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Large deviations of density in the non-equilibrium steady state of boundary-driven diffusive systems

Soumyabrata Saha, Tridib Sadhu

TL;DR

The paper tackles the longstanding problem of characterizing density fluctuations in the non-equilibrium steady state (NESS) of boundary-driven diffusive systems. It develops and applies a local transformation within Macroscopic Fluctuation Theory to obtain exact solutions for two solvable classes, clarifying when the density large-deviation functional (ldf) is local versus nonlocal and illustrating how rare fluctuations unfold along optimal paths. It demonstrates that nonlocality in the ldf generically arises at quadratic order in the boundary drive $O((\rho_a-\rho_b)^2)$ and links this to long-range equilibrium correlations, providing explicit formulas for fluctuation paths. The framework is then extended perturbatively to arbitrary dimensions and diffusivities, offering a unified methodology to compute nonlocal fluctuation features across a broad set of diffusive systems.

Abstract

A diffusive system coupled to unequal boundary reservoirs reaches a non-equilibrium steady state. While the full-counting-statistics of current fluctuations in these states are well understood for generic systems, results for steady-state density fluctuations remain limited to only a few integrable models. By obtaining an exact solution of the Macroscopic Fluctuation Theory, we characterize steady-state density fluctuations through large deviations for a wide range of boundary-driven diffusive systems. This allows us to identify two distinct classes of systems, one with only short-range correlations and another displaying long-range correlations. We also quantitatively describe the irreversible dynamical paths leading to these rare fluctuations in such systems. For very generic systems in arbitrary dimensions, we use a perturbation around the equilibrium state to solve for large deviations and the corresponding fluctuation paths. We find that non-locality in the large deviations emerges only at quadratic order in the perturbation, revealing non-trivial features of long-range correlations in non-equilibrium steady states.

Large deviations of density in the non-equilibrium steady state of boundary-driven diffusive systems

TL;DR

The paper tackles the longstanding problem of characterizing density fluctuations in the non-equilibrium steady state (NESS) of boundary-driven diffusive systems. It develops and applies a local transformation within Macroscopic Fluctuation Theory to obtain exact solutions for two solvable classes, clarifying when the density large-deviation functional (ldf) is local versus nonlocal and illustrating how rare fluctuations unfold along optimal paths. It demonstrates that nonlocality in the ldf generically arises at quadratic order in the boundary drive and links this to long-range equilibrium correlations, providing explicit formulas for fluctuation paths. The framework is then extended perturbatively to arbitrary dimensions and diffusivities, offering a unified methodology to compute nonlocal fluctuation features across a broad set of diffusive systems.

Abstract

A diffusive system coupled to unequal boundary reservoirs reaches a non-equilibrium steady state. While the full-counting-statistics of current fluctuations in these states are well understood for generic systems, results for steady-state density fluctuations remain limited to only a few integrable models. By obtaining an exact solution of the Macroscopic Fluctuation Theory, we characterize steady-state density fluctuations through large deviations for a wide range of boundary-driven diffusive systems. This allows us to identify two distinct classes of systems, one with only short-range correlations and another displaying long-range correlations. We also quantitatively describe the irreversible dynamical paths leading to these rare fluctuations in such systems. For very generic systems in arbitrary dimensions, we use a perturbation around the equilibrium state to solve for large deviations and the corresponding fluctuation paths. We find that non-locality in the large deviations emerges only at quadratic order in the perturbation, revealing non-trivial features of long-range correlations in non-equilibrium steady states.
Paper Structure (4 sections, 48 equations, 2 figures, 1 table)

This paper contains 4 sections, 48 equations, 2 figures, 1 table.

Table of Contents

  1. End Matter
  2. Perturbative ldf up to $\mathrm{O}\!\left((\rho_a-\rho_b)^2\right)$ in $d=1$
  3. Constant diffusivity systems
  4. Generalization to $d>1$ and arbitrary diffusivity systems

Figures (2)

  • Figure 1: Boundary-driven NESS: A one-dimensional system of length $L$ coupled to two unequal reservoirs (denoted by $\rho_a$ and $\rho_b$). The average density $\bar{\rho}(x)$ is the most-probable profile in the NESS while $r(x)$ denotes an atypical fluctuation.
  • Figure 2: The optimal evolution of density leading to a rare fluctuation for the SSIP with $K=3$ (see Table \ref{['system_list_transport_params']}), coupled to reservoirs of densities $\rho_a=2$ and $\rho_b=10$. The dotted black line denotes the average profile $\bar{\rho}(x)$, while the solid green line denotes the atypical profile $r(x)$.