A probabilistic framework for irreversible kinetics

A probabilistic framework for irreversible kinetics is proposed in which a constrained path functional mathcal J encodes constitutive physics and observations on the admissible history space mathcal Hrm ad, while a discrete Gibbs-type measure proportional to exp(-mathcal J/Theta) assigns probabilities to a candidate set mathcal Hsubseteqmathcal Hrm ad. The framework unifies forward-in-time evolution and inverse inference, which differ only through observations and how they constrain admissible histories. The parameter Theta controls epistemic uncertainty, and the measure is interpreted as a Bayesian posterior over histories. Maximizing this posterior is equivalent to simultaneous minimization of mathcal J over mathcal H, distinguishing the continuous minimizer hrm cont over mathcal Hrm ad from the discrete maximum a posteriori (MAP) history hrm MAP over mathcal H. As Theta→0, the posterior concentrates on the discrete MAP set. For generalized standard material(GSM)-type incremental energy--dissipation functionals, seven forward-in-time examples show that, despite using the same incremental functionals, causal GSM evolution is generally only incrementally optimal. When minimizers are unique, observations are absent, and hrm cont∈mathcal H, the strict ordering mathcal J(hrm cont)=mathcal J(hrm MAP)<mathcal J(hrm GSM) holds, showing that the GSM history does not minimize the cost of the entire history. Finally, an endpoint-conditioned inverse problem with nonconvex energy demonstrates the finite-Theta capability of the framework to infer unobserved states and quantify uncertainty over admissible histories.