Abstract
In this work, we consider non-linear corrections to the Langevin effective theory of a heavy quark moving through a strongly coupled CFT plasma. In AdS/CFT, this system can be identified with that of a string stretched between the boundary and the horizon of an asymptotically AdS black brane solution. We compute the Feynman-Vernon influence phase for the heavy quark by evaluating the Nambu-Goto action on a doubled string configuration. This configuration is the linearised solution of the string motion in the doubled black brane geometry which has been proposed as the holographic dual of a thermal Schwinger-Keldysh contour of the CFT. Our expression for the influence phase passes non-trivial consistency conditions arising from the underlying unitarity and thermality of the bath. The local effective theory obeys the recently proposed non-linear fluctuation dissipation theorem relating the non-Gaussianity of thermal noise to the thermal jitter in the damping constant. This furnishes a non-trivial check for the validity of these relations derived in the weak coupling regime.
Highlights
Non-linear Langevin theoryOur goal is to introduce the non-linear Langevin theory for the Brownian particle and its description within Schwinger-Keldysh formalism
The classic result of fluctuation dissipation theorem asserts that this variance of the fluctuation f 2 is directly proportional to the coefficient of linear drag/dissipation γ
On examining the influence phase computed from the string configuration, we find that the universal non-linear Langevin dynamics derived from weakly coupled intuition, continues to describe the non-linearities in this strongly coupled system
Summary
Our goal is to introduce the non-linear Langevin theory for the Brownian particle and its description within Schwinger-Keldysh formalism. We are interested in a heavy quark coupled to a CFTd plasma at a temperature T This quark feels a variety of forces, which at a macroscopic effective theory level result in energy/momentum dissipation and fluctuations. We immediately note that the system under consideration has a variety of symmetries including translational, rotational and reflection invariance These symmetries drastically reduce the number of non-linear corrections to the linear theory that can appear in general. The non-linear Langevin dynamics described by eq (2.1) will be shown to be dual to a Schwinger-Keldysh effective theory. We will argue that the ζγ-term in this equation represents the universal leading correction (consistent with the symmetries) to the standard linear Langevin dynamics
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