  In this study we derive a single-particle equation of motion, from
first-principles, starting out with a microscopic description of a tracer
particle in a one-dimensional many-particle system with a general two-body
interaction potential. Using a new harmonization technique, we show that the
resulting dynamical equation belongs to the class of fractional Langevin
equations, a stochastic framework which has been proposed in a large body of
works as a means of describing anomalous dynamics. Our work sheds light on the
fundamental assumptions of these phenomenological models.
