Non-Extensive Black Hole Thermodynamics estimate for Power-Law Particle Spectra

• Power-law tailed spectra and their explanations

• Abstract thermodynamics

• Event horizon thermodynamics

• Estimate from a wish

Talk by T.S.Biró at the 10. Zimányi School,Budapest, Hungary, November 30 – December 3, 2010

arXiV: 1011.3442

Power-law tailed spectra

• particles and heavy ions: (SPS) RHIC, LHC

• fluctuations in financial returns

• natural catastrophes (earthquakes, etc.)

• fractal phase space filling

• network behavior

• some noisy electronics

• near Bose condensates

• citation of scientific papers….

Heavy ion collision: theoretical picture

URQMD ( Univ. Frankfurt: Sorge, Bass, Bleicher…. )

Experimentalpicture … RHIC

Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra

SQM 2008, Beijing

with Károly Ürmössy

RH

IC d

ata

Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra

SQM 2008, Beijing

with Károly Ürmössy

RH

IC d

ata

Abstract thermodynamics• S(E) = max (Jaynes-) principle

• nontrivial composition of e.g. the energy E

• 0-th law requires: factorizing form

T1(E1) = T2(E2)

• This is equivalent to the existence and use of an

additive function of energy L(E)!

• Repeated compositions asymptotically lead to

such a form! ( formal logarithm )

• Enrtopy formulas and canonical distributions

Zeroth Law: (E1,…)=(E2,…)

Which composition laws are compatible with this?

empirical temperature

with Péter Ván

Zeroth Law compatible composition of energy with Péter Ván

Zeroth Law compatible composition of energy

same

function!with Péter Ván

Zeroth Law compatible composition of energy with Péter Ván

The solutionwith Péter Ván

An example

all L( ) functions are the same!

How may Nature do this?

In small steps!

Composition Laws

),x(hx

x)0,x(h

)y,x(hyx

1nn

:steps small init Do

)0,x(hdt

dx

y

hxx

)0,x(h),x(hxx

2

)0y,x(

1nn

1n1n1nn

1n

Composition Laws

)b(L)a(LL)b,a(

)x(L)x(L)x(L

)x(L)0,z(h

dzt

1

NNNN

x

02

2121

Formal logarithm:

Additive quantity:

Asymptotic composition rule:

Composition Laws: summary

Such asymptotic rules are:

1. commutative x y = y x

2. associative (x y) z = x (y z)

3. zeroth-law compatible

222111

212

1

112

2

22111212

T

1

E

S

)E(L

1

E

S

)E(L

1

T

1

E

S

)E(L

)E(L

E

S

)E(L

)E(L

dE)E(LdE)E(LdE)E(L

Lagrange method

Superstatistics

tE

tE

)E(L

0

Et

e

et)E(L

ee dt)t(w

Canonical Power-Law

ac,et

)c(

c)t(w

aE1e

aE1lna

1)E(L

ct1c

c

a)E(L

Footnote: w(t) is an Euler-Gamma distribution in this case.

TsallisRényiBoltzmann

Entropy formulas

Entropy formulas

i

q

iR

i

q

iiT

ii

iB

pln1q

1S

)pp(1q

1S

p

1lnpS

Tsallis

Rényi

Boltzmann

Function of Entropy

TR

i

q

iR

i

q

iiT

aS1lna

1S

q1apln1q

1S

)pp(1q

1S

.max)S(YS.maxS

Tsallis

Rényi

Rényi = additive version of Tsallis

Canonical distribution with Rényi entropy

1q

1

i

i

iq

j

1q

i

iii

q

i

q1

)E(1

Z

1p

Ep

pq

q1

1

maxEppplnq1

1

This cut power-law distribution is

an excellent fit to particle spectra

in high-energy experiments!

The cut power-law distribution is

an excellent fit to particle spectra

in high-energy experiments!

How to caluclate (predict) How to caluclate (predict)

T, q, etc… ?T, q, etc… ?

What is universal in collisons?

• Event HorizonHorizon due to stopping

• Schwinger formula + Newton + Unruh = Boltzmann

T/m

3p

T

q/m2

3p

T

2T

epd

dNE

2

aT,amq,e

pd

dNE

Dima Kharzeev

Horizon thermodynamics

• Information loss ~ entropy ~ horizon area

• Additive energy, non-additive horizon

• Temperature: Unruh, Hawking

• Based on Clausius’ entropy formula

Since the 1970 - s

Quantum and Gravity Units

Scales:

in c = 1 units

Unruh temperature

• entirely classical• special relativity suffices

An observer with constant acceleration Fourier analyses a monochromatic

EM - wave from a far, static system in terms of its proper time:

the intensity distribution is proportional to the

Planck distribution !Unruh

Unruh temperature• entirely classical• special relativity suffices

An observer with constant acceleration Fourier analyses a monochromatic

EM - wave from a far, static system in terms of its proper time:

