Nonextensive thermodynamics and stability of black holes

Nonextensive thermodynamics and stability of black holes

Viktor G. Czinner

Centro de Matematica, Universidade do Minho,Braga, Portugal

&HAS Wigner Research Centre for Physics,

Budapest, Hungary

Based on the work byT.S. Biro and V.G.Cz., Phys. Lett. B 726, 861 (2013);

Ongoing collaboration with Hideo Iguchi @ Nihon Unversity, Japan.

VII Black Holes Workshop, Aveiro University,

December 19, 2014

Viktor G. Czinner (CMAT U. Minho, Braga) Nonextensive Black Hole Thermodynamics VII BH Workshop, Aveiro, 12/19/2014. 1 / 20

Outline

1 Classical black hole thermodynamics and some of its issues

2 Nonextensive thermodynamics and a Zeroth Law compatible temperature

3 Nonextensive properties of a Schwarzschild black hole


Outline





The 4 Laws of Black Hole Mechanics and Thermodynamics

Zeroth Law:The surface gravity, κ of a stationary black hole is constant over the horizon.

The temperature, T is constant throughout the body in thermal equilibrium.

First Law:

dM = κ8πdA+ ΩHdJ + ΦdQ. ⇐⇒ dE = TdS + work terms.

Generalized Second Law:

δA+ δSm ≥ 0 in any process. ⇐⇒ δS ≥ 0 in any process.

Third LawIt is impossible by any physical process to achieve κ = 0 by a finite sequence of operations.

It is impossible by any physical process to achieve T = 0 by a finite sequence of operations. E ⇔M ; TH ⇔ κ2π ; SBH ⇔ A

4


Two Issues

The Planck-Nernst form of the third law requires that the S → 0 at lowtemperature, which is in contrast with traditional black hole thermodynamics,where the entropy is singular.

1 2 3 4 5 kBTMP

0.01

0.02

0.03

0.04

0.05

0.06

SkB

Stability Issue: In ordinary thermodynamics, local thermodynamical stabilityis linked to the dynamical stability of the system:

Def: Hessian(S) has no positive eigenvalues.

A positive mode in the Hessian means that at least some kind of smallfluctuations within the system are entropically favored, implying that thesystem is unstable against these.

The best known example of this failure is Schwarzschild black hole:A stable configuration with negative specific heat (i.e. a positive Hessian).


Black Holes are Nonextensive Objects

The above stability argument, however relies on the additivity of the entropy, aproperty which does not hold for black holes. Indeed, if two black holes coalesce,the area of the final event horizon is greater than the sum of the areas of theinitial horizons, i.e.

A12 > A1 +A2

Thus standard thermodynamical stability criteria fail in general when applied toblack holes because:

The entropy is nonadditive (scaling with A and not V ).

The long-range behaviour of the gravitational interaction.

In GR a local definition of mass and other familiar “should-be extensive”quantities is not possible.

BHs constitute elementary entities that cannot be subdivided into separatesystems, not even in any idealized manner.

Therefore BHs cannot be thought as made up of any constituent subsystems eachof them endowed with its own thermodynamics, and as such they are nonextensiveobjects.


Outline





Nonextensive thermodynamics

The basic variables of classical thermodynamics are the so-called extensive physicalquantities (X), like the energy (E), particle number of chemical components (Ni) and

the entropy (S). It is customary to assume that these quantities are added if we put two

thermodynamic bodies together:

X12 = X1 +X2.

Another important fundamental group of thermodynamic quantities are called intensives,

and their definition is related to thermodynamic equilibrium and to the zeroth and the

second laws of thermodynamics. The zeroth law of thermodynamics is formulated as the

requirement of transitivity of the thermodynamic equilibrium state. This transitivity

property implies the existence of an empirical temperature, which in standard

thermodynamics is defined as1

T=∂S

∂E.

However, any monotonic function of the entropy derivative is an equally usable empiricaltemperature, and in the above definition the additivity properties of the energy and theentropy are important ingredients.


A Zeroth Law compatible temperature

It can be shown, that in the case of nonadditive composition laws for the entropy(S) or the energy (E), or both, one can still define a temperature that is inaccordance with the zeroth and the second laws of thermodynamics:

1

T=∂L(S)

∂L(E).

Here the L and L functions are the so called formal logarithm of thenonextensives, which map the nonadditive composition laws to additive ones:

X12 6= X1 +X2 7−→

L12(S12) = L1(S1) + L2(S2),

L12(E12) = L1(E1) + L2(E2).

T.S. Biro and P. Van, Phys.Rev. E 83, 061147 (2013).


Outline





Nonextensive properties of a Schwarzschild black hole

Merging black holes.

The Bekenstein-Hawking entropy scales with the horizon area

S12 = S1 + S2 + 2√S1

√S2,

and we assume additivity in the energy

E12 ≈ E1 + E2.

