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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 ) + η ( t ) (1) where η ( t ) is Gaussian distributed, with h η i = 0 and h η ( t ) η ( t ) i = D τ e-| t- t | /τ . (2) (a) For a multi-variable Gaussian distribution P [ η ] = N- 1 e- 1 2 R dτdτ K ( τ,τ ) η ( τ ) η ( τ ) where N is a normalization constant and the kernel K ( τ,τ ) is symmetric and invertible, show that h η ( t ) F [ η ] i = Z dτK- 1 ( t,τ ) * δF δη ( τ ) + (3) for an arbitrary functional F [ η ]. Use F [ η ] = η ( t ) in Eq. (3) to find K- 1 in term of h ηη i given in (2). (b) Derive the Fokker-Planck equation which describes the time evolution of the probability density P ( x,t ) = h δ ( 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 τ → 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 ) · η ( 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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- Spring '10
- Equations, Distribution of wealth, Langevin equation, Fokker-Planck equation, Stratanovich dynamics