  The fluctuation-dissipation theorem is a central theorem in nonequilibrium
statistical mechanics by which the evolution of velocity fluctuations of the
Brownian particle under a fluctuating environment is intimately related to its
dissipative behavior. This can be illuminated in particular by an example of
Brownian motion in an ohmic environment where the dissipative effect can be
accounted for by the first-order time derivative of the position. Here we
explore the dynamics of the Brownian particle coupled to a supraohmic
environment by considering the motion of a charged particle interacting with
the electromagnetic fluctuations at finite temperature. We also derive
particle's equation of motion, the Langevin equation, by minimizing the
corresponding stochastic effective action, which is obtained with the method of
Feynman-Vernon influence functional. The fluctuation-dissipation theorem is
established from first principles. The backreaction on the charge is known in
terms of electromagnetic self-force given by a third-order time derivative of
the position, leading to the supraohmic dynamics. This self-force can be argued
to be insignificant throughout the evolution when the charge barely moves. The
stochastic force arising from the supraohmic environment is found to have both
positive and negative correlations, and it drives the charge into a fluctuating
motion. Although positive force correlations give rise to the growth of the
velocity dispersion initially, its growth slows down when correlation turns
negative, and finally halts, thus leading to the saturation of the velocity
dispersion. The saturation mechanism in a suparohmic environment is found to be
distinctly different from that in an ohmic environment. The comparison is
discussed.
