Numerical Integration of Langevin Equation

AI Thread Summary
The discussion centers on the conversion of the Langevin differential equation into a finite difference equation for numerical integration. The key point is the necessity of multiplying the Gaussian white noise term by the square root of the time differential to maintain the correct statistical properties of the noise in the discretized form. This adjustment ensures that the variance of the noise remains consistent with the continuous case. Participants are seeking clarification on the mathematical rationale behind this transformation between the specified equations. Understanding this aspect is crucial for accurate numerical simulations of systems described by Langevin dynamics.
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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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