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Analysis of trajectory entropy for continuous stochastic processes at equilibrium.
Kevin R Haas1, Haw Yang, Jhih-Wei Chu
1Department of Chemical and Biomolecular Engineering, University of California-Berkeley , Berkeley, California 94720, United States.
The Journal of Physical Chemistry. B
|May 1, 2014
Summary
This study derives trajectory entropy for the overdamped Langevin equation using two methods, confirming their equivalence and quantitative agreement with numerical solutions for stochastic processes.
Area of Science:
- Statistical Physics
- Stochastic Processes
- Physical Chemistry
Background:
- The Langevin equation describes systems influenced by random forces.
- Trajectory entropy quantifies the uncertainty in the path of a stochastic system.
- Understanding trajectory entropy is crucial for analyzing complex dynamics.
Purpose of the Study:
- To derive an analytical expression for trajectory entropy of the overdamped Langevin equation.
- To compare two distinct theoretical approaches for calculating trajectory entropy.
- To validate the analytical results against numerical simulations.
Main Methods:
- Derivation via the Fokker-Planck equation for probability density propagation.
- Derivation using the path integral formulation of Onsager-Machlup action.
- Comparison of analytical results with numerical solutions across various time resolutions.
Main Results:
- An analytical expression for trajectory entropy was successfully derived.
- Both the Fokker-Planck and path integral methods yielded consistent results in the continuum limit.
- Quantitative agreement between analytical and numerical results was observed, especially at finer time resolutions.
- The contributions of deterministic and stochastic forces to dynamics were elucidated.
Conclusions:
- The equivalence of partial differential equation and path integral formulations for trajectory entropy was confirmed.
- The derived analytical expression provides a reliable method for calculating trajectory entropy.
- The study enhances the understanding of dynamics in stochastic systems described by the Langevin equation.
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