Black hole in a finite size heath bath.

The leading order entropy composition law has a Tsallis type nonadditivity

S12 = S1 + S2 + aS1S2,

and we also consider additivity in the energy

E12 = E1 + E2.


Merging black holes

The standard thermodynamical functions of the Schwarzschild black hole.

SBH = 4πE2,1

TH= S′

BH(E) = 8πE, CBH =−S′2

BH(E)

S′′BH(E)

= −8πE2.

We map the nonadditive Bekenstein-Hawking entropy to its formal logarithm fordefining the thermodynamical analysis.

The additive entropy of a Schwarzschild black hole

S12 = S1 + S2 + 2√S1

√S2 ⇒ S ≡ L(SBH) = 2

√SBH .

New thermodynamical functions

S = 4√πE, T =

1

S′(E)=

1

4√π, C =

−S′2(E)

S′′(E)= −∞.


Thermodynamic functions & stability

1 2 3E/MP0

10

20

30

S/kB

(a) The entropy function

1 2E/MP0

1

4 π

1

2

kB T/MP

(b) The temperature function

1E/MP0

-50

-∞

C=-S'2/S''

(c) The heat capacity


Black hole in a finite size heat bath

Subsystem in a finite reservoir

T. S. Biró, Physica A 392, 3132 – 3139 (2013)T. S. Biró et. al., Eur. Phys. J. A 49, 110 (2013)

Subsystem1

Mutual information

large but finite reservoir 2

l=S1(E1)+S2(E2)−S12(E1+E2)≠0

Zero mutual information

lK=K1(S1(E1))+K2(S2(E2))−K12(S12(E1+E2))=0

Additive energy K (S)=λE+μ

Constant heat capacity1C0

=−S' '(E)

S '(E)2

S(E)=C0 ln(1+ EC0T0 )

K (S)≈SFor small S

K (S)=C0(eS/C0−1)

K (S)=∑i

PiK i(−lnPi )=C0∑i

Pi (e(−ln Pi )/C0−1)= 1

1−q∑i(piq−p1)=STsallis q=1− 1

C0

K-deform


Schwarzschild black hole in a finite heat bath

The standard thermodynamical functions of the Schwarzschild black hole.

SBH = 4πE2,1

TH= S′

BH(E) = 8πE, CBH =−S′2

BH(E)

S′′BH(E)

= −8πE2.

From the finite size correction of the heath bath in the canonical ensemble, thenonadditive composition law is the Tsallis formula in leading order, and theadditive entropy is the Renyi entropy.

The Renyi entropy of a Schwarzschild black hole

S12 = S1 + S2 + aS1S2 ⇒ L(SBH) ≡ SR =1

aln [1 + aSBH ] , a ≡ 1− q =

1

C0.

New thermodynamical functions

SR =1

aln

[1 + 4πaE2

],

1

TR= S′

R(E) =8πE

1 + 4πaE2, CR =

−S′2R (E)

S′′R(E)

=8πE2

4πaE2 − 1.


Thermodynamic functions & stability

0.0 0.5 1.0 1.5 2.0 2.5 3.0EMP05

1015202530SkB

E0

Rényi

Boltzmann

(a) The entropy functions

0.0 0.5 1.0 1.5 2.0 2.5 3.0EMP0.00.20.40.60.8

kBTMPE0

Rényi

Boltzmann

(b) The temperature functions

0.2 0.4 0.6 0.8 1.0 1.2 1.4 EMP

-100-50

050

100c=-S'2S''E0

Rényi

Boltzmann

(c) The heat capacities


The Schwarzschild-Renyi / AdS-Boltzmann similarity

0.0 0.5 1.0 1.5 2.0 2.5 3.0EMP05

1015202530SkB

E0

Sch.-Rényi

Sch.-Boltzmann

AdS.-Boltzmann

(a) The entropy functions

0.0 0.5 1.0 1.5 2.0 2.5 3.0EMP0.00.20.40.60.8

kBTMPE0

Sch.-Rényi

Sch.-BoltzmannAdS.-Boltzmann

(b) The temperature functions

0.2 0.4 0.6 0.8 1.0 1.2 1.4 EMP

-100-50

050

100c=-S'2S''E0

Sch.-Rényi

Sch.-Boltzmann

AdS.-Boltzmann

(c) The heat capacities


The Poincare method

Turnig point (Poincaré) method

d Zedy

y

Stability change

Less stable

More stable

O. Kaburaki, I. Okamoto and J. Katz, PRD47, 2234(1993)

No stability change

Massieu function Ze (y)

Control parameter y

Conjugate variable d Zedy


Stability and phase transition

microcanonical canonical

Stability curves

Massieu function

Control parameter

Conjugate variable

S

M

β(M)=∂S∂M

S−βM=−βF

β

−M(β)

Stability change


Thank you for your attention.


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Nonextensive thermodynamics and stability of black holes