holographic entropy production - mpp theory group · overview and repapration non-equilibrium and...

Overview and preparationNon-equilibrium and �uid dynamics

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Holographic entropy production

Yu Tian (田雨)1

1School of Physics, University of Chinese Academy of Sciences(中国科学院大学物理学院)

(Based on the joint work [arXiv:1204.2029] with Xiaoning Wu and HongbaoZhang, which received an honorable mention in the 2012 Essay Competition of

the Gravity Research Foundation)

Gauge/Gravity Duality 2013Max Planck Institute for Physics, 30 July 2013

Yu Tian (田雨) Holographic entropy production

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The problem

Perturb a thermodynamic system in equilibrium=⇒Various transport processes pull it back to equilibrium=⇒Production of entropy

Perturb a (static) black hole=⇒The black hole absorbs the energy of perturbations=⇒Increase of the black-hole entropy

Do the above two physical processes have direct relationship?

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The problem

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The problem

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The physical picture

Thanks to holography!Bulk: a black hole thateats everythingBoundary: transportationthat smoothes everything

Figure : A sketch map

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Outline

1 Overview and preparationHolography (bulk/boundary correspondence)In equilibrium: thermodynamics (and phase transition)

2 Non-equilibrium and �uid dynamicsIn non-equilibrium: transportation and entropy productionDiscussions

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Holography (bulk/boundary correspondence)In equilibrium: thermodynamics (and phase transition)

Outline

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Holography: a brief introduction

Early (rough) ideas of holographyG. 't Hooft (1993); L. Susskind (1995).

A more precise prescription: AdS/CFTJ. Maldacena (1998).S.S. Gubser et al (1998); E. Witten (1998).Basic principle (Euclidean):

ZBd+1 [φ + δ φ ] = ZBd+1 [φ ]

⟨exp

δ φOφ

⟩CFT

Generalization: bulk/boundary correspondenceAdS/QCD, AdS/CMT, holographic entanglement entropy,gravity/�uid, . . .

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ZBd+1 [φ + δ φ ] = ZBd+1 [φ ]

⟨exp

δ φOφ

⟩CFT

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ZBd+1 [φ + δ φ ] = ZBd+1 [φ ]

⟨exp

δ φOφ

⟩CFT

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The bulk/boundary correspondence

bulk: not necessarily (asymptotic) AdSboundary: not necessarily conformal (e�ective FT)[Takayanagi et al (2010), Strominger et al (2011), Maldacenaet al (2013), . . . ]

The general principle:[φ |bdry ↔ Non-dynamical (background) �eld φ ]

Zbulk[φ ] =∫Dψ exp(−IFT[φ ,ψ]) =⇒

Zbulk[φ + δ φ ] = Zbulk[φ ]

⟨exp

∫bdry

δ φOφ

√gddx

Oφ = −δ IFT[φ ,ψ]√gδ φ(x)

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bulk: not necessarily (asymptotic) AdSboundary: not necessarily conformal (e�ective FT)[Takayanagi et al (2010), Strominger et al (2011), Maldacenaet al (2013), . . . ]

The general principle:[φ |bdry ↔ Non-dynamical (background) �eld φ ]

Zbulk[φ ] =∫Dψ exp(−IFT[φ ,ψ]) =⇒

Zbulk[φ + δ φ ] = Zbulk[φ ]

⟨exp

∫bdry

δ φOφ

√gddx

Oφ = −δ IFT[φ ,ψ]√gδ φ(x)

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The general principle under classical approximation of the bulkgravity:

exp(−Ibulk[φ ]) =∫Dψ exp(−IFT[φ ,ψ])

with Ibulk[φ ] the on-shell action (Hamilton's principal function).

Variation with respect to φ gives

− δ Ibulk[φ ]√gδ φ(x)

=⟨Oφ (x)

Further variations give the correlations of Oφ on the boundary.

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The general principle under classical approximation of the bulkgravity:

exp(−Ibulk[φ ]) =∫Dψ exp(−IFT[φ ,ψ])

with Ibulk[φ ] the on-shell action (Hamilton's principal function).

Variation with respect to φ gives

− δ Ibulk[φ ]√gδ φ(x)

=⟨Oφ (x)

Further variations give the correlations of Oφ on the boundary.

