Sampling constrained stochastic trajectories using Brownian bridges

Area of Science:

• Computational Physics
• Statistical Mechanics
• Stochastic Processes

Background:

• Simulating systems under Langevin dynamics is crucial for understanding molecular processes.
• Sampling trajectories conditioned on specific endpoints is computationally challenging.
• Existing methods may lack accuracy or efficiency for certain systems.

Purpose of the Study:

• To introduce a new, accurate method for sampling conditioned trajectories under Langevin dynamics.
• To leverage Brownian bridges for improved trajectory simulation.
• To provide a computationally efficient approach for exploring system pathways.

Main Methods:

• Utilizing Brownian bridges to define conditioned trajectories.
• Reformulating bridge equations into a non-linear stochastic integro-differential equation.
• Approximating the equation for bundled trajectories (low temperature/transition paths).
• Solving the approximate equation iteratively via a fixed-point method.

Main Results:

• The proposed method accurately samples conditioned trajectories.
• The iterative fixed-point method provides an efficient solution for the approximate equation.
• Demonstrated performance on simple systems, validating the approach.
• The method is particularly effective for low-temperature or transition path sampling.

Conclusions:

• The Brownian bridge-based method offers a highly accurate way to generate conditioned trajectories.
• This technique enhances the study of systems evolving under Langevin dynamics.
• The method shows promise for applications in statistical mechanics and computational simulations.