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The dynamics of heavy quarks in quark-gluon plasma (QGP) formed in heavy ion collisions provide a unique window to characterize its properties. Existing approaches to describe heavy quarks in medium rely either on quasiparticle-based models of QGP, or on assuming that the momentum transfer from the medium follows Gaussian statistics. However, in the prototypical theory that is strongly coupled N=4 SYM, it has been long known there are no momentum-carrying quasiparticles, and furthermore, we have recently shown that the momentum transfer from the medium is far from being Gaussian. Since then, we showed that the asymmetry of said momentum transfer between energy loss and energy gain — which affects all moments of the distribution, not only its Gaussian characteristics — is, in fact, theory-independent. The deviations from Gaussianity can be substantial even at moderate Lorentz boost factors.

 In this talk, we present a new heavy quark transport framework that encodes all of the non-Gaussian features of the momentum transfer from the medium, all of which can be defined and in principle calculated in QCD, without relying on any assumptions regarding the strength of the coupling or the speed of the heavy quark. As a demonstrative example, we discuss how heavy quark stopping and equilibration takes place in N=4 SYM highlighting clear quantitative differences between the full theory and the Gaussian truncation, and the role of "survivor bias" effects. This paves the way towards extracting information about fundamental properties of QGP that have not been previously accessible.