A probabilistic framework for irreversible kinetics

A probabilistic framework for irreversible kinetics that unifies forward-in-time evolution and inverse inference, while quantifying epistemic uncertainty over admissible histories, is proposed. The forward-in-time evolution and inverse inference formulations differ only through observations and how they constrain admissible histories. All histories are ranked using a Gibbs-type probability measure $\exp(-\mathcal{J}/Θ)$, where $\mathcal{J}$ encodes constitutive physics and observations. $Θ$ controls epistemic uncertainty over the unknown histories, and the measure is interpreted as a Bayesian posterior over histories. Maximizing this posterior is equivalent to simultaneous minimization of $\mathcal{J}$ over the admissible history space. The framework is then specialized to generalized standard materials (GSM), for which $\mathcal{J}$ is constructed from incremental energy--dissipation functionals. This specialization reveals an important difference between the history resulting from simultaneous minimization over entire histories generally differs from the one generated by sequential GSM updates. Specifically, the causal GSM history incurs a higher history-level cost than the one obtained by simultaneous minimization, indicating that enforcing causality carries an additional cost at the history level. This difference is shown through seven representative examples: viscoelasticity, viscoplasticity, rate-independent plasticity, gradient damage, conserved and non-conserved phase transformations, and coupled thermoelectric evolution. Finally, the finite-$Θ$ formulation is demonstrated on an inverse problem with a non-convex free energy, where incomplete initial-state information is used to infer unobserved intermediate states and quantify uncertainty over admissible histories. As $Θ$ decreases, the posterior concentrates towards minimum-$\mathcal{J}$ histories.