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

Area of Science:

• Physics
• Computational Physics
• Statistical Mechanics

Background:

• Diffusive dynamics simulations often employ two distinct action functionals to define trajectory probability distributions.
• One common action functional requires calculating the potential's second derivative alongside the force.

Purpose of the Study:

• To reconcile the apparent paradox between two action functionals used in diffusive dynamics.
• To clarify the conditions under which these functionals are equivalent for trajectory sampling and reweighting.
• To determine the correct action functional for identifying the most probable path.

Main Methods:

• Theoretical analysis of action functionals in the context of diffusive dynamics.
• Consideration of the mathematical properties of Brownian paths, specifically their nondifferentiable nature.
• Incorporation of trajectory-associated "entropy" into the analysis.

Main Results:

• Both action functionals are demonstrated to be equivalent for trajectory reweighting and sampling.
• The action functional incorporating the second derivative term is identified as the correct one for determining the most probable path.
• The nondifferentiable nature of Brownian paths and trajectory entropy explain the observed equivalence and differences.

Conclusions:

• The choice of action functional in diffusive dynamics is critical for accurate most probable path determination.
• Understanding Brownian path properties and trajectory entropy resolves ambiguities in action functional application.
• This work clarifies fundamental aspects of simulating and analyzing diffusive processes.