Effective Temperature ofNon-equilibrium Steady States
in AdS/CFT Correspondence
Shin Nakamura (Chuo Univ.)
中村 真 （中央大学理工学部物理学科）

Refs.
S. N. and H. Ooguri, PRD88 (2013) 126003.
H. Hoshino and S. N. PRD91 (2015) 026009.
We employ the natural unit: kB=c=ℏ=1.

My personal view onquark-hadron physics• QCD: physics of quarks, hadrons,……
• “Condensed matter physics”• Phase diagram, etc. : statistical physics• Many people are working on/interested in
non-equilibrium physics.
What is the difference from condensed matter physics?

Relativistic system.
It is “high-energy condensed matter physics,”
or “relativistic condensed matter physics.”

Relativistic systems
out of equilibrium
Are there any non-equilibrium phenomena that are
peculiar to/vivid in relativistic systems of particles?

I would like to bring some results that can bea starting point for further discussion.
My method: AdS/CFT correspondence
Going to analyze non-equilibrium physics
in terms of general relativity.

Fundamental question
in statistical physics

What is temperature?

We have many answers.

Definitions of temperaturePeET1tt
, E E T Statistical distributionsdE  TdS

D  T
Diffusion const.

Mobility

Thermodynamics
Fluctuation-dissipation
relation
Two sides of thesame coin.

We have another definition of temperature: a a

Killing vectorb
Horizon 2T Hawkingbtemperature
HorizonPhoto taken from http://www.iqs.com/iqs-blog/common-ground-quality-management-risk-management/One side of the coin: general relativity
Gravitational force is coming fromthe curved spacetime.
Einstein-Hilbert action:

Curvature:∼combination of second
derivatives of the metricc3d 1
Sd x  g R  2   Sstar

16 GN
Cosmological constant
Einstein’s equation:

Metric: defines unit length in the geometry

8 GN (star)
1R  R g   g  T
42c

2nd-order
differential equation

Picture:http://www.faculty.iu-bremen.de/course/fall02/c210101/students/BlackHoles/

Black hole
A solution to the Einstein’s equation.
“strongly curved”“weakly curved”
radial direction

Light cannot escape(trapped region)

Horizon(Apparent horizon)

Light can escape(un-trapped region)

Important quantities:• Area (A)• Surface gravity (κ)

Black hole mimics thermodynamics
Black holes obey:

Area of the horizondM  TH dSBH
Mass of the black holekBc3 AS BH GN 4

Hawking temperature

Newton’s constantThis resembles of the first-law of thermodynamics

dE  TdS
This is not only an analogy. A black hole radiates a black-body radiation with
Hawking temperature: we can assign a temperature to a black hole.

Hawking, S. W. (1974). "Black hole explosions?". Nature 248 (5443): 30.

Temperature in black hole?

Black hole thermodynamics
Thermodynamics

Natural unit: kB=c=ℏ=1.
Black hole0-th law

T= const. at theequilibrium.

κ is constant in thestatic solution.

1st law

dE=T dS

dM=[κ/(8πGN)]dA

2nd law

Entropy never decreases.

The area of horizon (A) neverdecreases.

3rd law

We cannot reach T=0 in
any physical process.

κ cannot reach zero inany physical process.

κ: surface gravity (the gravitational acceleration at the horizon of the black hole)
GN: Newton’s constant,
M: mass of the black hole
A: area of the horizonκ and A mimic T and S, respectively.
TH= κ /2π, by Hawking.

AT, S2
4GNThis does not seem to be just a coincidence.

An answer from string theory:
AdS/CFT correspondence[Maldacena, 1997]

A quantum field theoryof gauge particles

A conjecture, but no contradiction
has been established so far.

Equivalent
When this correspondence
was discovered, the curved
spacetime was AdS and thegauge theory was CFT.
However, non-AdS/non-CFT
version has been also proposed.

A classical theory of gravity
on a curved geometry inhigher-dimensions.

An answer from string theory:
AdS/CFT correspondence[J. Maldacena, 1997]

A quantum field theoryof gauge particles

A classical theory of gravity
on a curved geometry

Finite temperature

Many-body systemof gauge particles
at temperature T

Black hole geometry(typically on AdS5)
at temperature T[E. Witten, 1998]

Mystery in gravity
Many-body systemof gauge particles
at temperature T

Black hole geometry(typically on AdS)at temperature T• AdS/CFT is a correspondence
at the level of microscopic
theory of gauge particles.

