Hydrodynamic fluctuations and QCD critical point M. Stephanov with Y. Yin, 1712.10305; with X. An, G. Basar and H.-U. Yee, 1902.09517, 1912.13456; M. Stephanov Fluctuations and QCD Critical Point WWND 2020 1 / 17
Critical point: intriguing hints Equilibrium κ 4 vs T and µ B : Where on the QCD phase boundary is the CP? 300 2760 200 The Phases of QCD √ s = 62.4 GeV 39 250 27 19.6 Quark-Gluon Plasma Temperature (MeV) 14.5 200 11.5 B E S - I I 9.1 150 7.7 1 s t O r d e r P h a s e T r a 100 n Critical s i t i o s n Point a G Color n o r d a H 50 Superconductor Nuclear Vacuum Ma � er 0 0 200 400 600 800 1000 1200 1400 1600 Baryon Chemical Potential μ B (MeV) “intriguing hint” (2015 LRPNS) Motivation for phase II of BES at RHIC and BEST topical collaboration. M. Stephanov Fluctuations and QCD Critical Point WWND 2020 2 / 17
Theory/experiment gap: predictions assume equilibrium, but Non-equilibrium physics is essential near the critical point. Challenge: develop hydrodynamics with fluctuations capable of describing non-equilibrium effects on critical-point signatures. Also notable: Fluctuations are the first step to extend hydro to smaller systems. M. Stephanov Fluctuations and QCD Critical Point WWND 2020 3 / 17
Stochastic hydrodynamics Hydrodynamic eqs. are conservation equations ( ∂ µ T µν = 0 ): ∂ t ψ = −∇ · Flux [ ψ ]; M. Stephanov Fluctuations and QCD Critical Point WWND 2020 4 / 17
Stochastic hydrodynamics Hydrodynamic eqs. are conservation equations ( ∂ µ T µν = 0 ): ∂ t ψ = −∇ · Flux [ ψ ]; J 0 ) are local operators Stochastic variables ˘ ψ = ( ˘ T i 0 , ˘ coarse-grained (over “cells” b : ℓ mic ≪ b ≪ L ): more � � ∂ t ˘ Flux [ ˘ ψ = −∇ · ψ ] + Noise (Landau-Lifshitz) M. Stephanov Fluctuations and QCD Critical Point WWND 2020 4 / 17
Stochastic hydrodynamics Hydrodynamic eqs. are conservation equations ( ∂ µ T µν = 0 ): ∂ t ψ = −∇ · Flux [ ψ ]; J 0 ) are local operators Stochastic variables ˘ ψ = ( ˘ T i 0 , ˘ coarse-grained (over “cells” b : ℓ mic ≪ b ≪ L ): more � � ∂ t ˘ Flux [ ˘ ψ = −∇ · ψ ] + Noise (Landau-Lifshitz) Linearized version has been considered and applied to heavy- ion collisions (Kapusta-Muller-MS, Kapusta-Torres-Rincon, . . . ) Non-linearities + point-like noise ⇒ UV divergences. In numerical simulations – cutoff dependence. M. Stephanov Fluctuations and QCD Critical Point WWND 2020 4 / 17
Deterministic approach Variables are one- and two-point functions: ψ = � ˘ ψ � and G = � ˘ ψ ˘ ψ � − � ˘ ψ �� ˘ ψ � – equal-time correlator Nonlinearities lead to dependence of flux on G . ∂ t ψ = −∇ · Flux [ ψ, G ]; (conservation) ∂ t G = L [ G ; ψ ] . (relaxation) In Bjorken flow by Akamatsu et al , Martinez-Schaefer. For arbitrary relativistic flow – by An et al (this talk). Earlier, in nonrelativistic context, – by Andreev in 1970s. M. Stephanov Fluctuations and QCD Critical Point WWND 2020 5 / 17
