Non-equilibrium Deterministic Dynamics

• Non-equilibrium deterministic dynamics are systems governed by non-conservative ODEs that exhibit unique steady states, macroscopic currents, and chaotic attractors.
• Analytic methods from dynamical systems, statistical mechanics, and optimization enable precise characterization of phase transitions, entropy production, and stability in these complex flows.
• Applications in physical, biological, and engineered systems use coarse-graining and data-driven regression to reveal emergent macroscopic laws from underlying non-reversible dynamics.

Non-equilibrium deterministic dynamics describe the evolution of systems governed by time-continuous, non-stationary, non-gradient, or non-reversible driving forces in the absence of stochastic noise or with noise that vanishes in the macroscopic limit. Such dynamics underpin the emergence and characterization of steady states, macroscopic currents, and pattern formation in physical, biological, and engineered systems. A deterministic system is formally expressed as an ordinary differential equation (ODE) $\dot{x} = F(x)$ with $x \in \mathbb{R}^n$ and a non-conservative vector field $F(x)$. Unlike equilibrium dynamics, these flows generally cannot be represented as the negative gradient of a scalar potential and lack detailed balance, leading to complex steady-state and transient behaviors. Analytic methods drawn from dynamical systems, statistical mechanics, and optimization theory provide complementary approaches to analyzing non-equilibrium steady states, transitions, and their stability.

1. Mathematical Formulations of Non-Equilibrium Deterministic Dynamics

The prototypical form is the autonomous ODE

$\dot{x} = F(x), \quad x \in \mathbb{R}^n,$

where $F(x)$ is typically non-gradient and can embed non-reversible or dissipative structure. Fixed points $x^*$ satisfy $F(x^*) = 0$. These serve as candidate steady states, but their stability and robustness describe only a subset of possible long-time behaviors, especially when $F(x)$ is non-normal or supports cyclic/chaotic attractors.

For random high-dimensional systems—such as recurrent neural networks

$F_i(x) = -x_i + \sum_{j=1}^N J_{ij} \phi(x_j),$

steady-state analysis may be recast as a global optimization problem. Introducing a "quasi-potential" or speed-cost function,

$$E(x) = \frac{1}{2}\|F(x)\|^2 + \frac{\eta}{2}\|x\|^2,$$

allows the search for steady states to be reformulated as finding global minima of $E(x)$, with $\eta \ge 0$ a regularizer to ensure well-posedness. In the absence of gradient structure, such formulations offer a tractable approximation to the landscape of fixed points and facilitate analytic progress via optimization and statistical mechanics techniques (Qiu et al., 2024).

In projected or coarse-grained descriptions, such as for active matter or interacting particles, hydrodynamic equations at the deterministic macroscopic level are inferred from stochastic micro-dynamics. Systematic coarse-graining or data-driven regression can yield deterministic PDEs,

$$\partial_t \rho + \nabla \cdot J(\rho, \mathbf{w}, ...) = 0,$$

where the current $J$ or the evolution of velocity/polarization fields $\mathbf{w}$ may contain terms arising from both equilibrium-like and fundamentally non-equilibrium effects (Maddu et al., 2022).

2. Steady-State Structure and Stationary Distributions

A distinguishing feature of non-equilibrium deterministic systems is the existence and structure of steady states—both at the level of phase-space densities and macroscopic observables. Unlike equilibrium, where stationary distributions are given by Boltzmann measures, in the non-equilibrium case, one may construct an artificial stochastic dynamics by adding thermal noise,

$dx = -\nabla E(x)\, dt + \sqrt{2T} dW(t),$

with associated Fokker–Planck equation

$$\frac{\partial P}{\partial t} = \nabla \cdot [P \nabla E(x)] + T \Delta P,$$

such that the steady-state (SS) measure is

$P_T(x) = Z_T^{-1}\exp[-E(x)/T].$

In the zero-temperature limit $T \to 0$, $P_T$ concentrates on minima of $E(x)$, thus recovering deterministic steady states of the original ODE system (Qiu et al., 2024).

For systems that couple deterministic relaxation to large-deviation theory, the steady-state self-information $\mathcal{I}(x) = -\log P_{ss}(x)$ connects phase-space occupation with entropy production along deterministic trajectories. Remarkably, in the macroscopic (large-system) limit, the most probable paths are given by deterministic ODEs—the mean-field equations—emerging from underlying stochastic processes governed by large-deviation rate functions (Freitas et al., 2021). This forms the basis for an "emergent second law" for NESS, tightly relating non-equilibrium entropy production $\Sigma$ and the change in self-information along deterministic trajectories: $\Sigma + k_B \Delta \mathcal{I} \geq 0,$ which sharpens the conventional second law and links deterministic flows to fluctuations (Freitas et al., 2021).

3. Fluctuation Relations, Currents, and Detailed Balance

Non-equilibrium deterministic dynamics generally break detailed balance, resulting in non-zero stationary probability currents and non-trivial fluctuation relations. In extended spaces, such as non-reversible deterministic maps (e.g., the baker map), fluctuation relations for phase-space contraction rates $\Lambda$ persist even when time-reversal symmetry is lost in certain directions. The Gallavotti–Cohen relation for the scaled variable $e_n = \overline\Lambda_n/\langle\Lambda\rangle$ holds under broad conditions, implying that the structure of macroscopic fluctuation statistics is robust to microscopic irreversibility (Colangeli et al., 2011).

Transport coefficients (e.g., linear response or Green–Kubo formulas) are equally insensitive to certain classes of microscopic irreversibility, provided the projection onto relevant variables leaves the dynamics unchanged. Detailed balance in the coarse-grained (reduced) description survives as long as irreversible operations do not alter projected transition matrices, thus allowing equilibrium-like stochastic behavior to emerge from fundamentally non-equilibrium deterministic flows (Colangeli et al., 2011).