This is a solution to theEinstein’s equation(2nd-order differentialequation).

• What we have done in the gravity
side was just solving the differentialequation.

Who did the coarse graining to reachthe thermodynamics?

Mystery
Benefit in gravity
Many-body systemof gauge particles
at temperature T

Black hole geometry(typically on AdS)at temperature T• AdS/CFT is a correspondence
at the level of microscopic
theory of gauge particles.

This is a solution to theEinstein’s equation(2nd-order differentialequation).

• What we have done in the gravity
side was just solving the differentialequation.

Somehow, the coarse graining is“automatic.”

Realization of non-equilibrium systems•

Usually, we begin with an equilibrium system.

•Then, we drive the system into non-equilibriumby acting an external force.

In the gravity dual:• We have already prepared an equilibrium systemin terms of black hole.
• The external force corresponds to a particularboundary condition in Einstein’s equation.
We can solve the equation, and the solution describes
the non-equilibrium state.

Our system to consider
Langevin system
A test particle immersed ina heat bath is driven bya constant external force.

test particle

heat bath

The friction produces dissipation:
the system is driven to out of equilibrium.
The heat bath: gluons in the deconfinement phase at T.

The test particle: a heavy quark.

Strategy
A strongly-interacting
quantum gauge theory

Heat bath(gluons)
Test particle(quark)in the heat bath

AdS/CFT
equivalent

A classical gravity(general relativity)
on a curved spacetimeof one dimension higher.

Black HoleA string in theblack hole geometry

Picture of many-body system is taken from internet.
Picture of black hole is taken from https://www.kahaku.go.jp/exhibitions/vm/resource/tenmon/space/theory/theory06.html

A cartoon in the gravity dual
stringboundary[Gubser, 2006][Herzog et al., 2006]

5-th direction r

Energy-momentum tensor of string

T0r＝energy flow into the black hole in unit time: dissipation＝Work in unit time by the force acting on the test particle

Lf  0 at v  0.
 ( r x) boundary[Gubser, 2006][Herzog et al., 2006]

Computation of drag force[Gubser, 2006], [Herzog et al., 2006]



Lstring  ( tension )  det  a X  b X g X (t , r )  vt  x(r )L f ( r x )

L
r
0
 ( r x )g rr ( g tt )  g xx v 2( r x )  f
 g tt g rr ( g tt ) g xx  f 2
2

Let us define a point r=r* by( g tt ) g xx  f 2

r*

02

( g tt )  g xx v

2r*If f satisfies this,𝜕𝑟 𝑥 can be real.

 0.

Right-hand-side can be negative.

f is given as
a function of v.

A special pointstringboundary

5-th direction r

There is a special point (r=r* ).

What is this special point?

“Special point” r=r*
It is a “horizon” on the string worldsheet
seen by the small fluctuations.
(See also [Gubser 2008, Kim-Shock-Tarrio 2011, Sonner-Green 2012])

We call it “effective horizon.”

How to see it?
• We find a curved string in the presence of external force.
• What is the “metric” that governs the small fluctuationof the string?

Langevin system See also, [Gubser, 2008]
Horizon at r=r* on theinduced geometry.

boundary
5-th direction r

Equation of motion for small fluctuation δX of the string:aequation  g~ g~   X   0, Klein-Gordon
on a curved spacetime
abbg~ab   a X   a X  g 

given by the induced metric.

Now we have two temperatures
r=rH

r=r*

Black hole horizon
gives the temperatureof the heat bath.

rboundary

The effective horizon on the string
gives a different “Hawking temperature”
that governs the fluctuations of thetest particle.

We call this effective temperature Teff of NESS.

If the system is driven to NESS,rH<r* at the order of v2 .
Two temperatures appear only in the non-linear regime.

Computations ofeffective temperature[S. N. and H. Ooguri, PRD88 (2013) 126003]
We have computed Teff for wide range of models.

• Heat bath:Near-horizon geometry of Dp-brane solution at T.
[Itzhaki-Maldacena-Sonnenschein-Yankielowitcz, 1998]

ds  r27 p2

 1 r07 p

22
dt  dx r 7 p

dr 2r7 p
2

1 

r07 pr 7 pp 32

 r d82 p• Test particle: probe D(q+1+n)-brane or F1 string
wrapped on n-sphere• Charged particles: probe D(q+1+n)-braneBehavior of effective temperature
Can never been understoodas a Lorentz factor.

