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Unformatted text preview: Phys 210B Fall 2010 Problem Set #3: Fokker-Planck equation and Multiplicative Noise Due date: Wednesday November 10 1. Fokker-Planck equation with continuous-time dynamics Consider the following Langevin equation dr dt = f ( r ) + ( t ) (1) where ( t ) is Gaussian distributed, with h i = 0 and h ( t ) ( t ) i = D e-| t- t | / . (2) (a) For a multi-variable Gaussian distribution P [ ] = N- 1 e- 1 2 R dd K ( , ) ( ) ( ) where N is a normalization constant and the kernel K ( , ) is symmetric and invertible, show that h ( t ) F [ ] i = Z dK- 1 ( t, ) * F ( ) + (3) for an arbitrary functional F [ ]. Use F [ ] = ( t ) in Eq. (3) to find K- 1 in term of h i given in (2). (b) Derive the Fokker-Planck equation which describes the time evolution of the probability density P ( x,t ) = h ( x- r ( t )) i by taking the time derivative of P ( x,t ) and then directly performing statistical average of the resulting expression in the limit 0. (c) Repeat the above for the case of multiplicative noise, i.e., for Langevin equation of the form dr dt = f ( r ) + g ( r ) ( t ) , (4) and the same noise as that characterized by Eq. (2). Show that forand the same noise as that characterized by Eq....
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This note was uploaded on 11/20/2010 for the course PHYS 250a taught by Professor Hwa during the Spring '10 term at UCSD.
- Spring '10