the intensity distribution is proportional to the

Planck distribution !Unruh

Max Planck

Unruh temperature

2

g1)c/g(chg

c)(x

)c/g(shg

c)(t

:Solution

gaa,1uu,0ua,d

dua

:onacceleratiConstant

22

2

Galilei

Rindler

Unruh temperature

0)(z,)(z

z

dz

g

cd

z1g

cc/xt,ez

:leNew variab

dee)ˆ(F

:amplitude wavePlane

c/g

ˆi)c/xt(i

Unruh temperature

gˆicg2/ˆcg/)gclnˆ(ic

0

g/zic1g/ˆicg/ic

eeg

c

ezdzeg

c)ˆ(F

:z in Integral

Unruh temperature

1e

1ˆg

c2

eg

c)ˆ(F

:factorIntensity

g/ˆc2

gˆic

gˆicg/ˆc

2

22

Interpret this as a black body radiation: Planck distribution of the frequency

Unruh temperature

2

LgM

2

g

cTk

2

gT

Tk

ˆˆ

g

c2

P

PB

B

Planck-interpretation:

Temperature inPlanck units:

Temperature infamiliar units:

Unruh temperature

2

2

P

2

B

2

R

L

2

McTk

R

GMg

small isit gravity NewtonianFor

On Earth’ surface it is 10^(-19) eV

Unruh temperature

2

mc1Tk

mc1

L

c1g

smallnot isit collision ionheavy a For

2

B

32

Stopping from 0.88 c to 0 in L = ħ/mc Compton wavelength distance:

kT ~ 170 MeV for mc² ~ 940 MeV (proton)

Clausius’ entropy

dET

1dS

Bekenstein-Hawking entropy

• Use Unruh temperature at horizon

• Use Clausius’ concept with that temperature

T

)Mc(dS

2

Hawking

Bekenstein

Acceleration at static horizons• Maupertuis action for test masspoints

• Euler-Lagrange eom: geodesic

• Arc length is defined by the metric

)r(fKcr

1fc/rtfKtf

fc/rtfmcL,dmcI

dc

r

)r(fc

drdt)r(fd

222

222

22222

2

2

2

2

2

22

Maupertuis

Acceleration at static horizons

)r(f4

1

c

Tk

g)r(f2

cr

r)r(fcrr2

tau wrspderivative theTake

B

2

2

This acceleration is the

red-shift corrected

surface gravity.

BH entropy inside static horizons

PP

)r(f2

2cPM

2PL

2

B

2

PP

P

P

B

M

dM

L

dr)M,r(f4

)Mc(d

k

S

.0)r(f while

)r(f2

cM

2

LgM

2

LTk

This is like a shell in a phase space!

BH entropy for static horizons

PPB

M

dM

L

dr)M,r(f4

k

S

This is like a shell in a phase space!

BH entropy: Schwarzschild

2

PB

2

P

2

P

P

PPB

2

2

P

P

2

L

A

4

1

k

S

L

RRdR

L2

M

ML

4S

k

1R2

cg,

R

1)R(f,

r

R)r(f

horizon 0)R(f

r

R1

r

L

M

M21

rc

GM21)r(f

This area law is true for all cases when f(r,M) = 1 – 2M / r + a( r ) !!!

Hawking-Bekenstein result

Schwarzschild

Schwarzschild BH: EoS

8E

Sc

E8E

S

T

1

E4S

2/RME

2

2

2

Hawking-Bekensteinentropy instable eos

S

E

T > 0c < 0

Planck units: k = 1, ħ = 1, G = 1, c = 1B

Schwarzschild BH: deformed entropy

22

2

2

2

Ea41

1Ea48c

Ea41

E8

T

1

Ea41lna

1aS1ln

a

1S

Tsallis-deformed HB

entropy for large E stable eos

☻S

E

T > 0c < 0

T > 0c > 0

a = q - 1a = q - 1

arXiV: 1011.3442

Schwarzschild BH: quantum zero point

rays) cosmic 9/2a(2

a

EE8

M

R22

LM

2

a2

ME

2

0

0

2

PPP

P

0

EoS stability limit is at / below the

quantum zero point motion energy

☻S

E

T > 0c < 0

T > 0c > 0

STAR, PHENIX, CMS: a ~ 0.20 - 0.22

inflection point

E0

arXiV: 1011.3442

Bekenstein bound

Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra

SQM 2008, Beijing

with Károly Ürmössy

RH

IC d

ata

Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra

SQM 2008, Beijing

with Károly Ürmössy

RH

IC d

ata

Blast wave fits and quark coalescence

SQM 2008, Beijing

202642.12

1q2

with Károly Ürmössy

Summary

• Thermodynamics build on composition laws

• Deformed entropy formulas

• Hawking entropy: phase space of f ( r ) = 0: horizon ‘size’

• Schwarzschild BH: Boltzmann entropy unstable eos

• Rényi entropy: stable BH eos at high energy ( T > Tmin )

• Estimate for q: from the instability being in the Trans-

Planckian domain

4

1

s,

21q

2