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

Chapter 20. Step processes: master equations 197
The notation should be clear: ‘i’ and ‘j’ designate the states, a the particular eigenvalue and
corresponding particular solution of the differential equation system.
For non-symmetric matrices (and most of our models are of that type), we also have to
consider left hand eigenvectors z a I such that:
∑
j
zaM ( E ) z
j
ji
a
a
i
The y and z eigenvectors are orthogonal to each other, which means:
∑
j
b
z ya 0 if a≠
b
j
j
They can always be normalised so that:
za
j ya
∑ j 1
j
Next, consider the formal solution presented above:
P( t) Ca
exp( E t)
i
∑
a
i
a
This can be expressed with the y eigenvectors:
P( t) C y exp( E t)
i
=∑ a i
a
a
The C-coefficients in this expression are independent of the states and denote the relative
weights of the particular solutions. When t 0, this becomes:
P( 0)
Caya
i
∑
a
i
This leads to the initial value problem. We assume that the values of the probabilities, P i ,
are known at the onset when t 0. This may mean that the system is in a particular state,