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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 WMAP 7 years 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 Molecular Dynamics Lubrication Equation Stochastic Lubrication Equation Fluctuations Near the Critical Point NSAC 2007 Long-range Plan Volume = 400 fm3 =(n-nc)/nc Incorporates correct critical exponents and amplitudes - Kapusta (2010) Static univerality class: 3D Ising model & liquid-gas transition But this is for a small system in contact with a heat and particle reservoir. Must treat fluctuations in an expanding and cooling system. Extend Landau’s theory of hydrodynamic fluctuations to the relativistic regime T Tideal T S , J nu J I Stochastic sources S and I S ( x) S ( y ) 2T h h h h 23 h h 4 ( x y ) S ( x) I ( y ) 0 nT 2 4 I ( x) I ( y ) 2 h ( x y) w 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 Mode coupling theory – diffusive heat and viscous are slow modes, sound waves are fast modes Fixman (1962) Kawasaki (1970,1976) Kadanoff & Swift (1968) Zwanzig (1972) Luettmer-Strathmann, Sengers & Olchowy (1995) together with Kapusta (2010) RDT c p DT c p (qD ) 6 = specific heat x Stokes-Einstein diffusion law x crossover function 1 n 1 (n, T ) 0 t 5 | | 2 nc / Critical exponent in reduced temperatu re t for is 0.61 Estimate 0 0.69 fm Dynamic universality class: Model H of Hohenberg and Halperin Luettmer-Strathmann, Sengers & Olchowy (1995) carbon dioxide Data from various experimental groups. ethane Excess thermal conductivity Will hydrodynamic fluctuations have an impact on our ability to discern a critical point in the phase diagram (if one exists)? Simple Example: Boost Invariant Model u sinh (s (s , )) 3 n ni i n(s , ) s( ) si i s(s , ) Linearize equations of motion in fluctuations noise I 3 s( ) f (s , ) sinh s Solution: ~ d ' ~ ~ X ( k , ) GX (k ; , ' ) f (k , ' ) i ' X n, s, response function f 2 n( )T ( ) 2 d C XY (s , f ) 3 ( ) GXY (s ; f , ) A i s( ) w( ) enhanced near critical point quarks & gluons critical point baryons & mesons Excess thermal conductivity on the flyby Fluctuations in the local temperature, chemical potential, and flow velocity fields u z sinh s (s , ) give rise to a nontrivial 2-particle correlation function when the fluid elements freeze-out to free-streaming hadrons. Magnitude of proton correlation function depends strongly on how closely the trajectory passes by the critical point. dN ( y2 ) dN ( y1 ) dN dy2 dy1 dy 2 dN dy 1 One central collision Pb+Pb @ LHC Zero net baryon density Noisy 2nd order viscous hydro Transverse plain Clint Young – U of M Matching looks straighforward… All hadrons in PDG listing treated as point particles. Order g5 with 2 fit paramters Q2 T2 ab 2 2 MS MS …but it is not. All hadrons in PDG listing treated as point particles. Order g5 with 2 fit paramters Q2 T2 ab 2 2 MS MS P(T ) 1 e (T0 / T ) 4 P (T ) e h Excluded volume I : Vex E ( p) (T0 / T ) 4 PpQCD (T ) , 0 (305 MeV)4 0 m Excluded volume II : Vex , 0 (361 MeV)4 0 Doing the matching at finite temperature and density, while including a critical point with the correct critical exponents and amplitudes, is challenging! Typically one finds bumps, dips, and wiggles in the equation of state. 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.