Relaxation to thermal equilibrium in the self-gravitating sheet model

TL;DR

This study investigates how the self-gravitating sheet model in one dimension relaxes to thermal equilibrium, revealing that relaxation time depends on particle number and initial conditions, and can be characterized by phase space order parameters.

Contribution

It demonstrates a clear method to detect and characterize relaxation in the sheet model using phase space moments and explores the dependence of relaxation time on initial conditions and particle number.

Findings

Relaxation time scales approximately linearly with particle number N.

Colder initial states relax faster than warmer ones.

Relaxation follows a stretched exponential temporal profile.

Abstract

We revisit the issue of relaxation to thermal equilibrium in the so-called "sheet model", i.e., particles in one dimension interacting by attractive forces independent of their separation. We show that this relaxation may be very clearly detected and characterized by following the evolution of order parameters defined by appropriately normalized moments of the phase space distribution which probe its entanglement in space and velocity coordinates. For a class of quasi-stationary states which result from the violent relaxation of rectangular waterbag initial conditions, characterized by their virial ratio R_0, we show that relaxation occurs on a time scale which (i) scales approximately linearly in the particle number N, and (ii) shows also a strong dependence on R_0, with quasi-stationary states from colder initial conditions relaxing much more rapidly. The temporal evolution of the order parameter may be well described by a stretched exponential function. We study finally the correlation of the relaxation times with the amplitude of fluctuations in the relaxing quasi-stationary states, as well as the relation between temporal and ensemble averages.