A probabilistic framework for irreversible kinetics

The gist A probabilistic framework for irreversible kinetics is proposed in which a constrained path functional J encodes constitutive physics and observations on the admissible history space Had, while a discrete Gibbs-type measure proportional to exp(−J /Θ) assigns probabilities to a candidate set H ⊆ Had.

General Probabilistic Framework

The framework unifies forward-in-time evolution and inverse inference, which differ only through observations and how they constrain admissible histories<ref:2601.10763#pg4>. The development begins by associating each history h in the continuous admissible history space Had, a cost through a constrained path functional J (h) that encodes the constitutive physics and constraints imposed by available observations<ref:2601.10763#pg5>. A discrete Gibbs-type probability measure is assigned to H proportional to exp(−J /Θ), where Θ quantifies epistemic uncertainty over H, interpreted as a Bayesian posterior over histories<ref:2601.10763#pg6>. Maximizing this posterior is equivalent to simultaneous minimization of J over H, distinguishing the continuous minimizer hcont over Had from the discrete maximum-a-posteriori (MAP) history hMAP over H<ref:2601.10763#pg8>.

Generalized Standard Materials (GSM)-type Formulation

The framework is specialized using a GSM-type path action functional along with observational constraints, where the constrained path functional J is defined as A(h) + C(h)<ref:2601.10763#pg9>. The Gibbs-type measure admits a product representation for the GSM-type constrained path functional<ref:2601.10763#pg11>. The framework allows applying the same GSM-type incremental energy–dissipation functional to inverse inference when the final state is prescribed and partial or no observation is available for the initial states<ref:2601.10763#pg6>.

Relationships Among Histories

The paper establishes relationships among continuous minimizing, discrete MAP, causal incremental, and GSM histories for forward-in-time evolution<ref:2601.10763#pg9>. The continuous history hcont minimizes J over Had, while the discrete MAP history hMAP minimizes the restriction of J to H<ref:2601.10763#pg8>. In general, hcont results in the lowest J (h)<ref:2601.10763#pg9>. The causal incremental history hinc is generated by sequentially maximizing unnormalized incremental Gibbs-type factors, and it generally does not attain the lowest cost over the entire history<ref:2601.10763#pg9>.

Deterministic Kinetics in the Limit

As Θ → 0, the Gibbs-type probability concentrates on MMAP, which is independent of Θ if J is independent of Θ<ref:2601.10763#pg10>. The selected deterministic history coincides with a continuous minimizing history when Mcont ∩ H ≠ ∅<ref:2601.10763#pg9>. When minimizers are unique and the continuous minimizing history is represented in H, hMAP = hcont<ref:2601.10763#pg9>.

Inverse Inference at Finite-Θ

The inverse inference capability is demonstrated using a 1D one-mode periodic Swift-Hohenberg system with a nonconvex free energy and prescribed final state<ref:2601.10763#pg24>. The finite-Θ posterior is used to infer unobserved initial and intermediate states and quantify uncertainty over admissible histories<ref:2601.10763#pg25>. The empirical MAP history heMAP closely follows the evolution of hcont, even though small fluctuations remain because it is the lowest-action history encountered within a finite Monte Carlo sample rather than being the exact discrete MAP minimizer<ref:2601.10763#pg33>. As Θ decreases, the posterior distribution of J contracts toward its low constrained path action edge, and the posterior mean and heMAP approach the continuous stationary reference hcont<ref:2601.10763#pg35>.

Comparison of Forward-in-Time Problems

Seven representative forward-in-time problems show that when minimizers are unique and hcont is represented in H, the causal GSM history results in a higher J than the history selected by simultaneous optimization<ref:2601.10763#pg9>. For example, in linear viscoelasticity (Maxwell model), J (hMAP) = J (hcont).

Conclusion and Perspectives

The proposed framework provides a history-level inference method that ranks entire admissible paths and need not select the causal GSM history as its MAP history<ref:2601.10763#pg35>. At finite Θ, the framework provides posterior statistics over complete histories, numerically demonstrated using a Metropolis-Hastings algorithm applied to a Swift-Hohenberg non-convex energy-based inverse problem with the final state prescribed and only partial information about the initial state<ref:2601.10763#pg32>. The proposed framework admits several extensions to be explored<ref:2601.10763#pg36>.

