Numerical study of path probability for stochastic motion of non dissipative systems

TL;DR

This paper numerically investigates the probability distribution of paths in stochastic non-dissipative systems, revealing an exponential decay of path probability with action and how randomness influences this decay.

Contribution

It introduces a numerical approach to study path probabilities in stochastic non-dissipative systems and demonstrates the exponential decay relation with action.

Findings

Path probability decays exponentially with action.

Decay rate increases as Gaussian randomness decreases.

Smooth sample paths enable accurate evaluation of physical quantities.

Abstract

The path probability of stochastic motion of non dissipative or quasi-Hamiltonian systems is investigated by numerical experiment. The simulation model generates ideal one-dimensional motion of particles subject only to conservative forces in addition to Gaussian distributed random displacements. In the presence of dissipative forces, application of this ideal model requires that the dissipated energy is small with respect to the variation (work) of the conservative forces. The sample paths are sufficiently smooth space-time tubes with suitable width allowing correct evaluation of position, velocity, energy and action of each tube. It is found that the path probability decays exponentially with increasing action of the sample paths. The decay rate increases with decreasing Gaussian randomness.