Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 20. Step processes: master equations 195
by “loading” extra energy.) We remark that there are examples of processes where states have
other interpretations, and where detailed balance is not necessarily valid.
20C
Formalism: matrix method
The various states are numbered in some way 0, 1, 2, 3… and we denote the probability for
state ‘i’ as P i . Here we use a rate terminology and speak about a transition rate k ji from
state ‘i’ to ‘j ’, rather than a transition probability. (This is only a change of terminology.)
Seen in the previous terms, with probability function P i , the probability of a transition from
‘i’ to ‘j’ per time unit is k ji P i .
Thus, there can be a network of states, connected to other states through the transition
rates. If there are paths connecting every state with all others, we have an ergodic situation.
In particular, that means that there is a well-defined equilibrium distribution, which doesn’t
change with time and which involves all states. For an ergodic system, any starting distribution
will lead to the equilibrium distribution after sufficiently long times.
Equilibrium between connected pairs means that
k P
ji
eq
i
k P
jj
eq
j
Detailed balance means that the rate constants are such that this relation is valid for all
pairs. If there are no loops, this is always possible, and when there are loops, this means
that the products of all rates going in one direction along the loop, k ij k jl k lm … is equal to the
product of rates in the opposite direction: k ml k lj k ji .
As already said, it can be possible to have one or several absorbing states, which also can
involve a group of states. By that we mean states or group of states, from which there are no
connecting transition rates to other states of the system. Formally, an absorbing state ‘j’
means that there are transition rates of type k ji leading to that state, but that the reverse rates k ij
are lacking (we may say equal to zero). There are transition possibilities that lead to these
states, and the system get stuck in these states. There can be several absorbing states or group
of states. There can be a group of states between which there are mutual transition rates, but
no transitions to states outside that group. With absorbing states, the system will eventually get
stuck in any of these, and with several such possibilities, there are several possible end states.
The probability for the final destination will depend on the initial conditions.
The master equation can then be written as:
dP
dt
i
∑
j
⎛ ⎞
kP ij j
∑kji
Pi
⎝⎜
j ⎠⎟
(20.5)
As the coefficients are constants, the solutions are simple exponential functions. A general
solution of the equation is:
Pi() t C
Et
i,
e
a
(20.6)
∑
a