Non‐Equilibrium Fluctuations of Horizons
( 23 July, 2010 @ Riken, Wako )
(26 July, 2010 @ Journal club, KEK)
Theory Center , KEK & Sokendai
Satoshi Iso
based on a collaboration (gr‐qc/1008.1184) with Susumu Okazawa & Sen Zhang (KEK, Sokendai)
岡澤晋
張森
Planck units
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Plan of the talk
Chap. 1 Physics of horizons
Thermodynamic Laws (Classical vs. Quantum) of Black holes
Chap. 2 Non‐equilibrium thermodynamics
Crooks fluctuation theorem & Jarzynski equality
Chap. 3 Fluctuation theorem for Black Hole horizons
・Transition amplitude for changing area of horizon
・Proof of the Generalized second law and its microscopic violation
Chap. 4 Conclusions and Discussions
・speculations Satoshi Iso
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Chapter 1
Physics of Horizons
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Horizon: Null Hypersurface (light‐like surface) Any information can come from the other side of the horizon (classically)
Even light cannot come out of the BH horizon.
Escape velocity from
a star M at radius R
→ c (speed of light)
Critical radius (Schwarzshild radius)
Black hole horizon (including acoustic BH)
F
observer
dependent
L
R
P
de Sitter horizon
= cosmological horizon
(accelerating universe)
Rindler horizon =accelerating horizon
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horizon
Classically causality plays an important role to characterize the horizon.
F
L
R
L
R
Left modes (ingoing modes) are
decoupled from the outer world.
P
Horizon thermodynamics
Quantum mechanically, gravitational and gauge anomaly appears.
Flux of the Hawking radiation saves it. Black hole:
κ =surface gravity
at horizon
Accelerating
observer
Unruh effect
a =acceleration
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Bardeen Carter Hawking (73) + Hawking(76) + Beckenstein (74) + . . . Stationary black holes satisfy Equilibrium Thermodynamics Laws
0th law : Surface gravity is constant over the horizon
(temperature is constant in an equilibrium state)
1st law : Energy conservation between the horizon and r=∞
Local version of 1st law
: Energy flow across horizon
Killing vector generating horizon
It is easily proved by using Raychaudhuri eq. and Einstein eq.
Note that the proof makes no reference to spatial infinity, and applicable to local horizon.
(Normalization of killing vector is cancelled in dE and T. )
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2nd law : (generalized 2nd law)
no matter what happens (including negative energy flow into BH)
Various proofs : (a proof by Wald)
Outer region is described by Hartle‐Hawking state (= thermal).
BH
Energy:
Entropy:
small perturbation
Clausius relation (entropy is maximized for the thermalized state)
And integrate over the horizon (using 1st law)
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Chapter 2 Non‐equilibrium Thermodynamics
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Equilibrium thermodynamics
1st law (energy conservation)
2nd law (entropy increase) Einstein (Brownian motion)
Fluctuations around the equilibrium
Hydrodynamics Based on “local equilibrium”
and “equation of states”
In order to prove the reality of atoms,
Einstein proposed many ways to estimate
Avogadro Number using fluctuations. transport coefficients
Fluctuation – Dissipation Theorem
Linear response theorem
(Non‐equilibrium) fluctuation theorem
(Jarzynski, Crooks, … )
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Non‐equilibrium Fluctuation theorem and Jarzynski equality
Classifications of out‐of‐equilibrium states
(1) NETS (non‐equilibrium transient state) equilibrium state → switch on an external perturbation → system returns to a new equilibrium (2) NESS (non‐equilibrium steady‐state) :steady current in constant electric field
driven by external force (like a constant electric field)
→ energy is dissipated (heat) like Joule heat of electric current
→ stationary entropy production
(3) NEAS (non‐equilibrium aging state) slow relation = very small heat dissipation glassy system etc. Fluctuation theorems can be applied to these cases.
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Jarzynski Equality (97) W: mechanical work exerted on the system by perturbation change of parameter
Equilibrium
at t=0
out of equilibrium
Surprise (1) : lhs = an average of work by a non‐equilibrium process over various initial states
rhs = difference of free energy at equilibrium. Surprise (2)
Dissipated work (entropy production)
2nd
or
law of thermodynamics (entropy increase law) Satoshi Iso
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Surprise (3) : An averaged dissipated work is positive.
But, if we look at each microscopic process, negative dissipated work is necessary ! ( violation of 2nd law at the microscopic level )
Surprise (4) Linear response th., FDT, or Onsager reciprocal relation etc. are derived!
