Onsager–Machlup Principle for Active Systems

• Onsager–Machlup theory for active systems extends classical variational formulations by introducing an active Rayleighian that incorporates mechanical power injection.
• The framework enforces covariance and uses generalized coordinates to ensure that dissipation, free-energy rates, and active power transform consistently under smooth coordinate changes.
• It unites deterministic path minimization and stochastic path weighting for applications such as active Brownian particles, revealing nonreciprocal and odd elastic behaviors.

Searching arXiv for the core papers and closely related work on Onsager–Machlup formulations for active and nonreciprocal systems. Search result 1: "Covariant Onsager and Onsager-Machlup principles for active and inertial dynamics" (Yasuda et al., 24 Apr 2026) Search result 2: "Onsager's variational principle for nonreciprocal systems with odd elasticity" (Lin et al., 2022) Search result 3: "Neural optimization of the most probable paths of 3D active Brownian particles" (Zheng et al., 20 Nov 2025) Search result 4: "Statistical formulation of Onsager-Machlup variational principle" (Yasuda et al., 2024) Search result 5: "Non-equilibrium Onsager-Machlup theory" (Peredo-Ortiz et al., 2023) The Onsager–Machlup principle for active systems denotes a path-variational formulation in which actively driven mesoscopic trajectories are weighted by an action built from an active Rayleighian, rather than from purely passive relaxational dynamics. In its most direct current formulation, activity enters through the mechanical power term $\dot W=X_i(\mathbf q)\dot q^i$ in

$$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$

with deterministic drift obtained by minimizing $R$ and stochastic trajectory weights assigned through the excess Rayleighian $L=R-R_{\min}$ and the path probability

$$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$

This construction extends classical Onsager–Machlup theory to systems with nonconservative active driving and, in the covariant formulation, keeps the theory geometrically consistent under smooth coordinate transformations (Yasuda et al., 24 Apr 2026).

1. Formal setting and departure from passive Onsager theory

Classical Onsager theory treats irreversible dynamics as the balance between a free-energy gradient and linear dissipation. That structure is insufficient for active matter because active systems are driven out of equilibrium by forces or propulsion mechanisms that cannot be reduced to a free-energy gradient. In the active formulation, the mesoscopic state is described by generalized coordinates

$\mathbf q=(q^1,q^2,\dots),$

equipped with a metric $g_{ij}(\mathbf q)$, and thermodynamic quantities are required to be scalars on the reduced state space. A key definition is the mesoscopic entropy

$$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$

where $\rho(\mathbf q)$ is the scalar probability density and $P(\mathbf q)$ the coordinate PDF. This yields a free energy $$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$0 that is itself a scalar under point transformations $$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$1 (Yasuda et al., 24 Apr 2026).

The requirement of covariance is not ornamental. In reduced descriptions of soft and active matter, the relevant variables are often curvilinear or collective coordinates, such as angles, shape modes, surface coordinates, or defect coordinates. A noncovariant formulation can depend spuriously on the chosen parametrization. The covariant active Onsager–Machlup construction therefore insists that the dissipation function, free-energy rate, active power, and resulting equations all transform consistently under smooth one-to-one coordinate changes (Yasuda et al., 24 Apr 2026).

A complementary non-equilibrium extension shifts emphasis away from equilibrium and toward stationarity. In that formulation, the essential generalization is from a globally stationary Ornstein–Uhlenbeck process to a globally non-stationary but locally stationary, piecewise-stationary stochastic process. The authors explicitly argue that stationarity, not equilibrium, is the essential mathematical requirement behind the generalized Langevin structure. This does not itself yield an active-matter theory, but it clarifies that equilibrium is not the only admissible reference structure for Onsager–Machlup-type reasoning (Peredo-Ortiz et al., 2023).

