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Related Experiment Videos

Symmetric path integrals for stochastic equations with multiplicative noise

Arnold1

  • 1Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|November 23, 2000
PubMed
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This study details converting multiplicative noise Langevin equations into path integrals using a time-symmetric discretization. It corrects a common assumption about Stratonovich Langevin equations with dependent noise amplitudes.

Area of Science:

  • Physics
  • Stochastic Processes
  • Mathematical Physics

Background:

  • Langevin equations with multiplicative noise are crucial for modeling systems where noise amplitude depends on the system's state.
  • Converting these equations into path integrals is essential for applying powerful tools of quantum field theory and statistical mechanics.
  • Existing methods for this conversion often rely on approximations or specific discretization conventions.

Purpose of the Study:

  • To derive the correct path integral representation for Langevin equations with multiplicative noise.
  • To investigate the impact of time discretization conventions on the path integral formulation.
  • To identify and correct a common misconception regarding the conversion of Stratonovich Langevin equations to path integrals.

Main Methods:

Related Experiment Videos

  • Derivation of path integral from a Langevin equation using a time-symmetric discretization scheme.
  • Analysis of the coordinate and time derivative definitions in the discretization process.
  • Comparison of the derived path integral with existing methods, particularly for the Stratonovich convention.

Main Results:

  • A correct path integral formulation is derived for Langevin equations with multiplicative noise using a time-symmetric discretization.
  • The derivation highlights the critical role of the discretization convention in defining the path integral.
  • It is demonstrated that the common shortcut for Stratonovich Langevin equations fails when the noise amplitude is state-dependent.

Conclusions:

  • The time-symmetric discretization provides a robust method for converting multiplicative noise Langevin equations to path integrals.
  • Accurate path integral formulation requires careful consideration of discretization choices, especially for state-dependent noise.
  • The findings necessitate a re-evaluation of previous analyses that employed simplified conversion techniques for Stratonovich Langevin equations.