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Important examples (with nµ the unit normal to the boundary)Fields Bulk Boundary

EM Aµ −nµFµa|bdry Current 〈Ja〉

Grav. gµν Brown-York tab|bdry Stress tensor⟨T ab

⟩Additional dictionaryBlack holes ↔ Thermal �eld theory[Euclidean partition function = partition function of the(grand) canonical ensemble]

{Local Hawking temperature = Temperature

Bekenstein-Hawking entropy = Entropy

Macroscopic investigation: Thermodynamic and hydrodynamicdescriptions under the low-frequency/long-wavelength limit,where

⟨Oφ (x)

⟩FT are just the macroscopic physical quantities.

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⟨Oφ (x)

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⟨Oφ (x)

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Why macroscopic investigation?

Figure : A prestigious example: Quark-gluon plasma produced in LHC

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The holographic prediction η

4πfor QGP in N = 4 SYM

(Policastro, Son & Starinets, 2002) is in qualitative agreementwith the RHIC data.

We hope to know how general holography can be, throughmacroscopic investigation.(Tentative answer: As general as gravitation!)

We hope to know the general features of holography, throughmacroscopic investigation.

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The holographic prediction η

4πfor QGP in N = 4 SYM

(Policastro, Son & Starinets, 2002) is in qualitative agreementwith the RHIC data.

We hope to know how general holography can be, throughmacroscopic investigation.(Tentative answer: As general as gravitation!)

We hope to know the general features of holography, throughmacroscopic investigation.

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Outline

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Thermodynamics

Equilibrium thermodynamics from the bulk point of view(Brown & York, 1993)

The Brown-York tensor for Einstein's gravity

tab =1

8πG(Kgab−K ab)

in static con�gurations has a form

tab = εuaub +phab, hab = gab +uaub

of the (relativistic) perfect �uid.

=⇒ dE +pdV = TdS + µdQ (the 1st law of thermodynamics)

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Thermodynamics

The holographic interpretation of the Brown-Yorkthermodynamics

In more general gravitational theories: Hamilton-Jacobi-likeanalyses

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In non-equilibrium: transportation and entropy productionDiscussions

Outline

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Transportation in non-equilibrium thermodynamics

Small perturbations: Linear response theoryExample 1: Ohm's law

J i = σE i

Example 2: Newton's law of viscosity

T xy =−2ησxy

Type Response Driving Force Transport

Heat conduction Heat �ow Temp. gradient Energy

Shear viscosity Momentum �ow Gradient of v ‖ p‖Bulk viscosity Momentum �ow Gradient of v⊥ p⊥

Eletric conduction Electric current Potential gradient Charge

Table : Transportation

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Cross-transportation (such as the thermoelectricityphenomena):

JA = ∑B

where the matrix (LAB) of transport coe�cients is symmetricand positive de�nite (Onsager's reciprocal relation).

The holographic realization:

linear (grav. and material) perturbations around the static bulkspacetime (black hole)ingoing boundary conditions on the horizonsolving the linear perturbation equations in the bulk

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Cross-transportation (such as the thermoelectricityphenomena):

JA = ∑B

where the matrix (LAB) of transport coe�cients is symmetricand positive de�nite (Onsager's reciprocal relation).

The holographic realization:

linear (grav. and material) perturbations around the static bulkspacetime (black hole)ingoing boundary conditions on the horizonsolving the linear perturbation equations in the bulk

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gµν → gµν +hµν , nµhµν = 0

Type Driving Force Bulk perturbation

Heat conduction Temp. gradient ∇i1T

Grav. perturb. 1fcT

∂thti

Shear viscosity Gradient of v ‖ Grav. perturb. 1√fc

∂thij

Bulk viscosity Gradient of v⊥ Grav. perturb. 1√fc

∂thii

Eletric conduction Potential gradient Ei EM perturb. 1√fcFti (rc)

Table : Holographic realization of transp. (in certain gauge)