2. Active Rayleighian and deterministic phenomenological equations

The deterministic active Onsager principle is formulated through the Rayleighian

$$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$2

where the dissipation function is quadratic,

$$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$3

with $$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$4 symmetric and positive-definite, the free-energy rate is

$$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$5

and the active or external power is

$$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$6

The physical velocity is selected by minimizing $$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$7 with respect to $$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$8 at fixed $$R(\mathbf q,\dot{\mathbf q})=\Phi(\mathbf q,\dot{\mathbf q})+\dot F(\mathbf q,\dot{\mathbf q})-\dot W(\mathbf q,\dot{\mathbf q}),$$9,

$R$0

which gives the active phenomenological equation

$R$1

In the passive limit $R$2, this reduces to the usual relaxational balance between dissipation and thermodynamic force (Yasuda et al., 24 Apr 2026).

This extension is conceptually minimal but structurally decisive. Activity enters linearly as power injection into the Rayleighian, rather than as an effective modification of the free energy. The deterministic active drift is therefore not a gradient flow in general. A plausible implication is that the active Onsager equation preserves the variational architecture of passive soft-matter theory while abandoning the equilibrium identification of drift with free-energy descent.

The same logic can be enlarged to inertial dynamics. The generalized Rayleighian becomes

$R$3

with kinetic energy

$R$4

and covariant acceleration

$R$5

Minimization must then be carried out at fixed $R$6 and fixed covariant acceleration $R$7, which yields

$R$8

The emphasis on $R$9, rather than $L=R-R_{\min}$0, is the paper’s criterion for covariance in the inertial sector (Yasuda et al., 24 Apr 2026).

3. Stochastic path weight, most probable paths, and fluctuation theorems

Thermal fluctuations are incorporated by defining the Onsager–Machlup Lagrangian as the excess Rayleighian above its local minimum,

$L=R-R_{\min}$1

so that $L=R-R_{\min}$2 on the deterministic active drift. The Onsager–Machlup functional is then

$L=R-R_{\min}$3

and the conditioned path probability is postulated as

$L=R-R_{\min}$4

The most probable trajectory is the minimizer of $L=R-R_{\min}$5, equivalently the extremal of the Euler–Lagrange equation associated with the scalar Lagrangian $L=R-R_{\min}$6 (Yasuda et al., 24 Apr 2026).

A central consistency condition is the detailed fluctuation theorem. Defining entropy production by

$L=R-R_{\min}$7

the active Onsager–Machlup construction yields

$L=R-R_{\min}$8

The same expression is obtained from thermodynamic bookkeeping, so the path-probability formulation is explicitly consistent with stochastic thermodynamics. In this sense, active dynamics break equilibrium detailed balance without invalidating the fluctuation-theorem structure of forward and backward path weights (Yasuda et al., 24 Apr 2026).

For non-reciprocal active systems, a thermodynamically consistent Markov framework based on microscopic local detailed balance provides closely related ingredients even though it does not print a final Onsager–Machlup functional in the standard diffusion form. At the coarse-grained level, the deterministic evolution is identified as the “most likelihood path” of the Doi–Peliti action, namely the “Instanton,” and the mean entropy production decomposes into reciprocal, non-reciprocal, chemical, and self-propulsion sectors. In particular, the non-reciprocal contribution takes the form

$L=R-R_{\min}$9

so dissipation is directly tied to vorticity currents in macrostate space (Mohite et al., 10 Apr 2025). This suggests that, in active non-reciprocal matter, the time-antisymmetric part of an Onsager–Machlup action is naturally organized by circulating currents rather than by free-energy relaxation alone.

4. Hidden active variables, nonreciprocity, and odd elasticity

A distinct but closely related line of work adapts Onsager’s variational principle to active systems with odd elasticity. That formulation is deterministic, overdamped, and coarse-grained; it does not formulate stochastic path weights or an Onsager–Machlup action. Its relevance lies in showing how a hidden actively driven coordinate can generate nonreciprocal effective dynamics after elimination (Lin et al., 2022).

