NEASS: Non-equilibrium Almost-Stationary States

• NEASS are asymptotic states in driven systems that generalize equilibrium and steady state behaviors through nearly invariant dynamics.
• They are classified by distinct macroscopic attractors such as fixed points, limit cycles, tori, and chaotic attractors, with implications for stochastic and quantum models.
• Rigorous construction via Fokker–Planck, WKB ansatz, and quantum dressing enables precise computation of key observables like transport coefficients.

A Non-equilibrium Almost-Stationary State (NEASS) is a class of asymptotic states realized in systems—both classical and quantum—when perturbed away from equilibrium, typically under weak or slow external driving. These states generalize standard equilibrium and non-equilibrium steady states to include time-dependent periodic, quasi-periodic, or chaotic attractors that remain “almost invariant” under the perturbed dynamics up to timescales much longer than microscopic relaxation. NEASS theory provides a rigorous framework for describing, constructing, and classifying these states, with direct applications ranging from stochastic urn models and interacting electron systems under fields, to ordered non-equilibrium structures in long-range interacting Hamiltonian models.

1. Formal Definition and Mathematical Framework

The canonical construction of NEASS proceeds via the large-$N$ Fokker–Planck equation for a probability density $\rho(\vec{x}, t)$ over $D$ degrees of freedom:

$$\partial_t \rho = -\sum_i \partial_{x_i} [A_i(\vec{x}, t) \rho] + \frac{1}{2N} \sum_{i,j} \partial_{x_i} \partial_{x_j} [B_{ij}(\vec{x}, t) \rho]$$

In the thermodynamic limit ($N \to \infty$), a WKB ansatz, $\rho(\vec{x}, t) \propto \exp[N f(\vec{x}, t)]$, allows one to expand $f$ about an “optimal point” $\vec{\xi}(t)$ (solving $\partial_i f|_{\vec{\xi}(t), t} = 0$), yielding macroscopic dynamics $\dot{\vec{\xi}} = A(\vec{\xi}, t)$ and the covariance evolution equation for fluctuations. The nature of the asymptotic solution $\vec{\xi}^{\mathrm{as}}(t)$ classifies the NEASS subclass:

• Fixed point: equilibrium (if detailed balance holds), or NESS (if balance is violated)
• Limit cycle: non-equilibrium periodic state (NEPS)
• Quasi-periodic torus: NEQPS
• Chaotic attractor: NECS

Thus, NEASS is an umbrella term for all possible asymptotic macroscopic behaviors (fixed points, cycles, tori, chaos) in the thermodynamic limit (Cheng et al., 2023).

In quantum lattice systems, NEASS is constructed by “dressing” an equilibrium projector with a quasi-local unitary $e^{iS_n}$ expanding in the perturbation parameter $\epsilon$:

$$\Pi_n^\epsilon = e^{\,iS_n^\epsilon} \Pi_0 e^{-\,iS_n^\epsilon}$$

with $S_n^\epsilon = \sum_{j=1}^n \epsilon^j A_j$, and the almost-stationarity property $[H^\epsilon, \Pi_n^\epsilon] = O(\epsilon^{n+1})$ (Teufel, 2017, Mazzini et al., 29 Jan 2026).

2. Classification and Subclasses of NEASS

The taxonomy of NEASS includes:

Subclass	Macroscopic Attractor	Detailed Balance
Equilibrium (EQ)	Fixed point	Satisfied
NESS	Fixed point	Violated
NEPS	Limit cycle	N/A (dynamical)
NEQPS	Torus (quasi-periodic)	N/A (dynamical)
NECS	Chaotic attractor	N/A (dynamical)

This separation is realized, for example, in the interacting Ehrenfest-urn ring, where multistability, bifurcations (Hopf, saddle-node, infinite-period), and coexistence of attractors can be analytically traced (Cheng et al., 2023). NEASS formation also appears in ordered non-equilibrium states of long-range Hamiltonian systems, where post-kick periodic orbits display breaking of ergodicity and sharply organized phase space structure (Joyce et al., 2017).

