The nonequilibrium statistical mechanics of Markov interacting particles

Abstract

We consider coupled stochastic systems decomposed into exterior, boundary, and interior variables, with the boundary variables sometimes carrying the directed structure of a sensor and actuator. The central question is when the conditional law of histories factorises, and how this path space statement is detected by log likelihoods, by Girsanov changes of measure, and by information theoretic quantities used in nonequilibrium statistical physics. The basic object is a regular conditional probability on a path space. Under domination by clamped reference laws, the boundary property becomes multiplicative separation of a Radon--Nikodym derivative, or equivalently additive separation of a path log likelihood. For It\=o diffusions this log likelihood is computed by Girsanov's theorem; its expectation is the quadratic control energy appearing in the F\"ollmer entropy identity and in the stochastic control formulation of Schr\"odinger bridge problems. When exact factorisation fails, the remaining coupling is measured by conditional mutual information, namely the relative entropy between the true boundary-conditioned path law and the product of its conditional marginals. This gives a common language for boundary screening, path likelihood inference, controlled changes of path law, and the thermodynamic value of mutual information.