equations of state with a chiral critical point
DESCRIPTION
Equations of State with a Chiral Critical Point. Joe Kapusta University of Minnesota. Collaborators : Berndt Muller & Misha Stephanov ; Juan M. Torres-Rincon; Clint Young, Michael Albright . Fluctuations in temperature of cosmic microwave background radiation. WMAP picture. - PowerPoint PPT PresentationTRANSCRIPT
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Equations of State with a Chiral Critical Point
Joe KapustaUniversity of Minnesota
Collaborators: Berndt Muller & Misha Stephanov; Juan M. Torres-Rincon; Clint Young, Michael Albright
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WMAP picture
WMAP 7 years
Fluctuations in temperature of cosmicmicrowave background radiation
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Sources of Fluctuations in High Energy Nuclear Collisions
• Initial state fluctuations• Hydrodynamic fluctuations due
to finite particle number• Energy and momentum
deposition by jets traversing the medium
• Freeze-out fluctuations
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Molecular Dynamics
Lubrication Equation
Stochastic Lubrication Equation
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Fluctuations Near the Critical Point
NSAC 2007 Long-range Plan
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Volume = 400 fm3
=(n-nc)/nc
Incorporates correct critical exponents and amplitudes - Kapusta (2010)Static univerality class: 3D Ising model & liquid-gas transition
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But this is for a small systemin contact with a heat and
particle reservoir.
Must treat fluctuations in an expanding and cooling system.
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Extend Landau’s theory of hydrodynamic fluctuations to the relativistic regime
IJnuJSTTT ,ideal
IS and
)(2)()( 432 yxhhhhhhTySxS
0)()( yIxS
Stochastic sources
)(2)()( 42 yxhwnTyIxI
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Procedure
• Solve equations of motion for arbitrary source function
• Perform averaging to obtain correlations/fluctuations
• Stochastic fluctuations need not be perturbative
• Need a background equation of state
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Mode coupling theory – diffusive heat and viscousare slow modes, sound waves are fast modes
)(6
DD
pTp qTRcDc
/
10 ||5
21),(
tnnTnc
Fixman (1962) Kawasaki (1970,1976) Kadanoff & Swift (1968) Zwanzig (1972) Luettmer-Strathmann, Sengers & Olchowy (1995) together with Kapusta (2010)
= specific heat x Stokes-Einstein diffusion law x crossover function
61.0 is for t re temperatureducedin exponent Critical fm 69.0 Estimate 0
Dynamic universality class: Model H of Hohenberg and Halperin
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Luettmer-Strathmann, Sengers & Olchowy (1995)
carbon dioxide ethane
Data from various experimental groups.
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Excess thermal conductivity
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Will hydrodynamic fluctuationshave an impact on our abilityto discern a critical point in thephase diagram (if one exists)?
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Simple Example: Boost Invariant Model),(s)( ),(n )),((sinh3
s
iis
iiss
ssnnu
,, )',(~)',;(~''),(~ snXkfkGdkX X
i
),;()()()()()(2),(
2
3
fsXYfsXY G
wsTnd
AC
f
i
Linearize equations of motion in fluctuations
Solution:
response function
noise
enhanced near critical point
ssfsI sinh),()(3
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quarks & gluons
baryons & mesons
critical point
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Excess thermal conductivity on the flyby
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),( sinhuz ss
Fluctuations in the local temperature,chemical potential, and flow velocity fields
give rise to a nontrivial 2-particlecorrelation function when the fluidelements freeze-out to free-streaminghadrons.
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Magnitude of proton correlation function depends strongly on how closely the trajectory passes by the critical point.
12
1
1
2
2 )()(
dydN
dydN
dyydN
dyydN
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One central collision
Pb+Pb @ LHC
Zero net baryon density
Noisy 2nd order viscous hydro
Transverse plain
Clint Young – U of M
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All hadrons in PDG listingtreated as point particles.
Order g5 with 2 fit paramters
MSMS
TbaQ2
2
2
2
Matching looks straighforward…
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All hadrons in PDG listingtreated as point particles.
Order g5 with 2 fit paramters
MSMS
TbaQ2
2
2
2
…but it is not.
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)(e)(e1)(4
04
0 )/()/( TPTPTP pQCDTT
hTT
40
0
MeV) 305(,)( :I volumeExcluded pEVex
40
0
MeV) 361(, :II volumeExcluded mVex
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Doing the matching at finite temperatureand density, while including a criticalpoint with the correct critical exponentsand amplitudes, is challenging!
Typically one finds bumps, dips, andwiggles in the equation of state.
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Summary
• Fluctuations are interesting and provide essential information on the critical point.• Fluctuations are enhanced on a flyby of the critical point.• There is clearly plenty of work for both
theorists and experimentalists!
Supported by the Office Science, U.S. Department of Energy.