Work each of the following short parts directly from the definitions built for a…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Non-equilibrium statistical mechanics and stochastic thermodynamics

Error, Chance and the Machine, Ch Random processes in continuous time

Problem

Work each of the following short parts directly from the definitions built for a Poisson process, a two-state master equation, the stochastic simulation algorithm, the Ornstein-Uhlenbeck process and the mean first-passage time. The parts are independent except where stated. ((a)) A detector records clicks from two independent, unrelated sources with constant rates 30 s−130\,{\rm s^{-1}} and 45 s−145\,{\rm s^{-1}}. Find the probability that no click at all is recorded in a 20 ms20\,{\rm ms} window. ((b)) An ion channel opens at rate 150 s−1150\,{\rm s^{-1}} and closes at rate 350 s−1350\,{\rm s^{-1}}, independently of how long it has been in its current state. Find its equilibrium open probability and its relaxation time. ((c)) Using the combined source of part (a) (total rate 75 s−175\,{\rm s^{-1}}), give the probability that the next recorded click, whenever it occurs, came from the 30 s−130\,{\rm s^{-1}} source rather than the 45 s−145\,{\rm s^{-1}} source. ((d)) A silica sphere in water at T=300 KT = 300\,{\rm K} has friction coefficient γ=8.0×10−9 kg s−1\gamma = 8.0\times10^{-9}\,{\rm kg\,s^{-1}}. Use the Ornstein-Uhlenbeck result for the diffusion coefficient to find DD. ((e)) The same sphere starts 2 μm2\,\mu{\rm m} from one end of a channel of width 8 μm8\,\mu{\rm m} with absorbing walls at each end. Find its mean time to first touch a wall.

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