Non-equilibrium Phase Transitions

• Non-equilibrium phase transitions are collective phenomena occurring in systems driven away from balance, featuring abrupt macroscopic changes due to irreversible processes.
• They arise in diverse models such as driven-dissipative systems, billiard dynamics, and long-range interacting setups, highlighting novel universality classes and critical behaviors.
• Recent methodological advances, including trajectory-based analysis and the minimum action method, offer precise means to predict phase diagrams and critical dynamics.

Non-equilibrium phase transitions are collective phenomena where a system, maintained away from detailed balance, exhibits abrupt changes in macroscopic properties as control parameters are varied. Unlike equilibrium transitions governed by symmetry, dimensionality, and the minimization of free energy, non-equilibrium transitions are rooted in the interplay of irreversible processes, drive, dissipation, and dynamical constraints. Such transitions are observed across stochastic, classical, and quantum many-body systems, and give rise to emergent pattern formation, novel steady states, and criticality fundamentally distinct from that in equilibrium systems.

1. Theoretical Foundations and Distinction from Equilibrium

Non-equilibrium phase transitions are characterized by the absence of detailed balance and the breakdown of the fluctuation–dissipation theorem. In equilibrium, phase transitions are categorized via singularities in the free-energy landscape; the critical exponents and universality classes are determined solely by system symmetries and spatial dimensionality. In contrast, non-equilibrium transitions may feature unique universality classes, time-dependent steady states, and transitions not associated with free-energy minima, often requiring dynamical order parameters or trajectory-based observables (Roy et al., 2022). The control parameters can be drive amplitude, coupling strength, dissipative rates, kinetic thresholds, or biases, depending on the specific system.

2. Prototypical Models and Mechanisms

a. Driven-Dissipative Systems and Absorbing-State Transitions

Driven, dissipative systems with open boundaries or continuous energy input show non-equilibrium phase transitions between absorbing (inactive) and active states. A prime example is the quantum cellular automaton generalizing the Domany-Kinzel model, where a tunable entangling gate controls a transition in the 1D directed percolation universality class, even when quantum correlations are introduced (Gillman et al., 2020). In Rydberg gases driven by laser coupling, the interplay of loss and gain creates both directed-percolation-like continuous transitions and first-order discontinuous transitions depending on lattice geometry and resonance conditions (Everest et al., 2015).

b. Billiard and Urn Models: Deterministic and Stochastic Non-equilibrium

A system of non-interacting billiard particles, subject to a bounce-back rule at a threshold occupancy, displays a sharp first-order transition from a homogeneous to an inhomogeneous stationary state as the threshold is tuned. This transition is mirrored by a modified Ehrenfest urn model with density-dependent transfer probabilities, permitting exact analytic treatment. Both are non-dissipative and reversible yet display genuine non-equilibrium phase transitions, demonstrating that macroscopic irreversibility and phase separation can arise from purely dynamical constraints (Cirillo et al., 2020, Cheng et al., 2021).

c. Long-range Interactions and Quasi-stationarity

In models with long-range interactions, such as the generalized Hamiltonian Mean Field XY model, molecular dynamics simulations reveal non-equilibrium transitions between paramagnetic, ferromagnetic, and nematic phases. The resulting phase diagram diverges significantly from equilibrium Boltzmann–Gibbs predictions. Instead, kinetic theory (Vlasov dynamics) determines transition lines and the discontinuities in the order parameters, demonstrating the inadequacy of equilibrium statistical mechanics for such systems (Teles et al., 2012).

3. Signatures and Order Parameters

Non-equilibrium phase transitions may be first-order or continuous (second-order), and their criticality is identified by dynamical or structural order parameters:

• Absorbing state order parameter: the long-time density of active sites n(t), vanishing in the absorbing phase and finite in the active phase (Gillman et al., 2020).
• Spectral imbalance: for coupled optical resonators, the order parameter is the normalized difference in degenerate and non-degenerate mode occupations, exhibiting a discontinuous jump at a transition (Roy et al., 2022).
• Trajectory-based order parameters: the time-integrated concentration of locally favored structures serves as an order parameter in trajectory-space transitions in glassforming liquids (Turci et al., 2016).
• Fraction of energy carried by an impurity: in sheared granular mixtures, the steady-state energy fraction carried by a tracer transitions from vanishing to finite as the shear or mass ratio is tuned (Garzo et al., 2011).

