Relativistic Hydrodynamic Fluctuations M. Stephanov with X. An, G. Basar and H.-U. Yee, 1902.09517 M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 1 / 14
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 Relativistic Hydrodynamic Fluctuations QM 2019 2 / 14
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 note: Fluctuations are the first step to extend hydro to smaller systems . M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 3 / 14
Stochastic hydrodynamics Hydrodynamic eqs. are conservation equations: ∂ t ψ = −∇ · Flux [ ψ ]; M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 4 / 14
Stochastic hydrodynamics Hydrodynamic eqs. are conservation equations: ∂ t ψ = −∇ · Flux [ ψ ]; J 0 ) are stochastic and obey ψ = � ˘ ψ � , where ˘ ψ = ( ˘ T i 0 , ˘ more � � ∂ t ˘ Flux [ ˘ ψ = −∇ · ψ ] + Noise (Landau-Lifshitz) M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 4 / 14
Stochastic hydrodynamics Hydrodynamic eqs. are conservation equations: ∂ t ψ = −∇ · Flux [ ψ ]; J 0 ) are stochastic and obey ψ = � ˘ ψ � , where ˘ ψ = ( ˘ T i 0 , ˘ more � � ∂ t ˘ Flux [ ˘ ψ = −∇ · ψ ] + Noise (Landau-Lifshitz) Usually treated in linear order in fluctuations. Non-linearities + point-like noise ⇒ UV divergences. In numerical simulations – cutoff dependence. M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 4 / 14
Deterministic approach Variables are one- and two-point functions: ψ = � ˘ ψ � and G = � ˘ ψ ˘ ψ � − � ˘ ψ �� ˘ ψ � – equal-time correlator ∂ t ψ = −∇ · Flux [ ψ, G ]; (conservation) ∂ t G = Relaxation [ G → G (eq) ( ψ )] 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 Relativistic Hydrodynamic Fluctuations QM 2019 5 / 14
Deterministic approach Variables are one- and two-point functions: ψ = � ˘ ψ � and G = � ˘ ψ ˘ ψ � − � ˘ ψ �� ˘ ψ � – equal-time correlator ∂ t ψ = −∇ · Flux [ ψ, G ]; (conservation) ∂ t G = Relaxation [ G → G (eq) ( ψ )] 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 Relativistic Hydrodynamic Fluctuations QM 2019 5 / 14
General deterministic formalism An, Basar, Yee, MS, 1902.09517 To describe hydrodynamic fluctuations (critical and non-critical) in arbitrary relativistic flow in h.i.c. we develop a general (deter- ministic) formalism. M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 6 / 14
General deterministic formalism An, Basar, Yee, MS, 1902.09517 To describe hydrodynamic fluctuations (critical and non-critical) in arbitrary relativistic flow in h.i.c. we develop a general (deter- ministic) formalism. Important issue in relativistic hydro – “equal-time” in the definition of G ( x, y ) = � φ ( x + y/ 2) φ ( x − y/ 2) � . Addressed by constructing “confluent” derivative. Renormalization performed analytically , giving cutoff-independent equations. M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 6 / 14
Equal time We want evolution equation for equal time correlator G = � φ ( t, x + ) φ ( t, x − ) � . But what does “equal time” mean? “Equal time” in � φ ( x + ) φ ( x − ) � depends on the choice of frame. The most natural choice is local u ( x ) (with 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 Relativistic Hydrodynamic Fluctuations QM 2019 7 / 14