In the finite-dimensional toy model, the potential energy is

$$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$0

with $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$1 the particle displacement and $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$2 the angular coordinate of a driven disk. The dissipation function is

$$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$3

and the Rayleighian is $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$4. Minimization with respect to $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$5 yields coupled deterministic equations, and eliminating $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$6 produces an effective friction matrix $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$7 that remains symmetric together with an elastic matrix $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$8 whose off-diagonal entries are antisymmetric and proportional to the nonequilibrium drive $$\rho[\mathbf q(t)\mid \mathbf q_0]=N\exp\!\bigl(-O[\mathbf q(t)]\bigr),\qquad O[\mathbf q(t)]=\frac{1}{2k_{\rm B}T}\int_0^\tau dt\,L(\mathbf q(t),\dot{\mathbf q}(t)).$$9. In the small-displacement limit,

$\mathbf q=(q^1,q^2,\dots),$0

so the odd elastic part is

$\mathbf q=(q^1,q^2,\dots),$1

The odd coupling is therefore proportional to the nonequilibrium force $\mathbf q=(q^1,q^2,\dots),$2 and the ratio of friction coefficients $\mathbf q=(q^1,q^2,\dots),$3 (Lin et al., 2022).

The continuum construction makes the same point at field level. With displacement field $\mathbf q=(q^1,q^2,\dots),$4, hidden active variable $\mathbf q=(q^1,q^2,\dots),$5, and deformation basis

$\mathbf q=(q^1,q^2,\dots),$6

the free energy includes the active term $\mathbf q=(q^1,q^2,\dots),$7, the dissipation function contains quadratic couplings between $\mathbf q=(q^1,q^2,\dots),$8 and strain-rate sectors, and in the small-strain limit one obtains odd elastic moduli

$\mathbf q=(q^1,q^2,\dots),$9

The equilibrium limit $g_{ij}(\mathbf q)$0 removes these antisymmetric moduli and restores reciprocal passive dynamics (Lin et al., 2022).

The paper explicitly states that it does not discuss Onsager–Machlup path probabilities, stochastic action functionals, large deviations, or path integrals. Conceptually, it sits closer to the deterministic drift structure that a stochastic Onsager–Machlup theory would use. This distinction is important because “Onsager–Machlup principle for active systems” is sometimes used loosely to refer to any active Onsagerian formulation, whereas the odd-elasticity construction remains a deterministic Onsager variational framework rather than a path-probability theory.

5. Active Brownian particles and explicit most-probable-path computations

A direct application of the active Onsager–Machlup principle is the 3D active Brownian particle. The particle position is $g_{ij}(\mathbf q)$1, the self-propulsion velocity is $g_{ij}(\mathbf q)$2, and the orientation unit vector is parametrized by spherical angles,

$g_{ij}(\mathbf q)$3

The angular velocity satisfies $g_{ij}(\mathbf q)$4 with $g_{ij}(\mathbf q)$5, hence

$g_{ij}(\mathbf q)$6

For a free active Brownian particle the Rayleighian is

$g_{ij}(\mathbf q)$7

with

$g_{ij}(\mathbf q)$8

Minimization gives the deterministic active path

$g_{ij}(\mathbf q)$9

and

$$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$0

The Onsager–Machlup integral therefore becomes

$$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$1

with path probability

$$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$2

The most probable path is the minimizer of $$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$3 under the Dirichlet boundary conditions $$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$4, $$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$5, $$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$6, and $$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$7 (Zheng et al., 20 Nov 2025).

In dimensionless variables,

$$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$8

the action is

$$S_{\mathrm{mes}}(\mathbf q)=-k_{\rm B}\ln \rho(\mathbf q) =-k_{\rm B}\ln\!\left[\frac{P(\mathbf q)}{\sqrt{g(\mathbf q)}}\right],$$9

The Euler–Lagrange equations are nonlinear coupled boundary-value equations, so the paper minimizes the action directly with a neural-network ansatz. The resulting most probable paths exhibit geometric transitions between in-plane I-paths, planar U-paths, fully 3D H-paths, and, for winding boundary conditions, planar $\rho(\mathbf q)$0-paths. The transition structure is diagnosed by the curvature and torsion,

$\rho(\mathbf q)$1

and by direct comparison of the Onsager–Machlup integral (Zheng et al., 20 Nov 2025).

