Off-equilibrium Non-Gaussian Cumulants: criticality, complexity, and universality

Transcription

1 Off-equilibrium Non-Gaussian Cumulants: criticality, complexity, and universality Swagato Mukherjee SM, R. Venugopalan, Y. Yin: arxiv: & arxiv: June 2016, Wroclaw

2 hope: observe something universal static universality QCD critical point STAR 3-d Ising kurtosis (n 1) 2 2 eq κeq n ξ f eq n (θ) κ eq 4 (μ μ c ) / Δ μ θ r 5 /3 h r reduced temperature h reduced magnetic field Stephanov: arxiv:

3 why non-equilibrium? its necessary where is freeze-out w.r.t. critical point? memory effects needed to preserve remnant critical signatures unless accidental freeze-out very close to critical point 3

4 why non-equilibrium? its unavoidable dynamical universality QCD critical point τ eff ξzeq slow relaxation of critical mode critical mode out of equilibrium model-h Z=3 Son, Stephanov: arxiv:hep-ph/ τ eff : relaxation time for critical mode critical mode linear combination of chiral condensate & baryon current 4

5 simple extension to non-equilibrium Berdnikov, Rajagopal: arxiv:hep-ph/ Ansatz for evolution of correlation length: 1 1 τ ξ 1 = τ 1 ξ ξ eff [ eq ] z with dynamical universality: τ eff ξeq non-gaussian cumulants just as in equilibrium (n 1) 2 2 κn (τ) [ξ( τ)] scaling holds off-equilibrium? signs of non-gaussian cumulants? 5

6 real-time evolution of cumulants effective action 3-d Ising SM, R. Venugopalan, Y. Yin: arxiv: λ3 3 λ4 4 Ω0 (σ) = mσ (σ σ 0 ) + (σ σ 0 ) + (σ σ 0) σ : critical mode 1 ξeq mσ remain within scaling regime, but not at the critical point ϵ = ξ3 / V < 1 Lmicr < ξ < L mass term ~ σ 2 /ξ 2eq momentum dependence kinetic term σ 2 /L 2 neglected 6

7 Langevin dynamics Fokker-Planck evolution evolution of cumulants only soft critical mode out of equilibrium, hard modes are in equilibrium, soft mode receive random kicks from thermal bath of hard modes τ P (σ ; τ ) = 1 σ [ σ Ω0 (σ )+ V 1 4 σ ] P(σ ; τ) m τ eff 2 σ [ 3 τ eff ξeq τ f (σ) = 1 f (σ)ω 0 (σ ) V 1 [ 4 f (σ) ] 2 m σ τ eff systematic expansion in ϵ = ξ /V < ]

8 closed set of coupled time evolution equations τ κ n = n τ 1 ef Fn ( κ 1,, κ n ) +O( ϵ) ϵ<1 evolution of the higher cumulants couples only to lower ones Gaussian limit Ω0 (σ) = lower cumulatns relax back to equilibrium first 1 2 m σ (σ σ 0 )2 2 eq eq κ 3 =κ 4 =0 eq τ κn = n τ 1 κ κ [ eff n n ] for n=2 reduces to the old Berdnikov-Rajagopal Ansatz 1 1 τ ξ 1 = τ 1 ξ ξ [ eff eq ] 8

9 modeling heavy-ion collisions or introducing non-universality Ising thermodynamic variables: ( r,h ) (μb, T ) details of trajectory in ( μb, T ) -plane relaxation time of the critical mode: location of freeze-out in ( μb, T ) -plane τ eff 9

10 example: trajectory Type-A, vary τ eff h= T T c ΔT μ μ r = Δμ c 3-d Hubble-like expansion 2 3c s T ( τ) = T c ( τ/ τ c ) c2s =0.1 speed of sound τc time when trajectory crosses h=0, crossover, line 10

11 universality lost corr length τ eff skewness ~ τ / τc ~ τ / τc ~ τ = τ τc non-gaussian cumulants do not follow growth of the correlation length unlike equilibrium expectation 11

12 sign can be different from equilibrium one ( T T c )/ Δ T universality lost kurtosis (μ μc ) Δ μ τ eff 12

13 new idea: Kibble-Zurek (KZ) dynamics two competing time scales τ eff ( τ) SM, R. Venugopalan, Y. Yin: arxiv: τ quench (τ) ξ θ τ quench = min ( τ quench, τ quench ) τ ξ quench τ ξ eq ( τ) = τ ξ eq (τ) θ quench θ( τ) = τ θ( τ) (r,h) (ξ,θ) 13

14 emergent scales τ eff ( τ) τ quench (τ) τ KZ = τ eff (τ ) = τquench ( τ ) lkz = ξeq ( τ ) θkz = θeq (τ ) frozen fluctuation & magnetization emergent scaling (n 1) 2 2 KZ κ n ( τ ; Γ) l t=~ τ / τ KZ I f n ( t ;θ KZ) ~ τ = τ τc (n 1) 2 2 eq κeq n ξ f eq n (θ) I class of trajectories τ KZ (Γ), lkz (Γ), θkz (Γ) Γ non-universal variables 14

15 example: trajectory Type-A, vary τ eff τ quench = τξquench τ ξ quench ξ eq ( τ) = τ ξ eq ( τ) 15

16 universality regained (n 1) 2 2 KZ κn ( τ ; Γ) l f In (t ; θkz ) correlation length 16

17 universality regained (n 1) 2 2 KZ κ n ( τ ; Γ) l f In (t ; θkz ) skewness 17

18 more general example: trajectory Type-B τquench = τ θquench τ θ quench θ( τ) = τ θ( τ) 18

19 1 5 + (n 1) 2 2 KZ κn ( τ ; Γ) l f In (t ; θkz ) kurtosis 19

20 freeze-out kurtosis 20

21 summary off-equilibrium complexity (μ μ c )/ Δ μ equilibrium criticality off-equilibrium universality freeze-out 21