Classical model for diffusion and thermalization of heavy quarks in a hot medium: memory and out-of-equilibrium effects

We consider a simple model for the diffusion of heavy quarks in a hot bath, modeling the latter by an ensemble of oscillators distributed according to either a thermal distribution or to an out-of-equilibrium distribution with a saturation scale. In this model it is easy to introduce memory effects by changing the distribution of oscillators: we model them by introducing a Gaussian distribution, \begin{document}$ {\rm d}N/{\rm d}\omega $\end{document}, which can be deformed continuously from a \begin{document}$ \delta- $\end{document}function, giving a Markov dissipation, to a broad kernel with memory. Deriving the equation of motion of the heavy quark in the bath, we remark how dissipation comes out naturally as an effect of the back-reaction of the oscillators on the bath. Moreover, the exact solution of this equation allows to define the thermalization time as the time necessary to remove any memory of the initial conditions. We find that the broadening of the dissipative kernel, while keeping the coupling fixed, lowers the thermalization time. We also derive the fluctuation-dissipation theorem for the bath, and use it to estimate the kinematic regime in which momentum diffusion of the heavy quark dominates over drift. We find that diffusion is more important as long as \begin{document}$ K_0/{\cal E} $\end{document} is small, where \begin{document}$ K_0 $\end{document} and \begin{document}$ {\cal E} $\end{document} denote the initial energy of the heavy quark and the average energy of the bath, respectively.