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Entropy production

The general (non-relativistic) entropy production rate

Σ = ∑A

JAXA = ∑AB

XALABXB

Type Driving force Entropy production

Heat conduction Temperature gradient -

Viscosity Velocity gradient Friction heat/T

Electric condution Electric �eld Joule heat/T

Table : Entropy production of transport processes

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Entropy production

The boundary side: entropy production rate

Σ = Jq ·∇1

T− 1

TΠ : ∇u+

TJ ·E = J iq∇i

T− 1

σij +1

TJ iEi

with Πij the dissipation part of the stress tensor and

σij =1

2(∇iuj + ∇jui ) (∇ ·u = 0)

The bulk side: entropy variation

δS =δE

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Entropy production

The boundary side: entropy production rate

Σ = Jq ·∇1

T− 1

TΠ : ∇u+

TJ ·E = J iq∇i

T− 1

σij +1

TJ iEi

with Πij the dissipation part of the stress tensor and

σij =1

2(∇iuj + ∇jui ) (∇ ·u = 0)

The bulk side: entropy variation

δS =δE

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Entropy production

Consider the Q = 0 (chargeless black hole background) case,where it turns out that there is no cross-transportation, forsimplicity.

By construction of conserved currents relating the horizon andthe boundary, one can verify δS =

∫bdry Σ, the equality of

entropy increase of the bulk black hole and total entropyproduction of the boundary �uid (YT, X.-N. Wu & H.-B.Zhang, 2012).

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Entropy production

Consider the Q = 0 (chargeless black hole background) case,where it turns out that there is no cross-transportation, forsimplicity.

By construction of conserved currents relating the horizon andthe boundary, one can verify δS =

∫bdry Σ, the equality of

entropy increase of the bulk black hole and total entropyproduction of the boundary �uid (YT, X.-N. Wu & H.-B.Zhang, 2012).

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Outline

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Discussions

Non-equilibrium thermodynamics (boundary): energy isdissipated in irreversible processesThe holographic point of view (bulk): energy of perturbationsis absorbed by the black hole

In cases with bulk viscosity, it seems that the spatial isotropy isrequired for a holographic realization of entropy production.

The Q 6= 0 (charged black hole background) case (withcross-transportation)YT, X.-N. Wu and H.-B. Zhang, in preparation.

Cases for more general gravitational theories with variousmatter content

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Discussions

Non-equilibrium thermodynamics (boundary): energy isdissipated in irreversible processesThe holographic point of view (bulk): energy of perturbationsis absorbed by the black hole

In cases with bulk viscosity, it seems that the spatial isotropy isrequired for a holographic realization of entropy production.

The Q 6= 0 (charged black hole background) case (withcross-transportation)YT, X.-N. Wu and H.-B. Zhang, in preparation.

Cases for more general gravitational theories with variousmatter content

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Towards boundary �uid dynamics: The 1st way

Non-relativistic long-wavelength expansionI. Bredberg, C. Keeler, V. Lysov & A. Strominger,[arXiv:1101.2451].R.-G. Cai, L. Li & Y.-L. Zhang, JHEP 1107 (2011) 027[arXiv:1104.3281].C. Niu, YT, X. Wu & Y. Ling, Phys. Lett. B 711 (2012) 411[arXiv:1107.1430].

Works for boundary at �nite distanceBut only works for intrinsically �at boundary

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Towards boundary �uid dynamics: The 2nd way

Petrov-like boundary conditionV. Lysov & A. Strominger, [arXiv:1104.5502].T. Huang, Y. Ling, W. Pan, YT & X. Wu, JHEP 1110 (2011)079 [arXiv:1107.1464].T. Huang, Y. Ling, W. Pan, YT & X. Wu, Phys. Rev. D 85(2012) 123531 [arXiv:1111.1576].C.-Y. Zhang, Y. Ling, C. Niu, YT & X. Wu, Phys. Rev. D 86(2012) 084043 [arXiv:1204.0959].

Works for intrinsically curved boundaryBut only works for boundary approaching the horizon

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Open problems

Holographic entropy production in the far-from-equilibriumcase?

Holographic thermodynamics and entropy production withquantum corrections to the bulk gravity

Boundary �uid dynamics in more general cases?

Boundary �uid dynamics with higher-order transportcoe�cients

Holographic (non-linear) super�uid dynamics

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Open problems

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Open problems

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Thank you!

holographic entropy production - mpp theory group · overview and repapration non-equilibrium and...

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