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 H_{\rm ad}, while a discrete Gibbs-type measure proportional to \exp(-\mathcal J/Θ) assigns probabilities to a candidate set \mathcal H\subseteq\mathcal H_{\rm ad}. The framework unifies forward-in-time evolution and inverse inference, which differ only through observations and how they constrain admissible histories. The parameter Θ 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 h_{\rm cont} over \mathcal H_{\rm ad} from the discrete maximum a posteriori (MAP) history h_{\rm MAP} over \mathcal H. As Θ\to0, 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 h_{\rm cont}\in\mathcal H, the strict ordering \mathcal J(h_{\rm cont})=\mathcal J(h_{\rm MAP})<\mathcal J(h_{\rm 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-Θ capability of the framework to infer unobserved states and quantify uncertainty over admissible histories.