Numerical Integration of Langevin Equation

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SUMMARY

The discussion focuses on the numerical integration of the Langevin equation, specifically addressing the transformation of the Gaussian white noise term when converting the Langevin differential equation into a finite difference equation. The key point is that the Gaussian white noise term is multiplied by the square root of the time differential to maintain the statistical properties of the noise during discretization. This adjustment ensures that the noise remains consistent with its continuous counterpart, preserving the integrity of the stochastic process.

PREREQUISITES
  • Understanding of Langevin equations and stochastic processes
  • Familiarity with finite difference methods in numerical analysis
  • Knowledge of Gaussian white noise characteristics
  • Basic proficiency in differential equations
NEXT STEPS
  • Study the derivation of the Langevin equation in detail
  • Explore finite difference methods for numerical integration
  • Learn about the properties of Gaussian white noise in stochastic calculus
  • Investigate applications of Langevin dynamics in physical systems
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Researchers in computational physics, applied mathematicians, and anyone involved in numerical simulations of stochastic processes will benefit from this discussion.

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Can anyone explain to me why the gaussian white noise term is multiplied by the square root of the time differential when we turn the Langevin differential equation into a finite difference equation for the purposes of integration?

http://pre.aps.org/pdf/PRE/v50/i6/p4404_1

The step I don't understand is the change in the last term in going from equation 3 to 4.

Any help would be greatly appreciated, thanks!
 
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equation 3:
Ocr12TW.png


equation 4:
oeahOsB.png


Here are the two equations. Thanks for your help!
 

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