Measuring the temperature
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Transcript of Measuring the temperature
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Measuring the temperature• Thermometer (direct contact)
dilatation air pressure mechanical or electric stress
• Chemistry (mixture) color mass ratios hadronic composition
• Spectra (telemetrics) astronomy (photons) pT spectra of light and heavy particles multiplicity fluctuations
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Measuring the temperature• Thermometer (direct contact)
dilatation air pressure mechanical or electric stress
• Chemistry (mixture) color mass ratios hadronic composition
• Spectra (telemetrics) astronomy (photons) pT spectra of light and heavy particles multiplicity fluctuations
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Interpreting the temperature• Thermodynamics (universality of equilibrium)
zeroth theorem (Biro+Van PRE 2011) derivative of entropy Boyle-Mariotte type equation of state
• Kinetic theory (equipartition) energy / degree of freedom fluctuation - dissipation other average values
• Spectral statistics (abstract) logarithmic slope parameter observed energy scale dispersion / power -law effects
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Why do thermal models work?Are there alternatives?
T.S.Biró¹, M.Gyulassy², Z. Schram³¹ MTA KFKI RMKI ² Columbia University ³University of Debrecen
arXiV: 1111.4817 Unruh gamma radiation at RHIC ?
Constant acceleration = temperature ? Soft bremsstrahlung interpolates
between high k_T exponential and low k_T 1 / square
Zimányi School 2011, Dec. 1. 2011, KFKI RMKI Budapest, HU.EU
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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
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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
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Blast wave fits and quark coalescence
SQM 2008, Beijing
202642.121q2
with Károly Ürmössy
arXiV: 1011.3442
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All particle types follow power-law
E
L(E)
G O O D
B E T T E R !with Károly Ürmössy
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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• preferential connection in networks• turbulence, critical behavior • near Bose condensates• citation of scientific papers….
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Why do statistics work ?
• Independent observation over 100M events
• Universal laws for large numbers
• Steady noise in environment: ‚reservoir’
• Phase space dominance
• By chance the dynamics mimics thermal behavior
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𝑨𝒖+𝑨𝒖→𝜸+𝑿
Experimental motivation
RHIC: PHENIX
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Theoretical motivation
• Deceleration due to stopping• Schwinger formula + Newton + Unruh = Boltzmann
T/m
3p
Tq/m2
3p
T
2T
epd
dNE
2aT,amq,e
pddNE
Satz, Kharzeev, …
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Arxiv: 1111.4817
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Why Photons (gammas) ?
• Zero mass: flow – Doppler, easy kinematics
• Color neutral: escapes strong interaction
• Couples to charge: Z / A sensitive
• Classical field theory also predicts spectra
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• Jackson formula for the amplitude:
With ,
and the retarded phase
Soft bremsstrahlung
�⃗�
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• Covariant notation:
Soft bremsstrahlung
Feynman graphsIR div, coherent effects
The Unruh effect cannot be calculated by any finite number of Feynman graphs!
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Kinematics, source trajectory
• Rapidity: Trajectory:
Let us denote by in the followings!
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Kinematics, photon rapidity• Angle and rapidity:
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Kinematics, photon rapidity
• Doppler factor:
Phase:
Magnitude of projected velocity:
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Intensity, photon number Amplitude as an integral over rapidities on the trajectory:
Here is a characteristic length.
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Intensity, photon number Amplitude as an integral over infinite rapidities on the trajectory (velocity goes from –c to +c):
With K1 Bessel function!
Flat in rapidity !