Confluent derivative, connection and correlator Confluent derivative: ( ¯ ∇ u = 0 ) ∆ x · ¯ ∇ φ = Λ(∆ x ) φ ( x + ∆ x ) − φ ( x ) Confluent two-point correlator: G ( x, y ) = Λ( y/ 2) G ( x, y ) Λ( − y/ 2) T ¯ (boost to u ( x ) – rest frame at midpoint) µa y a ∂ ∇ µ ¯ ¯ G AB = ∂ µ ¯ ω C µA ¯ ω C µB ¯ ω b ∂y b ¯ G AB − ¯ G CB − ¯ G AC − ˚ G AB . Connection ¯ ω makes sure that only the change of φ A relative to local rest frame u is counted. Connection ˚ ω corrects for a possible rotation of the local basis triad e a defining local coordinates y a . The derivative is independent of e a . We then define the Wigner transform W AB ( x, q ) of ¯ G AB ( x, y ) . M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 8 / 14
Scales b : hydro cell size. We coarse grain operators over scale b ≫ ℓ mic ∼ c s /T to leave only slow modes for which quantum fluctu- atuations are negligible compared to thermal, i.e., � ω ≪ kT . L : hydrodynamic gradients scale. Must be L ≫ b . back M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 9 / 14
Scales b : hydro cell size. We coarse grain operators over scale b ≫ ℓ mic ∼ c s /T to leave only slow modes for which quantum fluctu- atuations are negligible compared to thermal, i.e., � ω ≪ kT . L : hydrodynamic gradients scale. Must be L ≫ b . back ℓ ∗ : equilibration length (characteristic scale of y ): diffusion length during evolution time (typically τ ev ∼ L/c s ) � ℓ ∗ ∼ √ γτ ev ∼ γL/c s q ∗ ≡ 1 /ℓ ∗ Flucts at longer wavelengths do not have time to equilibrate. M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 9 / 14
Scales b : hydro cell size. We coarse grain operators over scale b ≫ ℓ mic ∼ c s /T to leave only slow modes for which quantum fluctu- atuations are negligible compared to thermal, i.e., � ω ≪ kT . L : hydrodynamic gradients scale. Must be L ≫ b . back ℓ ∗ : equilibration length (characteristic scale of y ): diffusion length during evolution time (typically τ ev ∼ L/c s ) � ℓ ∗ ∼ √ γτ ev ∼ γL/c s q ∗ ≡ 1 /ℓ ∗ Flucts at longer wavelengths do not have time to equilibrate. ℓ mic ≪ L implies the hierarchy (and power-counting scheme): ( γq 2 ∼ c s k ) ℓ mic ≪ b < ℓ ∗ ≪ L or T/c s ≫ Λ > q ∗ ≫ k q ≫ k – similar to kinetic theory (where Wigner function ≡ p.d.f.) M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 9 / 14
Matrix equation and diagonalization After many nontrivial cancellations we find evolution eq.: u · ¯ −{ Q , W − W (0) } ∇ W = − i [ L , W ] + K ◦ W expand � �� � � �� � � �� � background oscillation relaxation to W (0) Ideal hydro → L ∼ c s q, Noise/dissipation → Q ∼ γq 2 , and Background → K ∼ ∂ µ u ν , ∇ . W is relaxing to equilibrium W → W (0) at a rate 2 γq 2 disturbed by background hydrodynamic gradients of order k . M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 10 / 14
Matrix equation and diagonalization After many nontrivial cancellations we find evolution eq.: u · ¯ −{ Q , W − W (0) } ∇ W = − i [ L , W ] + K ◦ W expand � �� � � �� � � �� � background oscillation relaxation to W (0) Ideal hydro → L ∼ c s q, Noise/dissipation → Q ∼ γq 2 , and Background → K ∼ ∂ µ u ν , ∇ . W is relaxing to equilibrium W → W (0) at a rate 2 γq 2 disturbed by background hydrodynamic gradients of order k . The slowest W modes ( ω ≪ c s | q | ) are 4 “diagonal” ones in the basis of ideal hydro modes – sound-sound, shear-shear – and can be isolated by time-averaging over faster modes. see equations M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 10 / 14
Sound-sound and phonon kinetic equation � � ∇ + f · ∂ ( u + v ) · ¯ = − γ L q 2 ( W + − Tw ) + K ′ W + W + ���� ���� ∂q � �� � ∼ ∂ µ u ν , a µ W (0) L + [ W + ] expand M. Stephanov Relativistic Hydrodynamic Fluctuations QM 2019 11 / 14
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