General solution for the density large deviation functional in macroscopic fluctuation theory

Determine the large deviation functional F({ρ(x)}) and the associated generating functional G({A(x)}) for boundary-driven one-dimensional diffusive systems with arbitrary transport coefficients D(ρ) and σ(ρ) by solving the macroscopic fluctuation theory Euler–Lagrange system (the coupled equations (Hρ)) for general D and σ, thereby providing an explicit, self-contained characterization of F({ρ(x)}) and/or G({A(x)}) beyond the special solvable cases.

Background

The macroscopic fluctuation theory (MFT) expresses the probability of observing a space–time density/current history via an action functional, and the steady-state large deviation functional F({ρ(x)}) emerges from a variational problem dominated by optimal trajectories solving a coupled PDE system (Eq. (Hρ)).

For specific models such as the SSEP and certain solvable processes, explicit nonlocal formulas for F({ρ(x)}) or closed-form expressions for generating functionals G({A(x)}) are known. However, for general transport coefficients D(ρ) and σ(ρ), the explicit solution of the optimal-trajectory PDEs and the resulting F({ρ(x)}) or G({A(x)}) are not available.

Resolving this problem would generalize known exact results to broad classes of diffusive systems and clarify the structure (e.g., locality vs. nonlocality) of steady-state fluctuations under arbitrary constitutive relations.