Homework Set 2 MCC011. Non-equilibrium processes in physics, chemistry and biology Deadline: May 6, 2015, before class 1 Warm-up. Shot Noise (6p) Shot noise occurs in electrical circuits and photodetectors and is a result of the discrete nature of particles. For example, in a simple digital camera at low-light conditions, photons arrive one at a time. The rate of detection of one photon per unit sensor area per unit time is very low. Let us call it q. We can model this detection as a Poisson process. It is a one-step Markov chain that governs the probability pn (t) of n rare events (photon detection here) occurring by time t > 0. It can be defined by: rn = 0, gn = q, q n-1 pn (0) = δn,0 q n q n+1 Figure 1: Schematics of the Poisson process (a) Write down the master equation for the Poisson process. (b) Solve the master equation for pn (t). Hint: You might want to rewrite it in terms of a probability generating ∞ P function: G(t) = sn pn (t), s = const n=0 1 (c) Find the so-called signal-to-noise ratio (SNR), which is given by the ratio of the the signal average to its standard deviation: hni SNR = p Var[n] If you get it right, you will see that the SNR is better for larger qt, thus increasing the exposure time or your camera’s sensor size will help you take a better picture at night (considering shot noise only). http://en. wikipedia.org/wiki/Shot_noise#/media/File:Photon-noise.jpg 2 The mean-field approximation (6p) For the mean-field approximation discussed in the class, assume the first jump moment a1 (hY i) to be no longer a linear function of hY i. Expand it in hY i to obtain the first non-zero correction to the mean-field equation. You should obtain: y˙ = a1 (y) + 21 σ 2 a001 (y) d 2 dt σ where y(t) = hY i(t); a01 = 3 = a2 (y) + 2σ 2 a01 (y) , d dy a1 ; a001 = d2 a ; dy 2 1 σ 2 = Var[Y ]. Transition matrices (6p) (I) Viral networks In this example we will create a simple model for a bird flu spread ( based on http://www.pnas.org/content/112/1/172.full.pdf). Consider migration of birds between three geographical regions, characterized by the migration rates w. The individual transition rates are given by w1,2 = w2,3 = w3,1 = 10 birds/day and w2,1 = w3,2 = w1,3 = 1 bird/day. Looking at it on a yearly scale, we can model it as a nonreversible continuous time Markov cycle. (a) Write down the transition rate matrix W . (b) Find the stationary probability distribution of the system. (c) Check if the system obeys detailed balance. Yearly virus outbreaks seem to be associated with the persistent circulation of birds since the time lags of the outbreaks in different areas correlate with bird flight duration. (II) Population dynamics Great Britain’s native red squirrel has been displaced by a gray squirrel introduced there in the late 19th century. The areal populations across the islands can be modeled as a discrete time Markov chain with four states: R (red squirrels only), G (gray only), B (both) and N (none). The real data collected can be assembled in the transition matrix (RGBN) over one year:   0.88 0.04 0.05 0.00  0.02 0.80 0.00 0.01   P =  0.10 0.03 0.88 0.05  0.00 0.13 0.06 0.94 Note: The stochastic matrix describes the probability of moving from one state to another or staying in the same state over one time step. Thus it is positive real and its rows (or columns) sum up to 1. (a) Is the matrix ergodic? Is it regular? (b) Which species will dominate in the long run? How often would region R be repopulated? (find the mean recurrence time) Hint: Look for eigenvalue =1.