Stochastische Processen in de Fysica: verschil tussen versies
| Regel 9: | Regel 9: | ||
===17 juni 2015=== | ===17 juni 2015=== | ||
*Vraag 1: Consider a continuous time Markov process with state space K = {1, 2, . . . , M} and | *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? | 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: | *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 <\math> \xdot <\math> | |||
Versie van 20 aug 2015 13:52
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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 <\math> \xdot <\math>