Stochastic actions for diffusive dynamics: Reweighting, sampling, and minimization

TL;DR

This paper compares two action functionals used in diffusive dynamics, showing their equivalence for reweighting and sampling, and clarifies the nature of the most probable path considering path non-differentiability and entropy effects.

Contribution

It demonstrates the equivalence of two commonly used actions in diffusive dynamics and clarifies the conditions for identifying the most probable trajectory.

Findings

Both actions are equivalent for reweighting and sampling.

The most probable path is given by the global minimum of the second derivative action.

Brownian path non-differentiability influences the action's interpretation.

Abstract

In numerical studies of diffusive dynamics, two different action functionals are often used to specify the probability distribution of trajectories, one of which requiring the evaluation of the second derivative of the potential in addition to the force. Here it is argued that both actions are equivalent prescriptions for the purposes of reweighting and sampling trajectories, whereas the most probable path is more generally given by the global minimum of the action involving the second derivative term. The answer to this apparent paradox lies in the non-differentiable character of Brownian paths, as well as in the "entropy" associated with a given trajectory.