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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
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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….
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Heavy ion collision: theoretical picture
URQMD ( Univ. Frankfurt: Sorge, Bass, Bleicher…. )
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Experimentalpicture … RHIC
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Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra
SQM 2008, Beijing
with Károly Ürmössy
RH
IC d
ata
Page 6
Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra
SQM 2008, Beijing
with Károly Ürmössy
RH
IC d
ata
Page 8
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
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Zeroth Law: (E1,…)=(E2,…)
Which composition laws are compatible with this?
empirical temperature
with Péter Ván
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Zeroth Law compatible composition of energy with Péter Ván
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Zeroth Law compatible composition of energy
same
function!with Péter Ván
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Zeroth Law compatible composition of energy with Péter Ván
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The solutionwith Péter Ván
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An example
all L( ) functions are the same!
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How may Nature do this?
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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
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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:
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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
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Superstatistics
tE
tE
)E(L
0
Et
e
et)E(L
ee dt)t(w
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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.
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TsallisRényiBoltzmann
Entropy formulas
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Entropy formulas
i
q
iR
i
q
iiT
ii
iB
pln1q
1S
)pp(1q
1S
p
1lnpS
Tsallis
Rényi
Boltzmann
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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
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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!
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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… ?
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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
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Horizon thermodynamics
• Information loss ~ entropy ~ horizon area
• Additive energy, non-additive horizon
• Temperature: Unruh, Hawking
• Based on Clausius’ entropy formula
Since the 1970 - s
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Quantum and Gravity Units
Scales:
in c = 1 units
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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
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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
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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
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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
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Unruh temperature
gˆicg2/ˆcg/)gclnˆ(ic
0
g/zic1g/ˆicg/ic
eeg
c
ezdzeg
c)ˆ(F
:z in Integral
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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
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Unruh temperature
2
LgM
2
g
cTk
2
gT
Tk
ˆˆ
g
c2
P
PB
B
Planck-interpretation:
Temperature inPlanck units:
Temperature infamiliar units:
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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
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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)
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Clausius’ entropy
dET
1dS
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Bekenstein-Hawking entropy
• Use Unruh temperature at horizon
• Use Clausius’ concept with that temperature
T
)Mc(dS
2
Hawking
Bekenstein
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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
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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.
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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!
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BH entropy for static horizons
PPB
M
dM
L
dr)M,r(f4
k
S
This is like a shell in a phase space!
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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
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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
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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
Page 49
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
Page 50
Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra
SQM 2008, Beijing
with Károly Ürmössy
RH
IC d
ata
Page 51
Tsallis quark matter + transverse flow + quark coalescence fits to hadron spectra
SQM 2008, Beijing
with Károly Ürmössy
RH
IC d
ata
Page 52
Blast wave fits and quark coalescence
SQM 2008, Beijing
202642.12
1q2
with Károly Ürmössy
Page 53
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