Fans of the network theory series might like to look at this paper:
This paper tackles a basic question we never got around to discussing: how the probabilistic description of a system where bunches of things randomly interact and turn into other bunches of things can reduce to a deterministic description in the limit where there are lots of things!
Mathematically, such systems are given by ‘stochastic Petri nets’, or if you prefer, ‘stochastic reaction networks’. These are just two equivalent pictures of the same thing. For example, we could describe some chemical reactions using this Petri net:
but chemists would use this reaction network:
CO2 + NaOH → NaHCO3
NaHCO3 + HCl → H2O + NaCl + CO2
Making either of them ‘stochastic’ merely means that we specify a ‘rate constant’ for each reaction, saying how probable it is.
For any such system we get a ‘master equation’ describing how the probability of having any number of things of each kind changes with time. In the class I taught on this last quarter, the students and I figured out how to derive from this an equation saying how the expected number of things of each kind changes with time. Later I figured out a much slicker argument… but either way, we get this result:
Theorem. For any stochastic reaction network and any stochastic state evolving in time according to the master equation, then
assuming the derivative exists.
Of course this will make no sense yet if you haven’t been following the network theory series! But I explain all the notation in the paper, so don’t be scared. The main point is that is a vector listing the expected number of things of each kind at time The equation above says how this changes with time… but it closely resembles the ‘rate equation’, which describes the evolution of chemical systems in a deterministic way.
And indeed, the next big theorem says that the master equation actually implies the rate equation when the probability of having various numbers of things of each kind is given by a product of independent Poisson distributions. In this case is what people in quantum physics call a ‘coherent state’. So:
Theorem. Given any stochastic reaction network, let
be a mixed state evolving in time according to the master equation. If is a coherent state when then obeys the rate equation when
In most cases, this only applies exactly at one moment of time: later will cease to be a coherent state. Then we must resort to the previous theorem to see how the expected number of things of each kind changes with time.
But sometimes our state will stay coherent forever! For one case where this happens, see the companion paper, which I blogged about a little while ago:
• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks.
We wrote this first, but logically it comes after the one I just finished now!
All this material will get folded into the book I’m writing with Jacob Biamonte. There are just a few remaining loose ends that need to be tied up.