Critical exponents can be mean-field-like or in distinct universality classes, e.g., directed percolation (α ≈ 0.159 in 1D), with scaling laws governing the order parameter and susceptibility near criticality (Gillman et al., 2020, Zeng et al., 2018).

4. Macroscopic Fluctuation Theory and Lagrangian Phase Transitions

The macroscopic fluctuation theory provides a variational framework for describing large deviations in density and current in systems far from equilibrium. The dynamical large deviation functional defines an action for fluctuating trajectories, where the minimizer yields a non-equilibrium free energy (quasi-potential). Non-equilibrium phase transitions, termed Lagrangian phase transitions, correspond to singularities where competing time-dependent minimizers exist for the action functional, producing non-analyticities—typically cusps—in the quasi-potential (Bertini et al., 2010).

A compelling example is the boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a Lagrangian transition when an external field competes with boundary reservoirs leading to multiple global minimizers for the large deviation rate function.

5. Non-equilibrium Transitions in Driven Quantum and Topological Systems

Non-equilibrium driving in quantum systems can produce novel phase transitions not accessible in equilibrium:

• Floquet-driven Ising chains exhibit non-equilibrium quantum phase transitions, where the stroboscopic (“Floquet ground”) state changes topology or symmetry at critical driving parameters. Nielsen’s geometric circuit complexity serves as an order parameter, with dynamical freezing (coherent destruction of tunneling) marking non-equilibrium transition points (Camilo et al., 2020).
• Driven ultracold fermions in optical lattices realize non-equilibrium topological phase transitions, with the critical line determined by band inversion in Floquet quasi-energies. The resulting topological phases are classified by $\mathbb{Z}_2$ invariants and manifest robust edge modes not present in the underlying equilibrium system (Nakagawa et al., 2013).
• Holographic frameworks reproduce universal scaling exponents for non-equilibrium steady-state transitions, demonstrating that mean-field criticality can persist in time-periodically driven, strongly interacting systems (Zeng et al., 2018).

6. Methodological Developments: Pathwise and Minimum Action Approaches

First-order non-equilibrium phase transitions, especially in extended systems, require trajectory-level analysis. The Minimum Action Method extends Arrhenius’ law for rare-event rates to non-reversible, non-Gaussian settings: transition rates are governed by the minimum of a time-asymmetric action, distinguishing forward and backward instantons. The numerical implementation, via a saddle-point solution of a path-space min-max problem, allows explicit calculation of phase diagrams and critical nuclei in systems such as modified Ginzburg–Landau fields and stochastic reaction–diffusion networks (Zakine et al., 2022).

7. Broader Implications and Applications

Non-equilibrium phase transitions underlie dynamic reorganization in biological tissues (e.g., Voronoi models of cell colonies: transitions between jammed, active, and clustered phases) (Miotto et al., 12 Nov 2024), active matter, network systems (finite-time dynamical phase transitions with hyperbolic divergence in network observables) (Liu et al., 9 Dec 2024), and granular flows. The paradigm has substantial impact on developing sensors with enhanced sensitivity leveraging discontinuous transitions in driven photonic systems (Roy et al., 2022), and in understanding the formation and melting of glasses, where trajectory-space first-order transitions provide a route to unified dynamical and thermodynamic theories (Turci et al., 2016).

A unique aspect of non-equilibrium transitions is the emergence of new types of universality, the essential role of fluctuations and path-dependence, and the violation of equilibrium thermodynamic constraints, permitting the realization of phases, critical points, and dynamic patterns inaccessible in equilibrium statistical mechanics.