Fluctuating hydrodynamics and correlation lengths in a driven granular fluid

TL;DR

This paper derives analytical expressions for structure factors in a driven granular fluid using fluctuating hydrodynamics, compares them with simulations, and explores how correlation lengths grow with density.

Contribution

It introduces a theoretical framework combining external and internal noise to analyze velocity and density correlations in driven granular fluids.

Findings

Static velocity structure factors show two plateaux at different scales.

Correlation lengths increase with packing fraction.

Decay of dynamical structure factors slows as density increases.

Abstract

Static and dynamical structure factors for shear and longitudinal modes of the velocity and density fields are computed for a granular system fluidized by a stochastic bath with friction. Analytical expressions are obtained through fluctuating hydrodynamics and are successfully compared with numerical simulations up to a volume fraction $\sim 50%$. Hydrodynamic noise is the sum of external noise due to the bath and internal one due to collisions. Only the latter is assumed to satisfy the fluctuation-dissipation relation with the average granular temperature. Static velocity structure factors $S_\perp(k)$ and $S_\parallel(k)$ display a general non-constant behavior with two plateaux at large and small $k$, representing the granular temperature $T_g$ and the bath temperature $T_b>T_g$ respectively. From this behavior, two different velocity correlation lengths are measured, both increasing as the packing fraction is raised. This growth of spatial order is in agreement with the behaviour of dynamical structure factors, the decay of which becomes slower and slower at increasing density.