Appendix A Analogy with GPS Routing

The analogy illustrates that probabilities are assigned to the ensemble of admissible histories through a Gibbs-type measure constructed from a prescribed constrained path functional<ref:2601.10763#pg39>. The deterministic solution is recovered as the history (or histories) with minimum path cost, while the probabilistic framework additionally quantifies the relative plausibility of all other admissible histories without introducing assumptions beyond those already encoded in the constitutive model and observational constraints.

Table 1 Simulation Parameters

The simulation parameters for the 1D one-mode Swift-Hohenberg problem are summarized in Table 1<ref:2601.10763#pg30>. The proposal acceptance rate is ∼ 0.4, which is considered good in that it avoids nearcomplete rejection and near-complete acceptance<ref:2601.10763#pg31>. The integrated autocorrelation time is estimated using τˆcorr = 1 + 2X L∗l=1 ρˆ(l), where the summation is truncated at the first L∗ for which ρˆ(L∗ + 1) ≤ 0<ref:2601.10763#pg31>. The effective sample size (ESS) is estimated as ESS = M − b τˆcorr<ref:2601.10763#pg31>.

Figure 5 Comparison

Figure 5 compares the initial guess, the 95% credible interval, the posterior mean (expectation EΘ(a n)), hcont and heMAP<ref:2601.10763#pg33>. The continuous stationary history hcont attains a lower action than heMAP<ref:2601.10763#pg33>.

Figure 8 Posterior Distributions

Figure 8 shows the posterior J distributions and inferred histories for decreasing values of Θ, illustrating the concentration of the posterior toward its low-J region as Θ decreases<ref:2601.10763#pg36>. The results demonstrate that the probabilistic framework infers multiple histories compatible with sparse endpoint information while quantifying uncertainty in the unobserved initial and intermediate states<ref:2601.10763#pg35>.

Figure 7 Empirical Phase Evolution

Figure 7 compares the initial guess, hcont, heMAP, and the posterior mean phase evolution for retained admissible histories with Θ = 0.05<ref:2601.10763#pg35>. The phase enters the constrained action only through the incremental dissipation, which penalizes changes in phase between successive time steps rather than the phase itself<ref:2601.10763#pg35>.

Figure 4 Autocorrelation

Figure 4 shows that the autocorrelations of J (h(m)) and a 10 decay to negligible values after approximately 3×10 4 local proposals iterations<ref:2601.10763#pg32>. The autocorrelation of a 0 and a N−1 decay at much lower iterations<ref:2601.10763#pg33>. The smallest estimated ESS among the representative observables is ∼ 1.28×10 4, which is still a substantial ESS for the posterior statistics presented below<ref:2601.10763#pg33>.

Figure 6 Posterior Distribution of J

Figure 6 shows the posterior distribution of J (h(m)) obtained from 4.5×10 7 retained histories at Θ = 0.05<ref:2601.10763#pg33>. The continuous stationary history hcont attains a lower action than heMAP<ref:2601.

1. Improved inference capability for unknown initial states: The framework allows inferring unobserved states and quantify uncertainty over admissible histories using a finite-Θ capability, enabling a Bayesian posterior over histories when only partial or no observation is available for the initial state, as demonstrated in Section 5.

2. History ranking and selection: The system can distinguish between different deterministic constructions by comparing them: "hMAP = hcont ≠ hinc = hGSM, J (hMAP) = J (hcont) < J (hinc) = J (hGSM), thereby allowing the framework to rank admissible histories based on their consistency with the prescribed physics and observations."

3. Uncertainty quantification over competing hypotheses: The system can quantify epistemic uncertainty through the parameter Θ, where the measure is interpreted as a Bayesian posterior over histories, enabling researchers to compare alternative low-cost histories by assigning them relative probabilities proportional to exp(−J /Θ).

4. Identification of optimal paths in non-Hamiltonian systems: The framework identifies the continuous minimizing history hcont, which is defined by the necessary conditions derived from simultaneous minimization of J over Had, providing a rigorous path beyond simple causal incremental updates when the continuous minimizing history is represented in H.

5. Robustness to deterministic approximations: The system can assess the quality of deterministic approximations by comparing them against the posterior statistics: heMAP is an approximation of the hMAP and need not coincide exactly with hMAP, allowing researchers to distinguish between a single low-action history and the bulk of the posterior mass which may be concentrated at moderately higher J values.

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 (-J/Θ), where 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 J over the admissible history space. The framework is then specialized to generalized standard materials (GSM), for which 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- J histories.