phys210bhw3

# phys210bhw3 - Phys 210B — Fall 2010 Problem Set#3...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

phys210bhw3 - Phys 210B — Fall 2010 Problem Set#3...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online