Application of AdS/CFT Correspondence to Studies on Non-equilibrium Steady States Shin Nakamura (Chuo University and ISSP, U. Tokyo) Refs. S. N. and H. Ooguri, PRD88 (2013) H. Hoshino and S. N., PRD91 (2015) H. Hoshino and S. N. in preparation.

Gravity 100 years after discovery of general relativity It turned out that GR is more useful than expected. Thermodynamics of a 5d black hole Thermodynamics of 4d gauge particles For example, AdS/CFT correspondence Non-perturbative description of strongly-interacting quantum gauge theories Gravity is not only for gravity:

Application to Non-equilibrium Physics Thermodynamics describes equilibrium systems. How bout non-equilibrium systems? Construction of “thermodynamics” for non-equilibrium systems is still a great challenge in modern physics. Can we get any information from gravity dual? A system of conductor with constant current along the constant electric field. Air J E Heat is produced: non-equilibrium. But, the system is time-independent. Simplest example of non-equilibrium systems Non-equilibrium steady state (NESS) Energy is dissipated into the heat bath.

Gravity dual of NESS 4d boundary horizon D-brane 5th-direction E Horizon for fluctuations of D-brane. Cartoon of gravity dual Heat bath: a black hole geometry at T in the bulk (Describes the gluon sector that plays a role of heat bath.) NESS: a probe D-brane on the BH geometry (Describes the quark sector that plays a role of charged particles.) See eg. [Karch and O’Bannon, 2007] for D3-D7 system. Non-linear conductivity can be computed.

Now we have two temperatures r=r H r boundary r=r * Black hole horizon gives the temperature of the heat bath. The horizon on the D-brane defines another “temperature” that is seen by the fluctuations in NESS, such as the current density fluctuations. We call this effective temperature T eff of NESS. fluctuationdissipation See also, [Gursoy et al.,2010] [S.N. and H. Ooguri, 2013]

Analogue black hole: “non-gravity” black hole Example: sonic black hole in liquid helium. SlowFast Sonic horizon where the flow velocity exceeds the velocity of sound. The sound cannot escape from inside the “horizon”. It is expected that the sonic horizon radiates a “Hawking radiation” of sound at the “Hawking temperature”. [W. G. Unrhu, PRL51(1981)1351] The fluctuations on the D-brane are based on the DBI theory: another example of analogue black hole.

What we can read from analogue black holes Effective temperature The Hawking temperature of the analogue black hole governs the fluctuation-dissipation relation of NESS. Average velocity of charged particles in NESS: The average velocity of the charged particles in NESS is given by the velocity of black hole. [H. Hoshino and S.N., to appear] [S.N. and H. Ooguri, 2013; H. Hoshino and S.N., 2015] Earlier works include: [Gursoy et al.,2010, Kim-Shock-Tarrio 2011, Sonner-Green 2012]

Questions to you What is the physical meaning of the area of the horizon of an analogue black hole? Entropy? If yes, entropy of what? Are there any “thermodynamics” associated with analogue black holes? Zero-th law, First law, Second law, Third law? Answering to these questions may open a new window to a generalization of thermodynamics in NESS. Analogue black holes may be useful than expected. Analogue black hole may have more information on non-equilibrium physics.