Dynamic crossover and aging of a particle coupled to a fractal heat bath

We report the study of the one-time and two-time quantum dynamics of the Brownian particle coupled to a thermal bath of fractal dimension 0<δ<2. The autocorrelation of the position can be defined only if the particle is localized (T=0,0<δ<1), in which case the position thermalizes. On another hand, the displacement and its autocorrelation function investigated in this work are defined at any δ in the range 0<δ<2 and at any temperature T⩾0. The dynamics is found to be particularly rich when δ<1 where a dynamic crossover from a quasi-localized state at short times to a sub-diffusive regime at large times is observed, under conditions which we have determined. The displacement never reaches equilibrium: it ages. In particular, we find that the concept of effective temperature deduced from the violation of the fluctuation–dissipation theorem for this variable is always valid in the long-time limit, but fails at short times (i.e., times smaller than the time at which the dynamic crossover takes place) in the localized and in the quasi-localized regime.