Heavy-quark thermalization in the quark-gluon plasma via a fractional Langevin equation
The dynamics of heavy quarks in the quark-gluon plasma is commonly described within a standard Langevin equation framework. In this work, we extend the conventional framework by using a fractional Langevin equation based on the Caputo derivative of order $α$ with $1<α\leq 2$. We solve the fractional Langevin equation and derive the corresponding {\textit {fractional fluctuation-dissipation theorem}} for the heavy quark thermalization in the quark-gluon plasma medium. We calculate the time evolution of the mean-squared momentum, mean-squared displacement, momentum distribution, energy loss, longitudinal momentum loss, transverse momentum broadening, and the nuclear suppression factor of the heavy quarks. The mean-squared momentum thermalizes to the same value for all $α$ at late times, in agreement with the fractional fluctuation-dissipation theorem, and the nuclear suppression factor shows a systematic $α$-dependence. The energy loss of an energetic heavy quark grows strongly with $α$. The time evolution of the heavy quark momentum distribution provides additional sensitivity to the fractional dynamics of the heavy quarks in the QGP medium.