Relaxation to thermal equilibrium in the self-gravitating sheet model

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 theirseparation. We show that this relaxation may be very clearly detected and characterized byfollowing the evolution of order parameters defined by appropriately normalized momentsof the phase space distribution which probe its entanglement in space and velocitycoordinates. For a class of quasi-stationary states which result from the violentrelaxation of rectangular waterbag initial conditions, characterized by their virial ratioR0, weshow that relaxation occurs on a timescale which (i) scales approximately linearly in the particle numberN and (ii) also shows astrong dependence on R0, 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 stretchedexponential function. We study finally the correlation of the relaxation times with theamplitude of fluctuations in the relaxing quasi-stationary states, as well as the relationbetween temporal and ensemble averages.