Deterministic approach Variables are one- and two-point functions: ψ = � ˘ ψ � and G = � ˘ ψ ˘ ψ � − � ˘ ψ �� ˘ ψ � – equal-time correlator Nonlinearities lead to dependence of flux on G . ∂ t ψ = −∇ · Flux [ ψ, G ]; (conservation) ∂ t G = L [ G ; ψ ] . (relaxation) In Bjorken flow by Akamatsu et al , Martinez-Schaefer. For arbitrary relativistic flow – by An et al (this talk). Earlier, in nonrelativistic context, – by Andreev in 1970s. Advantage: deterministic equations. “Infinite noise” causes UV renormalization of EOS and transport coefficients – can be taken care of analytically (1902.09517) M. Stephanov Fluctuations and QCD Critical Point WWND 2020 5 / 17
Fluctuation dynamics near CP: Hydro+ Yin, MS, 1712.10305 Fluctuation dynamics near CP requires two main ingredients: Critical fluctuations ( ξ → ∞ ) Slow relaxation mode with τ relax ∼ ξ 3 (leading to ζ → ∞ ) M. Stephanov Fluctuations and QCD Critical Point WWND 2020 6 / 17
Fluctuation dynamics near CP: Hydro+ Yin, MS, 1712.10305 Fluctuation dynamics near CP requires two main ingredients: Critical fluctuations ( ξ → ∞ ) Slow relaxation mode with τ relax ∼ ξ 3 (leading to ζ → ∞ ) Both described by the same object: the two-point function of the slowest hydrodynamic mode m ≡ ( s/n ) , i.e., � δm ( x 1 ) δm ( x 2 ) � . Without this mode, hydrodynamics would break down near CP when τ expansion ∼ τ relax ∼ ξ 3 . M. Stephanov Fluctuations and QCD Critical Point WWND 2020 6 / 17
Additional variables in Hydro+ At the CP the slowest new variable is the 2-pt function � δmδm � of the slowest hydro variable: � � δm ( x + ) δm ( x − ) � e i Q · ∆ x φ Q ( x ) = ∆ x where x = ( x + + x − ) / 2 and ∆ x = x + − x − . Wigner transformed b/c dependence on x ( ∼ L ) is slow and relevant ∆ x ≪ L . Scale separation similar to kinetic theory. M. Stephanov Fluctuations and QCD Critical Point WWND 2020 7 / 17
Relaxation of fluctuations towards equilibrium As usual, equilibration maximizes entropy S = � i p i log(1 /p i ) : � � � s (+) ( ǫ, n, φ Q ) = s ( ǫ, n ) + 1 log φ Q − φ Q + 1 ¯ ¯ 2 φ Q φ Q Q M. Stephanov Fluctuations and QCD Critical Point WWND 2020 8 / 17
Relaxation of fluctuations towards equilibrium As usual, equilibration maximizes entropy S = � i p i log(1 /p i ) : � � � s (+) ( ǫ, n, φ Q ) = s ( ǫ, n ) + 1 log φ Q − φ Q + 1 ¯ ¯ 2 φ Q φ Q Q Entropy = log # of states, which depends on the width of P ( m Q ) , i.e., φ Q : Wider distribution – more microstates φ ) 1 / 2 ; – more entropy: log( φ/ ¯ vs Penalty for larger deviations from peak entropy (at δm = 0 ): − (1 / 2) φ/ ¯ φ . - - - equilibrium (variance ¯ φ ) Maximum of s (+) is achieved at φ = ¯ —- actual (variance φ ) φ . M. Stephanov Fluctuations and QCD Critical Point WWND 2020 8 / 17
