Ergodic properties of the harmonic process

Abstract: In this paper we study detailed fluctuation results for a class of non-equilibrium steady states. The main example is the boundary driven harmonic model \cite{frassek2022exact}. In this model, the non-equilibrium steady state (NESS) is a mixture of products of geometric distributions, of which the local parameters are in turn distributed as uniform order statistics. For such a NESS, we prove law of large numbers, central limit theorem and large deviation results for fields of a general local functions (generalizing the density field). We also obtain quantitative results on the deviation from local equilibrium.

• The paper derives strong statistical results—LLN, CLT, and LDP—for the boundary-driven harmonic process using a mixing of geometric measures based on uniform order statistics.
• It employs explicit moment formulas and concentration inequalities to quantify deviations from local equilibrium and reveal long-range correlations in non-equilibrium steady states.
• The analysis provides a rigorous framework that can guide the development of advanced sampling algorithms and the study of more complex interacting particle systems.

Ergodic Properties and Large Deviations in the Boundary-Driven Harmonic Process

Introduction

The paper "Ergodic properties of the harmonic process" (2604.17469) provides a rigorous investigation of the statistical structure and fluctuation properties of a class of non-equilibrium steady states (NESS) associated with exactly solvable boundary-driven harmonic particle systems. Central to this analysis is the explicit description of NESS as mixtures of product measures with geometric marginals, where the mixing is expressed in terms of the order statistics of uniform random variables. This framework allows for direct derivation of strong statistical results, including laws of large numbers (LLN), central limit theorems (CLT), and large deviation principles (LDP) for general fields of local observables. The study also quantifies deviations from local equilibrium and characterizes the long-range correlations inherent in these NESS.

Model Setting and NESS Structure

The main example is the boundary-driven harmonic process on a finite chain, where reservoirs with different parameters at the boundaries induce a nontrivial stationary profile. For this process, the NESS can be represented as a mixture of product measures, where the single-site marginals are geometric distributions. The key technical observation, previously established in [frassek2022exact, carinci2024solvable], is that the local parameters of these geometric marginals are distributed as the order statistics of $N$ i.i.d. uniform random variables over $[\theta_L, \theta_R]$, with $\theta_L < \theta_R$ denoting the reservoir parameters. This parametric mixing structure enables closed-form calculation of expectations, correlations, and generating functions for a broad class of observables.

Fluctuation Results for Order Statistics and Random Parameters

A substantial portion of the analysis relies on detailed properties of uniform order statistics:

• Closed-form moments: The product moments and single-moment formulas for order statistics of uniforms are utilized (cf. [mathai2002product]), e.g.,

$$\mathbb{E}[U_{r:N}^k] = \frac{r (r+1)\dots (r + k-1)}{(N+1)\dots (N+k)}.$$

• Ergodicity and LLN: For any local function of the random parameters (i.e., functions depending on a finite set of coordinates), sums of such functions exhibit a strong law of large numbers. The limiting profiles are deterministic and determined by the continuous density profile interpolating between the reservoir parameters.
• CLT for fields of local functions: At the next order, central limit theorems are established for the same class of sums, with explicit formulas for the limiting variances, which encode the spatial correlations induced by the order statistics. The limiting covariance exhibits a Brownian bridge structure, reflecting the nontrivial long-range correlations in NESS.
• Uniform concentration: The paper proves concentration inequalities for order statistics, ensuring that the random parameters deviate from their deterministic linearly spaced expectations by at most $O(N^{-1/2})$ uniformly with high probability.
• Large deviations: Using results from [duffy2011sample], a full sample-path large deviation principle is proved for the vector of order statistics, with the rate function given by a variational formula involving the entropy of the derivative of the path with respect to the uniform measure.

Local Equilibrium and Its Quantitative Refinements

A cornerstone of hydrodynamic approaches is the concept of local equilibrium, i.e., the idea that the NESS, on mesoscopic scales, resembles a product measure with slowly varying parameters. The paper proves that the mixed product NESS indeed exhibits local equilibrium in the sense that, for any fixed macroscopic coordinate $x \in (0,1)$ and local function $g$, the expectation of $g$ localized at site $\lfloor xN \rfloor$ converges to its value under the geometric measure with mean $\rho(x) = \theta_L + x (\theta_R - \theta_L)$. The rate of convergence and uniformity of this approximation are also quantified, including $[\theta_L, \theta_R]$0 estimates for duality polynomials.

Quantitative Self-Duality

For self-duality polynomials, which play a fundamental role in exactly solvable exclusion and harmonic models, the paper shows the non-equilibrium correction to local equilibrium is of order $[\theta_L, \theta_R]$1, making the approximation robust for practical purposes.

Laws of Large Numbers and Fluctuation Fields for Local Observables

The explicit solvability allows the authors to advance beyond single-site observables, considering empirical fields of general local functions. For such fields:

• LLN: The empirical averages of general local observables converge almost surely to deterministic integrals against the macroscopic density profile.
• CLT: Fluctuation fields, properly rescaled, converge in distribution to Gaussian fields with variance given as the sum of two terms: a white-noise term associated with local equilibrium fluctuations, and a Brownian bridge term encoding the correlations from the mixing of the product structure. The Brownian bridge contribution is strictly nonzero off equilibrium and vanishes when the boundaries are equalized ($[\theta_L, \theta_R]$2).

Large Deviations for Local Fields

By leveraging Gibbsian LDPs for inhomogeneous product measures (see [georgii]) and the sample-path LDP for order statistics, the authors derive a variational principle for the large deviation rate function of the empirical field of a local function in the NESS. Specifically, the rate function is given by the infimum over profiles of the sum of the large deviation functional for geometric measures at fixed profile and the entropy cost for the mixing parameters. This extends and substantiates previous predictions from macroscopic fluctuation theory (see [bertini2007stochastic, carinci2025large]).

Discussion and Implications

The explicit probabilistic structure of the NESS for the harmonic process—mixed product of geometric distributions with uniform order statistics—enables a level of analytical control not available in generic non-equilibrium systems. The detailed fluctuation and large deviation results rigorously confirm features anticipated by non-equilibrium statistical mechanics, such as the coexistence of local equilibrium, nonlocal correlations, and the additivity principle for the rate function.

These findings underpin rigorous hydrodynamic limits and provide a blueprint for the analysis of more complex systems where nontrivial mixing measures appear. The approach is extendable to other exactly solvable models (including, with proper modifications, certain KMP-type models). Furthermore, the precise characterization of long-range correlations as arising from the mixing of order statistics could inform the design of advanced sampling algorithms or non-equilibrium generalizations in Markovian and non-Markovian settings.

From a theoretical perspective, the methodology strengthens the connections between interacting particle systems, integrable probability, empirical processes, and large deviations theory. Potential extensions include the investigation of hyperuniformity, edge behavior, and universality in solvable boundary-driven systems and their relevance for advancing microscopic foundations of non-equilibrium thermodynamics.

Conclusion

This paper delivers a comprehensive, rigorous analysis of ergodic and fluctuation properties for the boundary-driven harmonic process via the explicit mixture of product measures with geometric marginals parametrized by uniform order statistics (2604.17469). Strong numerical results include exact LLN, CLT, and LDP for general fields of local functions and precise characterizations of the spatial correlations and large deviation rate functions in NESS. The approach and results significantly enhance the analytical toolkit available for exactly solvable models in non-equilibrium statistical mechanics and establish a robust foundation for further investigations of NESS in interacting particle systems.