Nonequilibrium fluctuations and moderate deviations for the occupation time of the SSEP with Glauber dynamics

Abstract: We study the symmetric simple exclusion process with Glauber dynamics. When the process starts from a nonequilibrium measure, we prove central limit theorems for the occupation time in dimension two, and sample path moderate deviation principles in dimension one. For the fluctuations, we use the martingale method and the sharp relative entropy method from [Jara and Menezes, arXiv:1810.09526]. For the moderate deviations, the main idea is to relate the occupation time to the density fluctuation field by using the logarithmic Sobolev inequality from the Glauber dynamics.

• The paper establishes central limit theorems in two dimensions and derives sample-path moderate deviation principles in one dimension for occupation times in a non-stationary SSEP with spin-flip dynamics.
• Methodologically, it employs a martingale decomposition combined with entropy methods and a superexponential replacement lemma to control deviations in nonequilibrium settings.
• The findings extend fluctuation theory beyond stationary measures, highlighting dimensional limitations and offering insights for reaction-diffusion systems and related models.

Nonequilibrium Fluctuations and Moderate Deviations for Occupation Time of the SSEP with Glauber Dynamics

Introduction and Problem Formulation

The paper investigates the nonequilibrium fluctuation regime and moderate deviation principles (MDPs) for occupation times of the symmetric simple exclusion process (SSEP) on the discrete torus, augmented by Glauber-type creation and annihilation dynamics. This composite dynamics blends the conservative transport of SSEP with non-conservative spin-flip interactions dictated by local configuration-dependent rates.

Previous work on SSEP occupation times—both for large deviations and fluctuations—has relied crucially on the underlying duality and the simplifying properties of stationary (invariant) initial conditions. The present work departs from this tradition by treating dynamics initiated from non-invariant (nonequilibrium) product measures, thereby broadening the scope of fluctuation theory for interacting particle systems. Notably, the analysis establishes central limit theorems (CLTs) for occupation times in dimension two and delivers sample-path MDPs in dimension one with respect to the nonequilibrium law.

Model Description

The process is defined on the $d$-dimensional discrete torus $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$. At each site, the state indicates occupation by at most one particle. The generator combines the SSEP (nearest-neighbor exclusion moves at diffusive scaling) with Glauber dynamics—sitewise creation and annihilation with rates that depend linearly on local occupation. The generator is summarized as: $L_n = n^2 L_n^{\mathrm{ex}} + L_n^{\mathrm{r}}$ where $L_n^{\mathrm{ex}}$ captures the exclusion part and $L_n^{\mathrm{r}}$ the reaction (Glauber) part, with creation/annihilation rates parametrized by $a$, $b$ (creation, annihilation), and $\lambda$ (local interaction).

The process is started from a product Bernoulli measure at density $\rho_*$, where $\rho_*$ is tuned so that the expectation of the spin-flip “current” vanishes—yielding a reference measure similar to those used to describe nonequilibrium steady states, although this measure is not strictly invariant under the full dynamics.

The primary observable is the origin occupation time: $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$0 with normalization $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$1 chosen according to the typical variance scaling for fluctuations in SSEP.

Main Results

Nonequilibrium Fluctuations

Dimension Two

A central limit theorem is established for the occupation time at the origin: $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$2 as $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$3, for $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$4, where $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$5 is standard Brownian motion. The variance coefficient incorporates $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$6, the Bernoulli variance at the reference density.

This result is significant: the verification of a CLT from a non-invariant (nonequilibrium) law required exploiting the martingale decomposition of additive functionals and controlling non-stationary error terms via sharp relative entropy bounds derived from the underlying spin-flip (Glauber) dynamics.

Dimension One

While the main focus for strong path-space results is on the moderate deviation regime in dimension one, the methodology also recovers the process-level CLT for functionals of the density fluctuation field, extending methods developed for the exclusion process (notably, those using the local Boltzmann-Gibbs principle in the nonequilibrium context).

Moderate Deviations (MDP) in Dimension One

The main technical achievement is the derivation of a sample-path MDP for the rescaled occupation time: $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$7 with $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$8, establishing upper and lower bounds for probabilities of moderate deviations of the occupation time path. The rate function is quadratic, depending on the covariance structure of the limiting Gaussian (consistent with the MDP for Gaussian processes): $\mathbb{T}_n^d = (\mathbb{Z}/n\mathbb{Z})^d$9 where $L_n = n^2 L_n^{\mathrm{ex}} + L_n^{\mathrm{r}}$0 encodes the finite-dimensional covariance structure.

The proof leverages a logarithmic Sobolev inequality for the Glauber part and a superexponential replacement lemma, allowing a reduction to moderate deviations of the density fluctuation fields and application of the contraction principle. The methodology is robust for $L_n = n^2 L_n^{\mathrm{ex}} + L_n^{\mathrm{r}}$1 due to the strong mixing of the dynamics, but crucial entropy and correlation estimates break down in higher dimensions, highlighting limitations of available space-time replacement methods without further advancements in correlation bounds.

Methodological Contributions

• Martingale Method: The occupation time is decomposed into a martingale part (essential for Gaussian limits) and negligible terms. Higher-order corrections induced by the Glauber part are controlled via a combination of entropy methods and careful replacement and flow arguments.
• Relative Entropy and Logarithmic Sobolev Inequality: Key nonequilibrium bounds are established using entropy methods, exploiting the regularizing effect of the Glauber dynamics, which facilitates control of deviations from local equilibrium.
• Superexponential Replacement Lemma: Extending ideas from hydrodynamic and large deviation theory, the pathwise replacement is carried out at a “superexponential” scale, ensuring the validity of pathwise moderate deviation analysis.
• Dimensional Dependence and Limitation: The results are sharp in their dimensional restrictions, revealing technical obstructions to higher-dimensional extensions—primarily tied to the lack of sharp correlation (mixing) estimates for non-conservative interacting particle systems outside $L_n = n^2 L_n^{\mathrm{ex}} + L_n^{\mathrm{r}}$2.

Implications and Theoretical Impact

The findings push the boundary of additive functional fluctuation theory into genuinely nonequilibrium territory for interacting particle systems with both conservative and non-conservative elements. The work demonstrates that, at least in low dimensions, fluctuation–moderate deviation theory survives the removal of stationary initial conditions, provided a suitable mix of mixing/entropy controls is available. This suggests the universality of Gaussian fluctuation behavior for occupation times in nonconservative systems, even far from stationarity, under appropriate scaling regimes.

Practically, the results underline the capacity for sharp probabilistic description of time-averaged observables in models relevant to reaction-diffusion and population dynamics, including in settings where the generator lacks self-duality—a customary technical crutch in the field.

Theoretically, the methodology points to future work on:

• Extending such results to higher dimensions, possibly via novel approaches to multi-point space-time correlations or even hierarchy closure.
• Analyzing more general (nonlinear) additive functionals and their MDPs under nonequilibrium dynamics.
• Developing analogous fluctuation theories for other non-self-dual or more complex systems, e.g., inclusion of asymmetric interactions or systems with multiple conserved quantities.

Conclusion

The paper provides a comprehensive treatment of nonequilibrium central limit theorems and moderate deviation principles for the occupation time of SSEP with Glauber dynamics in low dimensions, foundationally enhancing the understanding of fluctuations in non-stationary settings for both conservative and dissipative stochastic models. The technical apparatus established here—martingale/entropy decompositions, replacement lemmas, and meticulous control of nonstationary effects—lays the groundwork for future theoretical expansions in non-equilibrium statistical mechanics and interacting particle systems.

For further mathematical detail and explicit proofs, see (2604.10509).