Informational Onsager–Machlup Principle

Updated 17 October 2025

The informational Onsager–Machlup principle is a framework that assigns a statistical weight to complete stochastic trajectories using an action functional that encodes dynamic and thermodynamic information.
It employs methods like path integrals, Girsanov transformation, and Fokker–Planck equations to analyze fluctuations across Gaussian, non-Markovian, and jump-diffusion systems.
The principle extends to feedback and constrained systems, enabling practical analysis of rare events, optimal transition paths, and the interplay between energy and information.

The informational Onsager–Machlup principle is a formalism for describing, quantifying, and predicting the likelihood of entire trajectories—rather than just instantaneous states—traced by systems governed by stochastic dynamics, especially when those systems are far from equilibrium. At its core, the principle states that the statistical weight of a history or path in configuration space is governed by an action functional, now called the Onsager–Machlup (OM) functional, which encodes both dynamic and thermodynamic information. The framework subsumes equilibrium fluctuation theory, nonequilibrium processes, systems with pathwise constraints (e.g., conservation laws, feedback, or control), and can accommodate general noise models including those with jumps and memory.

1. Historical Background and Fundamental Principle

The OM principle originated from the work of Onsager and Machlup, who in the 1950s sought to generalize Onsager’s reciprocal relations and minimum dissipation principles to properly account for the probability of fluctuations of time-dependent processes [28]. In the original setting—Gaussian, Markovian, near-equilibrium systems such as those governed by linear Langevin equations—the OM functional provided a rigorous path-probability density by upper-bounding the probability that a realization remains close (in a suitable metric) to a smooth reference trajectory over a finite time.

The essential formula, for an SDE $dx_t = f(x_t) dt + \sigma dW_t$ , prescribes the path probability as

$P[\phi(t)] \propto \exp\left( -\frac{1}{2} \int_0^T \left| \dot \phi(t) - f(\phi(t)) \right|^2 dt - \frac{1}{2} \int_0^T \mathrm{div} f(\phi(t)) dt \right)$

with the integrand defining the OM functional. This action quantifies the exponential rarity of a path as a function of its discrepancy from the “most probable” or noise-free solution.

2. Mathematical Structure: Derivation and Variants

The OM functional is derived through several methods—path integrals, change-of-measure (Girsanov transformation), or Fokker–Planck (Kolmogorov) equations. Its canonical form for continuous Itô diffusions is

$S_{OM}(\phi) = \frac{1}{2} \int_0^T \left| \frac{d\phi}{dt} - f(\phi) \right|^2 dt + \frac{1}{2} \int_0^T \mathrm{div} f(\phi(t)) dt$

and variants adjust this structure for:

Fractional (non-Markovian) noise, by replacing the Cameron–Martin norm with one induced by fBm, and including appropriate divergence corrections (Maayan, 2017).
Time-dependent or state-dependent diffusion, $g(t)$ or $g(x)$ , requiring the quadratic form and divergence to be weighted by $g^{-1}$ (Zhang et al., 5 Jul 2024).
Systems with jumps (Lévy noise), where the OM function acquires correction terms capturing the effect of finite or infinite activity, modifying the quadratic deviation as well as incorporating variance- or bias-like terms stemming from the Lévy intensity at the origin (Chao et al., 2018, Huang et al., 2 Sep 2024).

Discrete and infinite-dimensional settings use small-ball asymptotics or large deviation rate functions, often relying on measures over function spaces beyond classical Lebesgue measure (e.g., Gaussian or $\Phi^4$ measures in quantum field theory and SPDEs) (Gasteratos et al., 24 Sep 2025).

3. The Informational Principle and Extensions

The “informational” aspect refers to the connection with large deviation theory, information theory, and pathwise inference. The OM functional precisely characterizes the relative information (or action) associated with fluctuation paths: the log-probability cost for deviating from nominal or optimal trajectories is explicitly given by the OM action.

This principle now extends in several key directions:

Conditioned/feedback-constrained processes: Incorporation of memory or feedback variables (e.g., informational active matter with memory states) leads to a conditioned OM functional that integrates energetic, dissipative, and informational terms (Yasuda et al., 15 Oct 2025). The variational process—maximizing a modified OM functional including mutual information—yields cumulant generating functions and optimal paths for observables under feedback.
Pathwise variational formulations and numerical methods: The OM functional underpins advanced sampling and numerical schemes—such as OM-action-based path sampling in Fourier space, combined with replica-exchange methods for overcoming path trapping and efficiently sampling rare transitions in high-dimensional energy landscapes (Fujisaki et al., 2010); and deep learning-based variational solvers (DOMM, deep Onsager–Machlup method), which parametrically represent candidate solutions with DNNs and minimize the OM action numerically (Li et al., 2023).
Generalized metrics and geometric interpretations: The OM functional may be viewed as a generalized “density” or “rate function” on arbitrary metric measure spaces, with explicit transformation rules under reweighting of both the measure and the geometry (Selk, 12 Oct 2025). For instance, under conformal mappings or metric rescalings, the OM action shifts by an explicit function of the local scaling exponent and the reweighting potential.

