A probabilistic framework for irreversible kinetics

Abstract

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 $\exp(-\mathcal J/\Theta)$ 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 $\Theta$ 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 $h_{\rm cont}$ over $\mathcal H_{\rm ad}$ from the discrete maximum a posteriori (MAP) history $h_{\rm MAP}$ over $\mathcal H$. As $\Theta\to0$, 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-$\Theta$ capability of the framework to infer unobserved states and quantify uncertainty over admissible histories.