Nonlinear Langevin dynamics via holography
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Springer
Received: October 31, 2019 Accepted: December 26, 2019 Published: January 28, 2020
Nonlinear Langevin dynamics via holography
International Centre for Theoretical Sciences (ICTS-TIFR), Tata Institute of Fundamental Research, Shivakote, Hesaraghatta, Bangalore 560089, India
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] 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. Keywords: Holography and quark-gluon plasmas, Quantum Dissipative Systems, AdSCFT Correspondence ArXiv ePrint: 1906.07762
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)165
JHEP01(2020)165
Bidisha Chakrabarty, Joydeep Chakravarty, Soumyadeep Chaudhuri, Chandan Jana, R. Loganayagam and Akhil Sivakumar
Contents 1 Introduction
1
2 Non-linear Langevin theory 2.1 Stochastic path integral 2.2 Heavy quark in SK formalism 2.3 Thermality, time reversal and fluctuation-dissipation relations
5 6 7 9 10 10 14 17 19 20
4 Non-linear Langevin theory 4.1 Influence phase of the heavy quark 4.2 Influence phase in Keldysh basis and its derivative expansion
22 23 25
5 Discussion and conclusion
29
A Derivative expansion for Green’s function A.1 Retarded bulk to boundary Green’s function A.2 Derivative expansion of the linearised solution
31 31 33
B Exact solution in AdS3
33
1
Introduction
The Langevin theory, describing the motion of a particle coupled to a thermal bath, is the simplest example of an open quantum system. The linear version of this theory with a linear damping γ and Gaussian noise N i , described by the equation d2 q i dq i + γ = hf 2 iN i , dt2 dt
(1.1)
serves as a textbook example of non-equilibrium statistical mechanics. Here hf 2 i is the variance of the force per unit mass which controls the Gaussian ensemble for N i : ˆ hf 2 i i i i Probability[N ] ∼ exp − dt N N . (1.2) 2
–1–
JHEP01(2020)165
3 Holographic B
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