7 Markov Processes, Master Equation

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Potsdam University, Department of Physics WS 2009/2010 Stochastic Processes and Statistical Methods A. Pikovsky 7 Markov processes, Master equation Problem 7.1 (**) Prove that the stationary Gaussian Markov process is the Ornstein-Uhlenbeck process. 1) First express the transition probability density for a stationary Markov process via the correlation function 2) Insert this expression in the Chapman-Kolmogorov equation and derive the following equation for the correlation function: K(t + t′)= K(t)K(t′) 3) Write a differential equation for K(t) (Hint: consider the limit t′ → 0) and solve it. Problem 7.2 (*) Consider a chemical reaction (death process) k A −→ B Let x be the number of molecules of type A (k is the reaction rate). 1) Show that the master equation reads ∂P (x, t) = −kxP (x, t)+ k(x + 1)P (x +1, t), x =0, 1,...x0; P (x, o)= δ ∂t x,x0 2) Show that the generating function F (s, t)= P (x, t)sx obeys P ∂F ∂F = k(1 − s) ∂t ∂s 3) Use the Ansatz F = f(g(s)λ(t)) and demonstrate that a general solution reads ekt F (s, t)= f( ) 1 − s Show that the solution mit the given initial conditions is F (s, t)=(1 − e−kt + se−kt)x0 4) Show that the probability reads x0 P (x, t)= e−x0kt(1 − e−kt)x0−x x 5) Demonstrate that −kt 2 −kt −kt hxi = x0e , (x −hxi) = x0e (1 − e ) At which value of hxi is the variance maximal? Show that for x0 → ∞ the process is like a deterministic one. Problem 7.3 (***) In a random walk on a two-dimensional plane all steps are equally probable ((x, y) → (x±1,y), (x, y) → (x, y ± 1)), but the 180◦-reversals are forbidden (i.e. (x, y) → (x +1,y) → (x, y) is not possible). Find the diffusion constant of this random walk. Hints: a) The velocity is a Markov process that can be described with a Markov chain! Consider first only the x-component of the velocity. Define for the three possible states {−1, 0, 1} the transition matrix. A small help: 1/3 · · M = · ·· · ·· b) Find the eigenvectors and eigenvalues of the matrix. What is the stationary probability distribution? e) Find the correlation function. For this represent the two-point probability distribution through the transitional probability M N and the stationary one-point probability. d) Calculate the correlation function and the diffusion constant in x-direction. Find the diffusion constant on the 2-d plane and compare with the diffusion constant for a simple random walk..

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