Stochastische Processen in de Fysica

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Algemeen

Dit vak vervangt het deel van Wiskundige Methoden gegeven door Professor Maes over Markovketens. Het is daarnaast ook iets uitgebreider geworden met wat meer leerstof over Brownian Motion en een klein deel Non-Equilibrium Physics.

Examens Stochastische Processen

2014-2015

17 juni 2015

• Vraag 1: Consider a continuous time Markov process with state space K = {1, 2, . . . , M} and

with transition rates k(x, x + 1) = q except for x = M, k(x, x − 1) = p except for x = 1. All other transition rates are zero. Determine the stationary distribution as a function of p, q and M. Is there detailed balance? (Oefening 5 deel Continue Markovprocessen)

• Vraag 2: Lady Ann possesses 3 umbrellas which she employs in going from home to office

and back. If she is at home (resp. office) at the beginning (resp. end) of a day and it is raining, then she will take an umbrella with her to the office (resp. home), at least if there is one to be taken. If it is not raining, then she will not take an umbrella. Assuming that, independent of the past, it rains at the beginning (end) of a day with probability 1/3, what fraction of the time does Lady Ann arrive soaked at the office? (Oefening 15 deel Discrete Markovprocessen)

• Vraag 3: We consider the overdamped diffusion process with dx/dt = -V'(x_t) + \sqrt(2T)\ksi_t with the \ksi the standard white noise. Show the calculation that the distribution \rho(x) = exp(-V(x)/T)/Z is the only stationary distribution. Explain why a stationary distribution is an equilibrium distribution when it is symmetric under time reversal or satisfies the condition of detailed balance. Determine <x_t>.
• Vraag 4: Consider a network with four states (x, v) where x ∈ {0, 1}, v ∈ {−1, +1}. (Imagine x

to be a position and v like a velocity.) We define a Markov process in continuous time via transition rates that depend on parameter b > 0, k((1, +1),(1, −1)) = k((1, −1),(1, +1)) = k((0, +1),(0, −1)) = k((0, −1),(0, +1)) = 1, k((1, −1),(0, −1)) = k((0, +1),(1, +1)) = b All other transitions are forbidden. Make a drawing. Determine the stationary distribution on the four states as function of b. Is there detailed balance? (Oefening 8 deel Continue Markovprocessen)