Crooks Fluctuation Theorem (2000)
Forward
change of parameter
Jarzynski equality
Integrate over Reversed Surprise (5) There are several (mesoscopic) experiments to show theses equalities!
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Proof of Crooks fluctuation theorem
Very easy, almost trivial A system described by configuration See e.g.
Ritort’s review
(07)
Sequence of configurations
:Probability that the system is in config. at time Markovian process: transition prob
Probability for a sequence of config
Note: Ratio of transition prob. is related to the entropy difference;
Local detailed balance
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reversed change of parameter
reversed sequence of config. Ratio:
Initial condition Assume that the initial state is in an equilibrium state
Define the total dissipation rate as Satoshi Iso
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Probability to produce a total dissipation along the forward protocol is
q.e.d
Note:
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Chapter 3
Fluctuation Theorem for BH
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Fluctuation theorems for black hole horizons
BH
Area = A
outer system
(detector or matter)
entangled
ADM mass M
is fixed.
How can we describe such a state?
= Einstein Hilbert action
Gravity is a constrained system. → Does Hamiltonian always vanish? → NO
Wheeler DeWitt eq.
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Variations of the action do not vanish if there are boundaries.
Regge Teitelboim (74)
Teitelboim (95) Boundary variables must be considered as independent variables.
(1) Spatial infinity It vanishes if we fix the ADM mass.
But if we want to fix the time‐interval at spatial infinity,
we need to Legendre transform Satoshi Iso
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(2) Horizon is another boundary
If area of the horizon is fixed, it vanishes.
Gibbons Hawking
Black hole entropy
For Euclidean case, Satoshi Iso
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BH
Area = A
outer system
(detector or matter)
entangled
ADM mass M
is fixed.
What is the transition probability from one configuration to another?
We should always impose the ingoing boundary condition at horizon.
( = initial state is in the Unruh vacuum for the outgoing modes)
Regularity at the horizon
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Transition amplitude from configuration C with BH area A to C’ with A’ is evaluated by a first order perturbation of the interaction Hamiltonian, mediate the interaction via Hawking radiation
Kruskal coordinate
On the other hand, the reversed amplitude is given by
Massar ,Parentani (99)
Kraus Wilczek (95)
Note: if we use
ordinary result of Hawking radiation rate
: heat transfer to outer region
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Ratio of transition probabilities for sequences of configurations of black holes with an initial condition for the outer region
Fluctuation theorem for Black holes
Jarzynski equality for Black holes
Generalized 2nd law for Black holes
Entropy decreasing process is microscopically necessary.
(Microscopic violation of 2nd law)
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Chapter 4
Conclusions and . . . A Fantasy
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Conclusions
We have applied the fluctuation theorem and Jarzynski relation to Black hole systems (also applicable to any local horizon)
→ (1) ratio of the transition amplitude is given by Detailed balance → entropy of BH
Hawking radiation
(backreaction is included)
(2) Applying the method in proving Crooks fluctuation theorem, we have obtained
total entropy
(3) We have proved the generalized 2nd law of black holes
It is important, however, that the GSL should be violated for each
microscopic process to satisfy the Jarzynski relation.
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Discussions (or a fantasy? )
(1) What is an analog of Avogadro number in gravity?
number of particles per mol Ù number of microstates per unit area
Avogadro number
Planck number (or Beckenstein)
・ Avogadro number cannot be determined within (local) equilibrium thermodynamics.
Various methods (i) Maxwell distribution (Loschmidt) Ù D‐brane dynamics ??
(ii) Planck’ s radiation law Ù Hawking radiation
(iii) Brownian motion (Einstein) Ù Brownian motion near horizons
Fluctuation
Fluctuations of space‐time ?
Is the space‐time made of atoms ???
Is the gravity entropic force ????
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(2) Origin of irreversibility (or Origin of dissipation) H‐theorem, Loschmidt, Zermelo,
Maxwell daemon
Hawking, Beckenstein, …
“Information paradox “ Can we manipulate the ‘heat bath’
(source of dissipation) at our will?
Dissipation = loss of entanglement
How can we take the effect of backreaction?
Non‐equilibrium fluctuation theorem
2nd law and its microscopic violation
Entropy = entanglement?
(Even Rindler horizon can have entropy)
(3) Einstein equation is a kind of equilibrium thermodynamic equation.
T. Jacobson
Derivation of Einstein eq
Black hole is the maximum entropy state. (Bousso bound)
But, Clausius inequality Deviation from Einstein eq ??
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