Arnold Sommerfeld Center for Theoretical Physics
Ludwig-Maximilians-Universität München
Prof. Dr. U. Gerland
L. Reese, E. Bakker, P. Hillenbrand
Stochastic Processes in Physics and Biology– Exercise 4
Wintersemester 2010/2011
Deadline: 17.11.2010 at the end of the lecture
1. Linear Birth-Death Process
The linear birth-death process is given by the following reactions
µ
λ
A −→ ∅ and A −→ A + A.
The process describes birth and death of a single species with different rates and has a huge range of
applications not just in biology but also in other branches of statisics such as in e.g. queueing theory,
demography, and biochemical reactions . This can be modeled by a Master equation, which is of the
following form
dpn (t)
= µ[(n − 1)pn−1 (t) − npn (t)] + λ[(n + 1)pn+1 (t) − npn (t)].
dt
a) Determine the time evolution of hn(t)i where n is the number of particles and n0 the number
of particles at t = 0. Compare your result with the rate equation.
b) Derive the differential equation of hn2 (t)i with the initial condition n20 at time t = 0. Solve the
differential equation by using the variation of constants method.
c) How does the variance evolve in time?
2. A Different Birth-Death Process
We now take a different birth death process, this time defined by:
λ
∅ −→ A
µ
and A −→ ∅.
This process is more relevant to the modelling the number of proteins produced by and activated gene,
since the birth rate is no longer dependent on the number of the species present, though the death rate
still is.
a) Convince yourself that the masters equation for this process is:
dpn (t)
= λ[pn−1 (t) − pn (t)] + µ[(n + 1)pn+1 (t) − npn (t)].
dt
b) At which value of n would the distribution pn = 1 be a stationary distribution?(i.e. not changing
in time). What is a possible problem with this distribution physically?
We now introduce the generating function (thisP
will be encountered later in the lecture also).
The generating function is defined as G(s, t) = pn (t)sn . Convince yourself of the following properties.
G(1, t) = 1
dG(s, t)
=< n >
ds s=1
c) Find an equation for the generating function of this distribution
hint: think about summing both sides of the equation
d) Calculate the generating fuction in the case of a stationary distribution
(note: this is not the distribution stationary distribution above)
e) Calculate the stationary distribution exactly from this generating function
3. Random Walker Revisited
We have already encountered the random walker on exercise sheet 1. This time, again, we consider a
random walker with different probability of stepping to the right or to the left, yet we want to find a
solution of the full dynamics in terms of the time evolution of the probability. To this end we solve the
Master-equation analytically.
a) The random walker moves along an infinitly long chain with transition rates p and q stepping
to the right and to the left, respectively. The initial position of the random walker is n0 and
therefore pn (0) = δn,n0 . Derive the Master-equation for an infinitely long chain.
b) The Master-equation can again be solved by using the generating function. Deduce and solve
the differential equation for the generating function with the appropriate intitial condition.
c) Calculate pn (t) from the expression of the generating function.
d) Determine the variance σn2 (t).