Chemical Kinetics as Maximum Entropy Inference

At some point in the semester, I attempted to show that an exponential decay formula (e.g. for kinetics of unimolecular decay) could be motivated by MaxEnt reasoning, and quickly got confused trying to prove myself.

I have found that this is part of Chapter 15 in the book that I used to teach BIPH2000 this year. This is the first time I have seen this in an undergraduate textbook (or indeed any).

Here is how you do it. You are doing maximum entropy over a sample space of trajectories. Each trajectory is associated with a time-dependent probability p(t), indicating the probability that the reaction has taken place. The constraints are given as integrals over time. You find the probability distribution that maximizes the entropy under the following constraints:

1. Normalization (the integral of p(t) over all time is one)

2. Average lifetime (the integral of t*p(t) is equal to the lifetime, call it tau)

The result is a probability distribution that is exponentially decaying in time with a time constant given by the Lagrange multiplier in the optimization.

The idea of doing MaxEnt over trajectories is an idea ascribed to Jaynes (he called it "maximum caliber"): Jaynes (1980) Annu. Rev. Phys. Chem. 31. 579-600.

It has recently been revisited by e.g. Dill and Stock:

Ghosh, K., Dill, K., Inamdar, M., Seitaridou, E., & Phillips, R. (2006). Teaching the principles of statistical dynamics. American Journal of Physics, 74(2), 123–133.

Otten, M., & Stock, G. (2010). Maximum caliber inference of nonequilibrium processes. The Journal of Chemical Physics, 133(3), 034119–034118.

Stock, G., Ghosh, K., & Dill, K. A. (2008). Maximum Caliber: A variational approach applied to two-state dynamics. The Journal of Chemical Physics, 128(19), 194102. doi:10.1063/1.2918345