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Photon spectrum, limits Amplitude as an integral over infinite rapidities on the trajectory (velocity goes from –c to +c):
for
for
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Photon spectrum from pp backgroound, PHENIX experiment
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Apparent temperature
• High - infinite proper time acceleration:
Connection to Unruh:
proper time Fourier analysis of a monochromatic wave
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Unruh temperature
• Entirely classical effect• Special Relativity suffices
Unruh
Max Planck
11)(
)(
/2
2
0
1//
2
/)(1/)(1
gcgcigzic
dcVcVi
edzzeI
deI
Constant ‚g’ acceleration in a comoving system: dv/d = -g(1-v²)
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Unruh temperature
22
2
gc2
PPB
B
LgMgc
Tk
gT
Tk
Planck-interpretation:
The temperature inPlanck units:
The temperaturemore commonly:
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Unruh temperature
2
2P
2
B
2
RL
2McTk
RGMg
gravity Newtonianfor Small
On Earth’ surface ist is 10^(-19) eV, while at room temperature about 10^(-3) eV. 29
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Unruh temperature
2mcTk
mcL2
cg
collisions ionheavy in smallNot
2
B
32
Braking from +c to -c in a Compton wavelength:
kT ~ 150 MeV if mc² ~ 940 MeV (proton) 30
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Bekenstein-Hawking entropy• Unruh temperature at the event horizon• Clausius: 1 / T is an integrating factor
Hawking
Bekenstein
2PB
23
B
2
B
B
2
22
LA
41
kS
RG
cTk
)Mc(dkS
c2gTk
R2c
RGMg
cGM2R
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Connection to Unruh
Fourier component for the retarded phase:
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Connection to Unruh
𝑑𝑁𝑘⊥𝑑𝑘⊥𝑑𝜂 𝑑𝜓
=𝛼𝐸𝑀
2𝜋𝑘⊥2 cosh2𝜂
−∞
+∞
|𝑓 𝑘|2|𝑎𝑘|
2 𝑐ℓ𝑑𝑘2𝜋
Fourier component for the projected acceleration:
Photon spectrum in the incoherent approximation:
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Connection to Unruh
Fourier component for the retarded phase at constant acceleration:
KMS relation and Planck distribution:
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Connection to UnruhKMS relation and Planck distribution:
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Connection to UnruhNote:
𝑎𝑘=cosh𝜂 𝑘𝜋sinh𝑘𝜋 /2
It is peaked around k = 0, but relatively wide! (an unparticle…)
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Bessel-K compendium
1. 2.
3.
3 1 : expand Dirac delta as Fourier integral,3 2 : rewrite integral over difference and the sum bx = x1 + x2
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Transverse flow interpretation
Mathematica knows: ( I derived it using Feynman variables)
Alike Jüttner distributions integrated over the flow rapidity…
0
𝜋 𝑑𝜃sin 𝜃 𝐾 2( 𝑧
sin 𝜃 )=𝐾 12(𝑧2 ) 1
sin𝜃=cosh𝜂
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Finite time (rapidity) effects =
with
Short-time deceleration Non-uniform rapidity distribution; Landau hydrodynamics
Long-time deceleration uniform rapidity distribution; Bjorken hydrodynamics
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Short time constant acceleration
Non-uniform rapidity distribution; Landau hydrodynamics
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The reverse problem
• Photon spectrum amplitude• Amplitude Fourier component• Phase velocity• Velocity rapidity• Rapidity acceleration• Acceleration Path in t -- x
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Analytic resultsx(t), v(t), g(t), τ(t), 𝒜, dN/kdkd limit
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Large pT:
NLO QCD
Smaller pT:
plus direct gammas
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𝑨𝒖+𝑨𝒖→𝜸+𝑿Glauber model
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𝑨𝒖+𝑨𝒖→𝜸+𝑿Glauber model
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Summary
• Semiclassical radiation from constant accelerating point charge occurs rapidity-flat and thermal
• The thermal tail develops at high enough k_perp• At low k_perp the conformal NLO result emerges• Finite time/rapidity acceleration leads to peaked
rapidity distribution, alike Landau - hydro• Exponential fits to surplus over NLO pQCD results
reveal a ’’pi-times Unruh-’’ temperature
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Is acceleration a heat container?