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We have so far always first considered the rate equation and then talked about the master equation and given heuristic arguments on how they are different. I wanted to start from the master equation and show how the rate equation arises from this. That’s exactly what we are going to do in this post.
The master equation
We can also arrive at a master equation. Now say we have a probability,
of having things in state 1, things in state etc. We write
and , then the master equation says
Each transition corresponds to an operator. There is a term
In general, the master equation is written as
We will attempt to solve , by trying
So the master equation acting on becomes
We will assume that
From which it follows that
We will make use of the compact notation for the Hamiltonian as
This becomes
but from the assumption, the second term vanishes. Hence,
and we let
which yields a solution for non-zero as
which gives a solution to the rate equation, as asserted. Creation ex nihilo, something from nothing.
Some notation
Consider the nth derivative of the product of and .
and so and become short hand for the lth and mth derivative respectively. In the case we consider here, this takes the simpler form as
And so we are left with the scalar to the mth power times the lth derivative of the Hamiltonian operator. We will evaluate this at
Yielding
Note that the term will vanish iff we count over a value where . The same is true for the term .
This is the moment
We will consider the number operator acting times on
From above we know that
and also that
For now we will consider each term separately.
When we set we arrive at
- (Result) For we recover that the mean vanishes for appropriate choice of , given that the rate equation vanishes with appropriate choice of .
It is a straight forward application of the factor theorem to show that is a factor of so we have
We know from the rate equation that
Example of multiple equilibrium states
For the general terms and definitions used in this project see
Here we consider examples from Table 1 in the following paper.
P. M. Schlosser and M. Feinberg, A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions, Chemical Engineering Science 49 (1994), 1749–1767.
- By ‘just one equilibrium state’, I presume they mean just a 1-parameter family of equilibrium states. After all, if is an equilibrium solution of the rate equation, so is for any .
The chemical reaction network
The example we consider here is from Table 1 (7), giving the chemical reaction network defined by
The input and output functions then give.
and for
The Stochastic Petri Net
The chemical rate equation
The general form of the rate equation is
The population of species () will be denoted () and the rate of the reaction () as ().
and
The master equation
The general form of the master equation is
In our case, this becomes
Proof that if the rate equation vanishes then so does the mater equation
Since the rate equation is known to vanish from the Deficiency Zero Theorem, we have that
and
- (Algebraic independence) Let
be polynomials in
are called algebraically independent if there is no non-zero polynomial
such that
then its called the annihilating polynomial.
We consider a wave function of the form
We then calculate . We find solutions where this vanishes, for non-zero as
For this to vanish, each of the terms must vanish separately. Hence we arrive at a system of equations
In this case, and in every case I’ve considered so far, a solution of one of these equations, gives a solution of the second.