4. Connection to Nonequilibrium Statistical Physics

In nonequilibrium systems, the OM principle enables both theoretical formulation and computational prediction of rare events, transition pathways, and time-asymptotic properties:

In the context of nonequilibrium transport, relaxation, or driven steady states—where boundary fluxes or constraints fix some “forces” while others are free—the OM action can be decomposed so that only “free” components enter the dissipation principle (projection onto orthogonal subspaces using the transport coefficients as a metric) (Sonnino et al., 2015).
For aging materials and glassy dynamics, the non-equilibrium Onsager–Machlup theory adopts a piecewise-stationary approximation, coupling deterministic (slow) evolution to local (fast) stationary fluctuations, and leading to self-consistent kinetic equations that recover both equilibrium and dynamical arrest as natural outcomes (Peredo-Ortiz et al., 2023).

In rare event analysis (transition path theory, metastable transitions), the OM functional serves as a large deviation rate function: the probability of observing a fluctuation tube around a candidate trajectory decays exponentially with the OM action ( $P \sim e^{-S_{OM}/\epsilon^2}$ ). The most probable (i.e., minimum action) path solves a variational problem, formally analogous to classical mechanics, but with the OM functional as the effective “Lagrangian”.

5. Recent Developments: Lévy Noise, Degenerate Systems, and SLEs

Jump-diffusion and Lévy noise: For SDEs with jumps, the OM functional acquires explicit terms involving the Lévy intensity near the origin, and the functional may only admit a time-discrete form in the case of infinite activity. The probability flow approach overcomes technical obstacles by relating the jump-diffusion to an equivalent diffusion with adjusted drift (Huang et al., 2 Sep 2024). In degenerate cases (e.g., kinetic Langevin systems with noise only in some variables), the OM action is characterized via constrained variational problems, often requiring the Hamilton–Pontryagin principle to account for kinetic constraints (Chao et al., 9 Jan 2025).
Hamiltonian systems and KAM theory: For stochastic Hamiltonian systems, the OM functional is constructed over position-momentum space, providing the large deviation action for rare deviations from quasi-periodic motion; classical KAM theory ensures that, in the most probable sense, invariant tori remain stable under small noise, with the exponential decay rate for deviations given by the OM action (Zhang et al., 18 Mar 2025).
Stochastic Loewner Evolution (SLE) and random geometry: In SLE and its generalizations, the OM functional aligns with the so-called Loewner energy, quantifying the probability (at leading exponential order) that a random curve lies within a small neighborhood of a prescribed trace. This identification holds for multi-chordal and multi-radial SLEs, with explicit conformal deformation formulas, enabling a large deviation/OM perspective on random conformally invariant curves and their field-theoretic partition functions (Fan, 9 Aug 2025).

6. Physical and Conceptual Implications

The informational Onsager–Machlup principle unifies variational, stochastic, and informational views of nonequilibrium dynamics:

It provides a pathwise generalization of fluctuation-dissipation relations, encompassing both energy and entropy flows and, in feedback systems, directly quantifies the role of information via mutual OM functionals (Yasuda et al., 15 Oct 2025).
The variational structure naturally yields cumulant generating functions and higher-order statistics for observables, connecting thermodynamic quantities (such as entropy production or work) to optimal transitions and rare events.
In active or feedback-driven systems, it rigorously describes how information processing can be harnessed to sustain persistent nonequilibrium states, providing a foundation for informational thermodynamics and the design of feedback-powered engines.

7. Table: Key Formulas and Domains

OM Functional/Class	SDE/Noise Model	OM Action/Rate Function
Classical Itô Diffusion	$dx_t=f(x_t)dt+\sigma dW_t$	$\frac{1}{2} \int (\dot x - f(x))^2 dt + \frac{1}{2}\int \text{div} f(x) dt$
Fractional/Non-Markov	fBm, Hurst $H<1/2$	$-\frac{1}{2}(\\|h-b(h)\\|_{\mathcal{H}}^2 + \int \text{div} b(h) dt)$
Jump-Diffusion	Brownian + Lévy	$\frac{1}{2c^2}[\dot z - f(z) + \int_{\|ξ\|<1} ξ ν(dξ)]^2 + f'(z) - (\int_{\|ξ\|<1} ξ ν(dξ)/c)^2$
Degenerate SDE	Degenerate noise	$½\|[\dot φ_2 - f(φ_1,φ_2)+\int_{\|ξ\|<1} ξ ν(dξ)]/c\|^2 + ½ ∂_y f(φ_1, φ_2)$ (Chao et al., 9 Jan 2025)
SLE/Random Geometry	Chordal/Multi-chordal SLE	$OM = c(κ)/2 [ H(γ) - H(γ_0)] - F(γ)$ (difference in Loewner potentials plus correction)

References

Onsager–Machlup action-based path sampling (Fujisaki et al., 2010)
Minimum dissipation and projection methods (Sonnino et al., 2015)
Fractional noise and Cameron–Martin norm (Maayan, 2017)
Jump-diffusion OM functionals (Chao et al., 2018, Huang et al., 2 Sep 2024)
Deep Onsager–Machlup methods (Li et al., 2023)
Degenerate dynamical systems and Hamilton–Pontryagin principle (Chao et al., 9 Jan 2025)
Stochastic Hamiltonian OM functionals and KAM persistence (Zhang et al., 18 Mar 2025)
SLE Loewner energy and OM interpretation (Fan, 9 Aug 2025)
Generalized OM in metric measure spaces (Selk, 12 Oct 2025)
Feedback/information in thermodynamics (Yasuda et al., 15 Oct 2025)

The informational Onsager–Machlup principle thus serves as a foundational tool in both the theoretical and computational analysis of nonequilibrium and path-dependent statistical phenomena, encompassing classical thermodynamics, modern stochastic analysis, and the emerging interface with information theory, control, and active matter.