The same work also gives the thermal-bath entropy change through local detailed balance,

$\rho(\mathbf q)$2

with

$\rho(\mathbf q)$3

For the 3D active Brownian particle,

$\rho(\mathbf q)$4

This makes the path action, the most-probable-path problem, and the irreversibility measure part of the same active Onsager–Machlup structure (Zheng et al., 20 Nov 2025).

6. Broader generalizations, open directions, and current limitations

A broader non-equilibrium Onsager–Machlup theory has been proposed in which the classical globally stationary Ornstein–Uhlenbeck backbone is replaced by a globally non-stationary but locally stationary, piecewise-stationary stochastic process. The mean state evolves according to

$\rho(\mathbf q)$5

while fluctuations around the evolving mean obey, at fixed $\rho(\mathbf q)$6, a stationary non-Markov equation in the fluctuation time $\rho(\mathbf q)$7,

$\rho(\mathbf q)$8

The framework allows memory, additive noise, local fluctuation-dissipation structure, and a Lyapunov function $\rho(\mathbf q)$9, identified in thermodynamic applications with $P(\mathbf q)$0. It does not provide an explicit Onsager–Machlup path action, and it does not explicitly treat active matter, but it clarifies how a locally stationary stochastic description can extend Onsager–Machlup reasoning beyond equilibrium (Peredo-Ortiz et al., 2023).

The limitations of that extension are also clear and particularly relevant for active systems. The paper itself notes that active matter often lacks a thermodynamic entropy $P(\mathbf q)$1 whose negative is a Lyapunov function; active noise is usually non-thermal; self-propulsion variables are missing; non-reciprocal couplings and irreversible circulating currents are not developed explicitly; multiplicative noise is absent; and time reversal and entropy production are not formulated pathwise (Peredo-Ortiz et al., 2023). These are not peripheral issues. They identify the precise points at which a generic non-equilibrium Onsager–Machlup scaffold ceases to be a ready-made active-matter theory.

A different statistical extension reformulates the Onsager–Machlup variational principle as a tool for cumulant generating functions. Starting from

$P(\mathbf q)$2

it introduces the modified functional

$P(\mathbf q)$3

whose maximum approximates the cumulant generating function $P(\mathbf q)$4. That construction is demonstrated for a Brownian particle in a viscous fluid and in steady shear flow, and the authors explicitly propose the active extension

$P(\mathbf q)$5

where $P(\mathbf q)$6 is the power generated by active elements due to energy input or chemical reactions. The same paper also states that the present statistical formulation “cannot deal with active non-thermal fluctuations” (Yasuda et al., 2024). This is therefore a formal route toward active Onsager–Machlup theory, but not yet a complete treatment of canonical active-noise models.

Taken together, the current literature supports a relatively precise statement. The active Onsager–Machlup principle is now explicit for thermally fluctuating active systems whose mesoscopic dynamics can be written through an active Rayleighian with direct power injection, and it has been worked out in detail for active Brownian particles (Yasuda et al., 24 Apr 2026). Deterministic Onsagerian treatments with hidden active variables show how nonreciprocity and odd elasticity can emerge by elimination, but they are not themselves stochastic Onsager–Machlup theories (Lin et al., 2022). More general non-equilibrium formulations based on local stationarity, statistical variational principles, Doi–Peliti coarse-graining, or non-reciprocal stochastic thermodynamics provide adjoining structures rather than a single universal active Onsager–Machlup formalism (Peredo-Ortiz et al., 2023). A plausible implication is that the subject is converging on a family of active Onsager–Machlup principles rather than on one unique canonical action, with the decisive distinctions set by covariance, hidden active coordinates, noise statistics, and the thermodynamic status of the driving.