculations, obtaining excellent agreement in the weak noise limit.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.
Highlights
More than 50 years ago, Ilya Prigogine and collaborators showed in an ensemble of fundamental papers [1,2,3,4,5] that macroscopic physics is mainly modelled by dissipative dynamical systems, i.e. systems that do not conserve volumes in phase space through their time evolution, contrary to what is the case in Hamiltonian mechanics, and they break time reversal and have attractors which are responsible for the rhythms and forms we observe in Nature at the macroscopic level
Modifying the boundary conditions imposed on the Fokker–Planck equation (FPE), namely imposing absorbing boundary conditions, we develop with our path integral approach a general numerical method to calculate exit times from local attractors which can be used for an arbitrary intensity of the noise, a novelty, as known theoretical approximations are usually limited to the weak noise limit
We have introduced a general numerical path integral method for the calculation of the probability density function for a multi-dimensional stochastic differential equations (SDEs) as well as an alternative application for the calculation of mean exit times
Summary
More than 50 years ago, Ilya Prigogine and collaborators showed in an ensemble of fundamental papers [1,2,3,4,5] that macroscopic physics is mainly modelled by dissipative dynamical systems, i.e. systems that do not conserve volumes in phase space through their time evolution, contrary to what is the case in Hamiltonian mechanics, and they break time reversal and have attractors which are responsible for the rhythms and forms we observe in Nature at the macroscopic level. Modifying the boundary conditions imposed on the FPE, namely imposing absorbing boundary conditions, we develop with our path integral approach a general numerical method to calculate exit times from local attractors which can be used for an arbitrary intensity of the noise, a novelty, as known theoretical approximations are usually limited to the weak noise limit. The PDF can be computed in an independent way (see the electronic supplementary material for a description of the stochastic Runge–Kutta algorithm chosen for the numerical work) Another approach to solve equation (2.1) is to start with the deterministic FPE [7] for the time evolution of the conditional probability density function, P(x, t | x , t ), which satisfies the Chapman–Kolmogorov relation [7]. We refer the reader to [14] for an exact formula in one-dimensional systems (see the electronic supplementary material)
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