Dynamics of Heavy Quarks in Strongly Coupled $\mathcal{N}=4$ SYM Plasma

We calculate the probability distribution $P({\bf k})$ for a heavy quark with velocity $v$ propagating through strongly coupled $\mathcal{N}=4$ SYM plasma in the 't Hooft limit at a temperature $T$ to acquire a momentum ${\bf k}$ due to interactions with the plasma. This distribution encodes the well-known drag coefficient $\eta_D$ and the transverse and longitudinal momentum diffusion coefficients $\kappa_T$ and $\kappa_L$. Furthermore, our calculation determines all of the higher order and mixed moments to leading order in $1/\sqrt{\lambda}$ for the first time. These non-Gaussian features of $P({\bf k})$ include qualitatively novel correlations between longitudinal energy loss and transverse momentum broadening at nonzero $v$. We demonstrate that these non-Gaussian characteristics can be sizable in magnitude and even dominant in physically relevant situations. We use these results to derive a Kolmogorov equation for the evolution of the probability distribution for the total momentum of a heavy quark that propagates through strongly coupled plasma. This evolution equation accounts for all higher order correlations between transverse momentum broadening and longitudinal energy loss, which we have calculated from first principles. It reduces to a Fokker-Planck (FP) equation when truncated to only include the effects of $\eta_D$, $\kappa_T$ and $\kappa_L$. Remarkably, while heavy quarks do not reach kinetic equilibrium with the plasma if evolved with this FP equation, we demonstrate that heavy quarks do reach kinetic equilibrium if evolved with the all-order Kolmogorov equation we have derived. Our results thus provide a dynamically complete framework for understanding the thermalization of a heavy quark that may be initially far from equilibrium in the strongly coupled $\mathcal{N}=4$ SYM plasma -- as well as new insight into heavy quark transport and equilibration in quark-gluon plasma.