Teff  1  v

c0 4
7 p

12 7 pBeyond the linear-response regime
12 2 1  Kv 1
2  2
4T T  K v
T
O(v)
2
7 p

1
p 3 , K  q 3 p n27 p This factor can be negative!

Teff < T can be realized.

For a conformally invariant model (N=4 susy YM), K=0:
Teff  T

<T[Gubser, 2008]  1 1v

2The temperature seen by the fluctuation can be made smaller by driving
the system into out of equilibrium.

Is Teff<T allowed?
It is not forbidden.
Some examples of smaller effective temperature:[K. Sasaki and S. Amari, J. Phys. Soc. Jpn. 74, 2226 (2005)][Also, private communication with S. Sasa]

Is it OK with the second law?
• The system is an open system. No contradiction.
• The second law of thermodynamics applies toa closed system.
• The definition of “non-eq.” entropy (beyond thelinear response regime) is not clear.

What is the physical meaning of Teff?
Fluctuation of string

Fluctuation of external force
acting on the test particle

Computations of correlation functions of fluctuations in the gravity dualis governed by the ingoing-wave boundary condition at the effective horizon.

Im G R ( ) dt  f (t ) f (0) v  0  2Teff 0 ,v0fluctuationdissipation
See also, [Gursoy et al.,2010]

The fluctuation-dissipation relation at NESS is characterized by
the effective temperature (at least for our systems).

What is temperature?
Definitions of equilibrium temperature:

Pe

ET

1t
t

, E E TdE  TdS

D  T
diffusion const.

mobility

Distributions

Thermodynamics
Fluctuation-dissipation
relation

We have another definition of temperature: a a

Killing vectorb
Horizon 2T Hawking
btemperatureHorizon

What is temperature?
Definitions of effective temperature:PeET1t
t
, E E TdE  TdSv
D  Teff fdiffusion const.

Thermodynamics

differential mobility

Fluctuation-dissipation
relation

They give the same temperature.

 a ab

Killing vector

Eff. Horizon

Distributions

 2Teff Hawking
btemperature
Eff. Horizon

What is temperature?
Definitions of effective temperature:PeET1t
t

, E E TdE  TdSv
D  Teff fdiffusion const.

Thermodynamics

differential mobility

Fluctuation-dissipation
relation

They give the same temperature.

 a ab

Killing vector

Eff. Horizon

Distributions

 2Teff Hawking
btemperature
Eff. Horizon

What is temperature?
Definitions of effective temperature:

PeE

Teff

1t

t

, E E TdE  TdSv
D  Teff fdiffusion const.

Thermodynamics

differential mobility

Fluctuation-dissipation
relation

They give the same temperature.

 a ab

Killing vector

Eff. Horizon

Distributions ofsmall fluctuations

 2Teff Hawking
btemperature
Eff. Horizon

Associated thermodynamics?
??

dE  Teff dS
It is highly nontrivial.
Hawking radiation (Hawking temperature) is moregeneral than the thermodynamics of black hole.
Hawking radiation:
It occurs as far as the “Klein-Gordon equation” of fluctuation
has the same form as that in the black hole.
Thermodynamics of black hole:
We need the Einstein’s equation. It relies on the theory of gravity.

Example of “non-gravity”
Hawking radiation
Sonic black hole in liquid helium.

Fast

Sonic horizon where the flow velocity
exceeds the velocity of sound.

Slow• The sound cannot escape from inside the “horizon”.
• It is expected that the sonic horizon radiates a “Hawkingradiation” of sound at the “Hawking temperature”.
[W. G. Unrhu, PRL51(1981)1351]

However, any “thermodynamics” associated with theHawking temperature of sound has not been established so far.
[See for example, M. Visser, gr-qc/9712016 ]Summary
For some examples of non-equilibrium steady states:• There exists two temperatures in the non-linear regime.
• The effective temperature appears in terms of theHawking temperature at the effective horizon.
• It agrees with the coefficient in the generalized
fluctuation-dissipation relation in NESS.

• Teff < T can happen for some cases.

Some more hint for non-equilibrium physics?

Any relevance in QGP physics?
For a conformally invariant model (N=4 susy YM), K=0:
Teff  T<T

[Gubser, 2008]



1
1 v

2

c2

“Relativistic” effect

2v
Teff  1  
1
7 p

c22v
1 K

c2

12

 41
2  v2v
T  T T  K 
 c2  Oc42
7 p

Another scale in general.

I cannot say that the test particle has to be relativistic.
But, the effect will be vivid if the particle is relativistic.

Teff ～0 if v～c, (at least for this model).