Lattice models for granular-like velocity fields: Finite-size effects

TL;DR

This paper investigates finite-size effects and long-range correlations in a 1d lattice model of granular fluids, revealing multiscaling behavior and providing analytical and numerical insights into velocity and energy fluctuations.

Contribution

It introduces a detailed analytical framework for understanding finite-size effects and correlations in granular-like velocity fields, including exact solutions and perturbative analysis.

Findings

Velocity correlations tend to a stationary value over time.

Total energy fluctuations exhibit multiscaling and diverge with system size.

Numerical simulations confirm theoretical predictions.

Abstract

Long-range spatial correlations in the velocity and energy fields of a granular fluid are discussed in the framework of a 1d lattice model. The dynamics of the velocity field occurs through nearest-neighbour inelastic collisions that conserve momentum but dissipate energy. A set of equations for the fluctuating hydrodynamics of the velocity and energy mesoscopic fields give a first approximation for (i) the velocity structure factor and (ii) the finite-size correction to the Haff law, both in the homogeneous cooling regime. At a more refined level, we have derived the equations for the two-site velocity correlations and the total energy fluctuations. First, we seek a perturbative solution thereof, in powers of the inverse of system size. On the one hand, when scaled with the granular temperature, the velocity correlations tend to a stationary value in the long time limit. On the other hand, the scaled standard deviation of the total energy diverges, that is, the system shows multiscaling. Second, we find an exact solution for the velocity correlations in terms of the spectrum of eigenvalues of a certain matrix. The results of numerical simulations of the microscopic model confirm our theoretical results, including the above described multiscaling phenomenon.