In actively driven, biophysical, or network systems, the deterministic drift can be decomposed into the gradient of a non-equilibrium potential and a curl (divergence-free) flux,

$F(x) = -D \nabla U(x) + J_{\rm curl}(x),$

with $J_{\rm curl}$ underpinning sustained macroscopic currents or probability circulation in phase-space, and vanishing only under detailed balance (equilibrium) conditions (Santo et al., 2018, Liverpool, 2018). Non-normality and reactivity in the Jacobian of $F$ generate enhanced susceptibility to noise, leading to noise-driven avalanche and burst phenomena in such systems.

4. Emergence and Learning of Deterministic Macrodynamics

Many-body stochastic dynamics can, under coarse-graining or suitable limits, be mapped to closed deterministic equations for macroscopic fields. In active matter and living systems, adaptive estimation of density, velocity, and polarization fields from trajectory data enables the construction of deterministic PDEs via group-sparse regression: $$\min_{\xi} \tfrac{1}{2}\|U_t - \Theta \xi\|_2^2 + \lambda \sum_g \|\xi_g\|_2,$$ where feature libraries encode physical symmetries and interactions. The result is a family of learned deterministic equations reproducing key phenomena (e.g., stripe formation, jet patterns), even when the underlying microscopic dynamics are far-from-equilibrium and stochastic (Maddu et al., 2022).

The deterministic limit also emerges in closed quantum systems after coarse-graining, as demonstrated in finite quantum spin ladders: the unitary Schrödinger evolution, when projected onto a coarse observable (e.g., magnetization difference), yields macroscopically deterministic and Markovian relaxation described by rate equations for a few variables, even though the underlying system is fully quantum and closed (Niemeyer et al., 2014).

5. Phase Transitions and Bifurcations in Deterministic Non-Equilibrium Systems

Non-equilibrium deterministic systems can exhibit sharp phase transitions and spontaneous symmetry breaking even under strictly measure-preserving, time-reversible dynamics. Canonical examples include billiard models with deterministic “bounce-back”—threshold rules on particle flux—which induce first-order transitions between homogeneous (zero current, symmetric) and inhomogeneous (current-carrying, symmetry-broken) steady states as a function of control parameters such as the bounce threshold $T/N$ (Cirillo et al., 2020, Cirillo et al., 2021).

Such systems display non-equilibrium phase transitions analogous to stochastic models with asymmetric transition rates, yet maintain exact time-reversibility and Liouville’s theorem at the microscopic level. The macroscopic current and phase diagram can be computed analytically via mean-field and kinetic estimates, corroborated by analogies to modified Ehrenfest urn models and large-deviation tools (Cirillo et al., 2020, Cirillo et al., 2021).

6. Chaos, Attractors, and Stability in Deterministic Non-Equilibrium Systems

The presence of non-gradient, non-equilibrium driving generically endows deterministic systems with complex dynamical behaviors, including chaos, fractal attractors, and non-ergodicity. Time-reversible thermostatted oscillator and particle models (e.g., Nosé–Hoover and hard-disk Galton-Board) maintain steady heat or momentum flows while developing attractors of zero phase-space volume (strange attractors) with Lyapunov instability and fractional correlation/information dimension (Hoover et al., 2015). In nonequilibrium steady states, the forward-time and time-reversed trajectories are fundamentally distinct: almost all probability concentrates on a forward, attracting fractal, reflecting irreversible macroscopic entropy production despite microscopically reversible laws.

Stability of deterministic non-equilibrium steady states is addressed via phase-space contraction analysis. For an average phase-space divergence $C(x)$, a sufficient criterion for global stability is $C(x)\ge0$ everywhere (Liverpool, 2018). Stronger exponential convergence bounds can be formulated if $C(x)$ is strictly positive.

7. Order Parameters, Criticality, and Edge Phenomena

Order parameters and critical phenomena in deterministic non-equilibrium systems are analytically tractable via optimization-based and replica-theoretic approaches. For random recurrent networks, the replica method applied to the quasi-potential $E(x)$ yields order parameters $q, Q$ (activity-type) and $r, R$ (response-type) that capture the continuous transition between trivial fixed-point phases and chaotic regimes (“edge of chaos”). The disorder-averaged free energy and saddle-point equations characterize this transition, with maximal response susceptibility at criticality (Qiu et al., 2024).

Quantities such as overlap matrices and their extrema encode the structure of the fixed-point manifold, linking deterministic dynamics, phase transitions, and statistical mechanics in high dimensions.

---

Key References:

• (Qiu et al., 2024): Optimization-based equilibrium measure and replica method for steady states in deterministic non-gradient neural dynamics.
• (Niemeyer et al., 2014): Macroscopically deterministic Markovian relaxation emerging from unitary quantum dynamics.
• (Maddu et al., 2022): Data-driven inference of deterministic hydrodynamic equations from stochastic micro-trajectories.
• (Colangeli et al., 2011): Fluctuation relations and robust equilibrium features for irreversible deterministic dynamics.
• (Cirillo et al., 2020, Cirillo et al., 2021): Non-equilibrium phase transitions in deterministic reversible billiard models.
• (Freitas et al., 2021): Emergent second law relating self-information to deterministic trajectories in NESS.
• (Liverpool, 2018): Generalized steady-state description combining stationary density and deterministic flow; existence and stability criteria.
• (Santo et al., 2018): Non-equilibrium potential theory, curl flux, and avalanche statistics in deterministic neural systems.
• (Xu et al., 2017): Non-equilibrium deterministic transport and collective observables in relativistic heavy-ion collisions.
• (Hoover et al., 2015): Deterministic chaos, Lyapunov spectra, and fractal attractors in nonequilibrium molecular models.