Abstract
We study the motion of a stochastic string in the background of a BTZ black hole. In the 1+1 dimensional boundary theory this corresponds to a very heavy external particle (e.g., a quark), interacting with the fields of a CFT at finite temperature, and describing Brownian motion. The equations of motion for a string in the BTZ background can be solved exactly. Thus we can use holographic techniques to obtain the Schwinger–Keldysh Green function for the boundary theory for the force acting on the quark. We write down the generalized Langevin equation describing the motion of the external particle and calculate the drag and the thermal mass shift. Interestingly we obtain dissipation even at zero temperature for this 1+1 system. Even so, this does not violate boost (Lorentz) invariance because the drag force on a constant velocity quark continues to be zero. Furthermore since the Green function is exact, it is possible to write down an effective membrane action, and thus a Langevin equation, located at a “stretched horizon” at an arbitrary finite distance from the horizon.
Highlights
We study the motion of a stochastic string in the background of a BTZ black hole
To the extent that it assumes that time scales are larger than the microscopic time scale it must fail for very small time scales
We have studied the Brownian diffusion of a particle in one dimension using the holographic techniques
Summary
The Langevin dynamics [22] will be reviewed in brief. Suppose in a viscous medium a very heavy ( compared to the masses of the medium particles ) particle is moving. We can define the partition function for a heavy particle in a heat bath using a Schwinger-Keldysh contour (fig.). For very heavy particle we can consider the forces on the particle is very small compared to inertial term so we can expand it in second order , take the average over bath and make it an exponentiate again to get [Dx1][Dx2] ei dt1MQ0 x 21 e−i e dt2MQ0 x 22. We introduce a new random variable which we call ξ in anticipation that it will turn out to be the random noise, by defining e− This partition function is an average over the classical trajectories for the heavy particle under the noise ξ. If the Green function is expanded for small frequencies the coefficient of ω2 i.e, d2 x(t) dt adds to the mass of the particle and the coefficient of ω i.e, dx(t) dt will contributes as the drag term.
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