InteractAvatar and Non-equilibrium Dynamics

• InteractAvatar is a framework for analyzing deterministic non-equilibrium dynamics, defined by nonlinear rules that yield irreversible macroscopic behavior from microscopically reversible systems.
• The approach employs optimization-based formulations, recasting steady-state search as a landscape minimization problem while integrating replica methods to handle disorder.
• It demonstrates how non-gradient dynamics can produce sustained currents and phase transitions without external noise, offering practical insights for computational modeling and theoretical physics.

Non-equilibrium deterministic dynamics encompasses a wide class of systems whose time evolution is governed by nonlinear, often non-gradient, deterministic rules and which exhibit persistent macroscopic irreversibility, sustained currents, phase transitions, or emergent regularities without recourse to external noise or stochastic driving. Although these systems may be microscopically reversible and conserve phase-space volume, their macroscopic or coarse-grained dynamics can show irreversible relaxation, symmetry breaking, long-lived steady currents, and nontrivial critical phenomena. The field bridges statistical mechanics, dynamical systems theory, condensed matter, nonequilibrium thermodynamics, and computational modeling.

1. Foundational Definitions and General Structure

Non-equilibrium deterministic dynamics typically refers to the evolution of high-dimensional systems according to an ordinary differential equation (ODE) of the form

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

where the vector field $F(x)$ is not generally the gradient of a potential; i.e., the dynamics is "non-gradient" or lacks detailed balance. The steady states—fixed points defined by $F(x^*)=0$—may be isolated or form a manifold. In such systems, analytic characterization of steady states, time-dependent behavior, or stability properties is challenging due to the absence of a scalar Lyapunov function or global potential. The regime of interest typically involves complex interactions, disorder, or constraints that break equilibrium-like behavior and produce macroscopic irreversibility or currents, despite microscopic determinism (Qiu et al., 2024, Liverpool, 2018, Cirillo et al., 2021, Cirillo et al., 2020).

The study of non-equilibrium deterministic dynamics includes both classical and quantum systems, covers a range of deterministic thermostats, multi-component conservative flows, interacting particle systems, and statistical descriptions of driven high-dimensional systems.

2. Optimization-Based and Replica Approaches

An important recent development is the recasting of steady-state search as an optimization problem via a "quasi-potential" or "speed-cost" functional: $$E(x) \equiv \frac{1}{2} \|F(x)\|^2 + \frac{\eta}{2} \|x\|^2$$ with $\eta \geq 0$ a small regularizer. The global minima of $E(x)$ correspond precisely to deterministic steady states ($F(x^*)=0$), with the minimization problem well-posed even for non-gradient $F$. This formulation enables the reinterpretation of steady-state analysis in terms of energy landscapes and opens analytical and computational tools otherwise reserved for equilibrium statistical mechanics (Qiu et al., 2024).

In systems with quenched disorder (e.g., random couplings in recurrent neural networks), the replica method is employed to average over disorder and compute macroscopic order parameters characterizing non-equilibrium steady states. This leads to a set of saddle-point equations involving overlap matrices (activity-like and response-like order parameters) which fully characterize the statistical structure and stability of the steady-state landscape, elucidating critical points such as the "edge of chaos." Notably, the transition from a trivial fixed-point phase to a chaotic phase is quantitatively captured by the nonzero emergence of these order parameters at the critical coupling (Qiu et al., 2024).

3. Mechanisms for Irreversibility and Steady-State Currents

Non-equilibrium deterministic systems can display steady irreversible currents and phase transitions without stochasticity or explicit phase-space contraction. For instance, deterministic billiard models with local "bounce-back" rules—where particle motion in a channel is reversed once a threshold is exceeded—yield non-equilibrium steady states with symmetry-broken particle distributions, stationary (uphill) currents, and first-order–like phase transitions in the $N\rightarrow\infty$ limit (Cirillo et al., 2020, Cirillo et al., 2021). Even though the underlying microdynamics is time-reversible and preserves global phase-space volume, macroscopic irreversibility and entropy production emerge at the coarse-grained level.

Analogous mechanisms appear in quantum spin systems, where unitary evolution within a closed Hilbert space leads to deterministic, autonomous relaxation of coarse-grained observables (e.g., magnetization differences) to a stationary profile, effectively described by Markovian master equations and independent of initial condition details—a form of emergent thermodynamic irreversibility (Niemeyer et al., 2014).