Hydro+ mode kinetics The equation for φ Q is a relaxation equation: � ∂s (+) � ( u · ∂ ) φ Q = − γ π ( Q ) π Q , π Q = − ∂φ Q ǫ,n γ π ( Q ) is known from mode-coupling calculation in ‘model H’. It is universal (Kawasaki function). γ π ( Q ) ∼ 2 DQ 2 for Q < ξ − 1 and ∼ Q 3 for Q > ξ − 1 . more Characteristic rate: Γ( Q ) ∼ γ π ( Q ) ∼ ξ − 3 at Q ∼ ξ − 1 . Slowness of this relaxation process is behind the divergence of ζ ∼ 1 / Γ ∼ ξ 3 and the breakdown of ordinary hydro near CP (frequency depedence of ζ at ω ∼ ξ − 3 ). M. Stephanov Fluctuations and QCD Critical Point WWND 2020 9 / 17
Towards a general deterministic formalism An, Basar, Yee, MS, 1902.09517,1912.13456 To embed Hydro+ into a unified theory for critical as well as non- critical fluctuations we develop a general deterministic ( hydro- kinetic ) formalism. We expand hydrodynamic eqs. in { δm, δp, δu µ } ∼ φ and then average, using equal-time correlator G ( x, y ) ≡ � φ ( x + y/ 2) φ ( x − y/ 2) � . What is “equal-time” in relativistic hydro? Renormalization. M. Stephanov Fluctuations and QCD Critical Point WWND 2020 10 / 17
Equal time We use equal-time correlator G = � φ ( t, x + ) φ ( t, x − ) � . But what does “equal time” mean? Needs a frame choice. The most natural choice is local u ( x ) ( x = ( x + + x − ) / 2 ). Derivatives wrt x at “ y -fixed” should take this into account: using Λ(∆ x ) u ( x + ∆ x ) = u ( x ) : ∆ x · ¯ ∇ G ( x, y ) ≡ G ( x + ∆ x, Λ(∆ x ) − 1 y ) − G ( x, y ) . not G ( x + ∆ x, y ) − G ( x, y ) . M. Stephanov Fluctuations and QCD Critical Point WWND 2020 11 / 17
Confluent derivative, connection and correlator Take out dependence of components of φ due to change of u ( x ) : ∆ x · ¯ ∇ φ = Λ(∆ x ) φ ( x + ∆ x ) − φ ( x ) Confluent two-point correlator: G ( x, y ) = Λ( y/ 2) � φ ( x + y/ 2) φ ( x − y/ 2) � Λ( − y/ 2) T ¯ (boost to u ( x ) – rest frame at midpoint) µa y a ∂ ∇ µ ¯ ¯ G AB = ∂ µ ¯ µA ¯ µB ¯ ∂y b ¯ ω C ω C ω b G AB − ¯ G CB − ¯ G AC − ˚ G AB . Connection ¯ ω corresponds to the boost Λ . Connection ˚ ω makes sure derivative is independent of the choice of local space triad e a needed to express y ≡ x + − x − . We then define the Wigner transform W AB ( x, q ) of ¯ G AB ( x, y ) . M. Stephanov Fluctuations and QCD Critical Point WWND 2020 12 / 17
Sound-sound correlation and phonon kinetic equation Upon lots of algebra with many miraculous cancellations we ar- rive at “hydro-kinetic” equations for components of W . The longitudinal components, corresponding to p and u µ fluctua- tions at δ ( s/n ) = 0 , obey the following eq. ( N L ≡ W L / ( wc s | q ⊥ | ) ) � � � � ∇ + f · ∂ T ( u + v ) · ¯ = − γ L q 2 N L N L − ∂q c s | q ⊥ | � �� � � �� � L [ N L ] – Liouville op. N (0) L Kinetic eq. for phonons with E = c s ( x ) | q ⊥ | and v = c s q ⊥ / | q ⊥ | f µ = − E ( a µ + 2 v ν ω νµ ) − q ⊥ ν ∂ ⊥ µ u ν − ¯ ∇ ⊥ µ E . � �� � � �� � inertial + Coriolis “Hubble” N (0) is equilibrium Bose-distribution. L M. Stephanov Fluctuations and QCD Critical Point WWND 2020 13 / 17
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