Localization of thermal packets and metastable states in the Sinai model

We consider the Sinai model describing a particle diffusing in a one-dimensional random force field. As shown by Golosov, this model exhibits a strong localization phenomenon for the thermal packet: all thermal trajectories starting from the same initial condition in the same sample remain within a finite distance of each other even in the limit of infinite time. More precisely, he has proved that the disorder average P(t)(y) of the distribution of the relative distance y=x(t)-m(t) with respect to the (disorder-dependent) most probable position m(t), converges in the limit t--> infinity, towards a distribution P(G)(y) defined as a functional of two independent Bessel processes. In this paper, we revisit this question of the localization of the thermal packet. We first generalize the result of Golosov by computing explicitly the joint distribution P( infinity )(y,u) of relative position y=x(t)-m(t) and relative energy u=U(x(t))-U(m(t)) for the thermal packet. Next, we compute the localization parameters Y(k), representing the disorder-averaged probabilities that k particles of the thermal packet are at the same place in the infinite-time limit, and the correlation function C(l) representing the disorder-averaged probability density that two particles of the thermal packet are at a distance l from each other. We, moreover, prove that our results for Y(k) and C(l) exactly coincide with the thermodynamic limit L--> infinity of the analog quantities computed for independent particles at equilibrium in a finite sample of length L. So even if the Sinai dynamics on the infinite line is always out-of-equilibrium since it consists in jumps in deeper and deeper wells, the particles of the same thermal packet can nevertheless be considered asymptotically as if they were at thermal equilibrium in a Brownian potential. Finally, we discuss the properties of the finite-time metastable states that are responsible for the localization phenomenon and compare with the general theory of metastable states in glassy systems, in particular as a test of the Edwards conjecture.