3. Construction Techniques and Series Expansions

In quantum systems, NEASS is generated by a systematic perturbative procedure using space-adiabatic methods and inversion of the Liouvillian superoperator, $\mathcal{L}_{H_0}(A) = [H_0, A]$. For a weak static electric field perturbation, the leading correction generator is

$$S = -\mathcal{L}_{H_0}^{-1}([X_2, \Pi_0]) = \frac{1}{2\pi} \oint_\gamma (H_0 - z)^{-1}[\Pi_0, X_2](H_0 - z)^{-1} dz$$

Order-by-order, higher corrections $A_j$ are recursively determined such that $$[H^\epsilon, U_n^\epsilon \Pi_0 U_n^{\epsilon*}] = O(\epsilon^{n+1})$$ (Mazzini et al., 29 Jan 2026, Marcelli et al., 2022). In classical particle systems, the NEASS may be attained by finite-time operator kicks that “organize” post-relaxation phase space into stable, stationary, microscopically ordered configurations (Joyce et al., 2017).

4. Thermodynamic Potentials and Laws in NEASS

For stochastic interacting urn models, thermodynamic functionals are assigned as follows:

• Boltzmann entropy: $$S(t) = -\sum_{\vec{n}} \rho(\vec{n}, t) \ln[\rho(\vec{n}, t)/\text{degeneracy}(\vec{n})]$$
• Internal energy: $$\beta E = \sum_{\vec{n}} \rho(\vec{n}, t) [g/2 \sum_i n_i(n_i-1)]$$
• Work rate: $\beta dW/dt = -\beta \mu \sum_i K_{i \to i+1}(t)$, $\mu \equiv \beta^{-1} \ln(p/q)$

The exact nonequilibrium first law (Eq.(14) (Cheng et al., 2023)) reads:

$$\frac{dS}{dt} = \frac{d_i S}{dt} + \beta \frac{dE}{dt} + \beta \frac{dW}{dt}$$

For stationary asymptotic states (NESS, NEPS), ensemble-averaged entropy and energy are constant, and dissipation is tied to maintained work input. Thermodynamic quantities and fluctuation-dissipation relations extend to limit cycles and chaotic attractors within the NEASS framework.

5. Physical Implications and Applications

NEASS construction provides rigorous, explicit states for the evaluation of transport coefficients in quantum systems. For instance, the NEASS accumulation yields the “double-commutator” Kubo formula for transverse (Hall) conductivity:

$$\sigma_{\mathrm{Hall}} = i \mathcal{T}\left( \Pi_0\left[ [\Pi_0, X_1], [\Pi_0, X_2] \right] \Pi_0 \right)$$

It is shown that in systems with incommensurate magnetic flux, all power-law corrections beyond linear response vanish, establishing exact linearity and topological quantization of the Hall current in the NEASS (Mazzini et al., 29 Jan 2026). Spin conductivities can be computed with explicit, albeit more involved, formulas, and in non-conserved spin systems, the NEASS carries no residual spin torque to any order in the perturbation (Marcelli et al., 2022).

In classical and stochastic models, NEASS formation plays a critical role in understanding the organization and relaxation of long-range interacting systems, chemical and biochemical oscillators, driven material systems, biological networks, and quantum thermodynamics of periodically driven machines (Cheng et al., 2023, Joyce et al., 2017).

6. Model Examples: Ehrenfest-Urn and Long-Range Systems

The interacting Ehrenfest–urn ring model exemplifies NEASS classification: for $M=3$ urns, the phase diagram presents uniform and non-uniform NESS, coexistence regions, NEPS, and their transitions. The NEPS emerges via supercritical Hopf bifurcation at $g=-3$ independent of hopping bias $p$; saddle-node and infinite-period bifurcations demarcate boundaries between stationary and periodic NEASS (Cheng et al., 2023). Long-range Hamiltonian systems, when kicked by weak non-Hamiltonian perturbations, organize into strictly stationary, microscopically ordered NEASS comprising rings (shells) in phase space with sharply peaked energy and period distributions. Energy-balance equations predict shell radii and global order parameters with accuracy (Joyce et al., 2017).

7. Contextualization and Extensions

NEASS generalizes the concept of stationarity and extends rigorous response theory even to situations lacking a spectral gap in the perturbed Hamiltonian (Teufel, 2017). The formalism unifies equilibrium, classical steady, and genuinely dynamical non-equilibrium asymptotic states under Fokker–Planck + WKB (or quantum dressing) approaches, bridging microscale stochastic/quantum dynamics to macroscale thermodynamic structure. The asymptotically invariant nature and systematic classification of NEASS facilitate the computation of a broad spectrum of observables and response coefficients, enabling detailed analysis of dissipative and non-dissipative transport phenomena in both classical and quantum many-body systems.