# Maximum Caliber

We show how to apply a general theoretical approach to nonequilibrium statistical mechanics, called Maximum Caliber,
originally suggested by E. T. Jaynes 30 years ago, to a problem of two-state dynamics. Maximum Caliber is a variational
principle for dynamics in the same spirit that Maximum Entropy is a variational principle for equilibrium statistical
mechanics. The central idea is to compute a dynamical partition function, a sum of weights over all microscopic paths,
rather than over microstates. We illustrate the method on the simple problem of two-state dynamics, A ↔ B, first
for a single particle, then for M articles. Maximum Caliber gives a unified framework for deriving all the relevant
dynamical properties, including the microtrajectories and all the moments of the time-dependent probability density.
While it can readily be used to derive the traditional master equation and the Langevin results, it goes beyond them in
also giving trajectory information. For example, we derive the Langevin noise distribution, rather than assuming it.
As a general approach to solving nonequilibrium statistical mechanics dynamical problems, Maximum Caliber has some advantages:

- It is partition-function-based, so we can draw insights from similarities to equilibrium statistical mechanics,
- it is trajectory-based, so it gives more dynamical information than population-based approaches like master equations; this is particularly important for few-particle and single-molecule systems,
- it gives an unambiguous way to relate flows to forces, which has traditionally posed challenges, and
- like Maximum Entropy, it may be useful for data analysis, specifically for time-dependent phenomena.

Graphical representation of the dynamical partition function of a two-state system, showing all possible trajectories
for N=3 time steps for a system stating in A.

Maximum Caliber: A variational approach applied to two-state dynamics,
J. Chem. Phys. 128, 194102-194102-12 (2008)

Maximum caliber inference of nonequilibrium processes,
J. Chem. Phys. 133, 034119 (2010)

Inferring Transition Rates of Networks from Populations in Continuous-Time Markov Processes,
J. Chem. Theory Comput. 11, 5464 (2015)