2026/01/31 by Manas V. Upadhyay
Physics and Astronomy · Mathematics · #cond-mat.stat-mech #stat.ML #msc:74A20 #msc:74D10
39 pages, 8 figures. 6 sections. 1 Appendix. Probabilistic formulation of irreversible kinetics
arxiv created 2026/08/03 · arxiv updated 2026/08/04
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⊆\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→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 h\rm cont∈\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.