[Paper Review] Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations

Simone Floreani, Frank Redig+1 more|arXiv (Cornell University)|Jul 16, 2020

Stochastic processes and statistical mechanics18 references24 citations

TL;DR

This paper establishes orthogonal polynomial duality for boundary-driven symmetric exclusion and inclusion processes on general graphs with edge and site disorder, extending classical duality to non-equilibrium steady states. It proves that n-point correlation functions and cumulants in the non-equilibrium stationary measure factorize universally into (θL − θR)^n times a geometry-dependent function, independent of reservoir parameters, revealing a universal structure in non-equilibrium correlations.

ABSTRACT

We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then we derive new orthogonal polynomial dualities. From the classical dualities, we derive the uniqueness of the non-equilibrium steady state and obtain correlation inequalities. Starting from the orthogonal polynomial dualities, we show universal properties of n-point correlation functions in the non-equilibrium steady state for systems with at most two different reservoir parameters, such as a chain with reservoirs at left and right ends.

Motivation & Objective

• To extend classical duality to boundary-driven particle systems with edge and site disorder, generalizing previous results on symmetric exclusion and inclusion processes.
• To derive orthogonal polynomial duality functions for non-equilibrium systems, building on symmetry in the dual absorbing system.
• To establish universal factorization of n-point correlation functions and cumulants in the non-equilibrium steady state, with explicit dependence on reservoir parameter difference (θL − θR).
• To prove that correlation functions at any time t > 0, starting from local equilibrium, retain the same (θL − θR)^n structure, independent of reservoir parameters.
• To relate the joint moment generating function of occupation variables to expectations in the absorbing dual, enabling macroscopic analysis of fluctuation fields and large deviations.

Proposed method

• Introduce a generalized framework for boundary-driven symmetric exclusion and inclusion processes on arbitrary graphs with site- and edge-dependent parameters (conductances, maximal occupancy, attraction parameters).
• Leverage classical duality with absorbing boundaries to derive correlation inequalities and uniqueness of the non-equilibrium steady state.
• Construct orthogonal polynomial duality functions in product form, where bulk site factors are orthogonal polynomials (e.g., Charlier, Meixner) and boundary factors depend on reservoir parameters θL, θR.
• Use a symmetry-based method to derive orthogonal duality from classical duality, exploiting the relation between classical and orthogonal duality functions.
• Apply the duality to express n-point correlations and cumulants as (θL − θR)^n times a universal function ψt depending only on the dual particle system dynamics and spatial configuration.
• Relate the joint moment generating function of occupation variables to an expectation in the absorbing dual system, enabling simulation and macroscopic limit analysis.

Experimental results

Research questions

• RQ1 Can classical duality be extended to boundary-driven particle systems with both edge and site disorder?
• RQ2 What is the structure of n-point correlation functions in the non-equilibrium steady state for such systems?
• RQ3 How does the orthogonal polynomial duality function emerge from the symmetry of the dual absorbing system?
• RQ4 Can the correlation functions at any time t > 0, starting from local equilibrium, be expressed in a universal form independent of reservoir parameters?
• RQ5 What is the role of the reservoir parameter difference (θL − θR) in shaping the universal factorization of correlations and cumulants?

Key findings

• The n-point correlation functions and cumulants in the non-equilibrium steady state are of the form (θL − θR)^n multiplied by a universal function ψ that depends only on the spatial configuration and dual dynamics, not on θL or θR.
• This universal factorization structure holds not only at stationarity but also at any finite time t > 0 when the system starts from a local equilibrium product measure.
• The joint moment generating function of occupation variables is expressed as an expectation in the absorbing dual, providing a non-perturbative tool for studying large deviations and fluctuation fields.
• The orthogonal polynomial duality functions are constructed via a symmetry-based method that relates them to classical duality functions, enabling their derivation in the boundary-driven setting.
• The non-equilibrium stationary measure is uniquely characterized by its moments, and the existence and uniqueness of the stationary measure are proven using duality and Carleman's condition.
• For systems with at most two distinct reservoir parameters (e.g., a one-dimensional chain with left and right reservoirs), the correlation structure is universal and independent of the specific values of θL and θR, depending only on their difference.