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Stochastic Langevin equations: Markovian and non-Markovian dynamics
R L S Farias1, Rudnei O Ramos, L A da Silva
1Departamento de Ciências Naturais, Universidade Federal de São João del Rei, 36301-160, São João del Rei MG, Brazil. ricardo@dft.if.uerj.br
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2009
Summary
This study compares non-Markovian Langevin equations to their Markovian approximations. It details when local approximations are valid for additive and multiplicative noise, considering dissipation.
Area of Science:
- Statistical Physics
- Nonlinear Dynamics
- Stochastic Processes
Background:
- Stochastic Langevin equations model complex systems.
- Non-Markovian effects and local approximations are crucial for accuracy.
- Understanding the validity of approximations is key in theoretical physics.
Purpose of the Study:
- To compare non-Markovian Langevin equations with their Markovian approximations.
- To analyze the validity of local approximations against nonlocal ones.
- To specify conditions under which local forms are accurate.
Main Methods:
- Detailed analysis of nonlocal vs. local approximations.
- Investigation of additive and multiplicative noise scenarios.
- Inclusion of system-dependent dissipation via fluctuation-dissipation theorem.
Main Results:
- Explicit conditions for the validity of local approximations are derived.
- The study clarifies the limitations of Markovian approximations.
- Both additive and multiplicative noise cases are analyzed.
Conclusions:
- Local approximations of Langevin equations can be valid under specific conditions.
- The findings provide guidance for simplifying complex stochastic models.
- Accurate modeling requires careful consideration of non-Markovian dynamics and noise types.
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