4. Connections with Stochastic and Hydrodynamic Limits

A central motif is the emergence of deterministic macrodynamics as the "typical flow" or average trajectory of an underlying stochastic process in the thermodynamic limit. Starting from stochastic dynamics, such as autonomous Markov jump processes or Langevin descriptions with non-gradient drift, large deviation theory provides a bridge: the steady-state distribution concentrates on the deterministic trajectory solving the mean-field ODE as fluctuations become negligible at large system size. This leads to the formal connection between macroscopic entropy production, self-information of steady states, and deterministic entropy bounds (emergent second law) (Freitas et al., 2021, Liverpool, 2018).

In active matter and collective biological systems, empirical model discovery frameworks infer deterministic continuum hydrodynamic equations (PDEs) directly from stochastic particle-level simulations. This demonstrates how non-equilibrium deterministic PDEs governing density, velocity, or polarization fields arise from coarse-graining microscopic, noise-driven agents, with validation against direct particle simulations confirming faithfulness of the inferred deterministic laws (Maddu et al., 2022).

5. Detailed Balance, Fluctuation Relations, and Linear Response

Deterministic non-equilibrium systems may or may not exhibit detailed balance at the microscopic (full phase-space) level. However, upon projection to "relevant" degrees of freedom—coarse-grained variables capturing observable thermodynamic quantities—detailed balance or its stochastic equivalent can be restored, and transport coefficients or fluctuation relations remain valid. The Gallavotti–Cohen fluctuation theorem and Green–Kubo response formulas hold under minimal reversibility assumptions; full phase-space reversibility is not strictly necessary as long as the projected observables satisfy the required symmetry (Colangeli et al., 2011). This robustness underscores the generality of non-equilibrium thermodynamics and its insensitivity to detailed microdynamics for a suitable choice of observables.

6. Critical Phenomena and Dynamical Phase Transitions

Deterministic non-equilibrium dynamics supports transitions that are structurally distinct from equilibrium counterparts. Both first-order–like (discontinuous jump of an order parameter) and second-order (continuous emergence of macroscopic activity or response) phase transitions are observed. Examples include:

• The spontaneous symmetry breaking and current-carrying states in billiard models under threshold feedback (Cirillo et al., 2020, Cirillo et al., 2021).
• The continuous onset of chaotic activity in high-dimensional neural network models as a function of control parameters (e.g., coupling strength), captured by the nonvanishing replica order parameters (Qiu et al., 2024).
• Nonequilibrium avalanche phenomenology in systems with non-normal drift structures and intrinsic noise, where deterministic geometry (e.g., reactiveness, nonlinearity in the vector field) governs the scaling of rare macroscopic events (Santo et al., 2018).

Deterministic, time-reversible thermostatted oscillator models (e.g., Nosé–Hoover–Holian, single-friction multi-moment controls) reveal, through their Lyapunov spectra and phase-space contraction, the fractal structure of non-equilibrium steady states and the emergence of macroscopic irreversibility from microscopic reversible rules (Hoover et al., 2015).

7. Analytical, Computational, and Data-Driven Methodologies

Recent methodologies leverage optimization-based formulations to map deterministic steady-state search onto global landscape minimization, facilitate disorder averaging via the replica trick, and deploy group-sparse regression and dictionary learning for PDE discovery in high-dimensional, noisy data contexts (Qiu et al., 2024, Maddu et al., 2022). The hybridization of deterministic continuous-time flows with discrete-time Metropolis–Hastings steps in large-scale combinatorial optimization interpolates between non-reversible biased search and equilibrium sampling, offering new frameworks for algorithmic acceleration and nuanced sampling of rugged energy landscapes (Leleu et al., 2024).

Large-deviation principles and Lyapunov analysis underpin both theoretical and computational characterization of relaxation rates, attractor geometries, and the stability and robustness of non-equilibrium steady states (Freitas et al., 2021, Hoover et al., 2015, Liverpool, 2018).

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References

• Optimization-based equilibrium measure and replica analysis: (Qiu et al., 2024)
• Non-equilibrium relaxation in quantum spin systems: (Niemeyer et al., 2014)
• Data-driven hydrodynamics from active particle models: (Maddu et al., 2022)
• Irreversible deterministic maps and detailed balance: (Colangeli et al., 2011)
• Deterministic reversible models with non-equilibrium transitions: (Cirillo et al., 2020, Cirillo et al., 2021)
• Emergent second law and deterministic entropy bounds: (Freitas et al., 2021)
• Generalization of steady states with macroscopic currents: (Liverpool, 2018)
• Hybrid continuous–discrete non-equilibrium sampling: (Leleu et al., 2024)
• Thermostatted oscillator models and fractal attractors: (Hoover et al., 2015)
• Non-normality/reactivity in neural systems and avalanches: (Santo et al., 2018)
• Relativistic heavy-ion non-equilibrium transport: (Xu et al., 2017)