Long time behaviour for a chain of colliding rotators

Abstract

2014 A simple model of a plastic crystal is constructed to study the long term effects of a cog-wheel mechanism. We present some arguments which lead us to conjecture that an initial distribution of rotators’ angular velocity diffuses according to a normal law with a t-1/2 decay of the angular velocity autocorrelation function of the rotators. This has been confirmed by a molecular dynamic calculation performed on a chain of 1 000 rotators with periodic boundary conditions. LE JOURNAL DE PHYSIQUE LETTRES TOME 39, ler JUILLET 1978, P Classification Physics Abstracts 66.90 Recent molecular dynamic studies on two dimensional fluid systems of diatomics [1] have shown at the longest calculated time a t-3 decay of the angular velocity autocorrelation function T~(~). The asymptotic t 2 predicted by hydrodynamics [2] can occur at longer times with very small amplitude, and it was argued in [1] ] that the observed t 3 was possibly due to the transfer of the intrinsic angular momentum by some kind of cog-wheel mechanism. Here, an attempt to understand the long-term effect of cogwhell coupling in one dimension is reported. Further work [3] will be devoted to two dimensional systems. To isolate the cog-wheel coupling, we shall consider a line of colliding rotators with fixed centres. In the simplest case i) no multiple collisions occur and, ii) a couple «(1)1’ w2) of angular velocities transform to ( (1)2’ (1)1) when molecules 1 and 2 collide. This corresponds to two adjacent sectors of angle less than n interacting as soon as they touch each other (Fig. 1). The exchange of angular momentum is then instantaneous and the two sectors have the new momenta until the next contact. This model is inspired by the one-dimensional system of hard rods solved by Jepsen and LebowitzPercus [4]. In both, the dynamics consist of neighbouring particles interchanging velocities at each collision. This forbids ergodicity and relaxation towards an equilibrium. FIG. 1. Collision (a) corresponds to W cu2 > 0 and is compatible with a rough disk model. Collision (b) corresponds to WI c~2 0 and can only take place by virtue of the rules defined in the main