# Large deviations conditioned on large deviations II: Fluctuating hydrodynamics

## TL;DR

This paper investigates the large deviation behavior of density fluctuations conditioned on current in diffusive particle systems, deriving a microscopic large deviation function and connecting it to fluctuating hydrodynamics via Hamilton-Jacobi equations.

## Contribution

It provides a microscopic calculation of the conditioned large deviation function and links it to a fluctuating hydrodynamics framework through Hamilton-Jacobi equations.

## Key findings

- Derived the conditioned large deviation function for density fluctuations.
- Connected microscopic results to fluctuating hydrodynamics via Hamilton-Jacobi equations.
- Applicable to general diffusive systems with boundary reservoirs.

## Abstract

For diffusive many-particle systems such as the SSEP (symmetric simple exclusion process) or independent particles coupled with reservoirs at the boundaries, we analyze the density fluctuations conditioned on current integrated over a large time. We determine the conditioned large deviation function of density by a microscopic calculation. We then show that it can be expressed in terms of the solutions of Hamilton-Jacobi equations, which can be written for general diffusive systems using a fluctuating hydrodynamics description.