QUARK MATTER 2014 Darmstadt, Germany, 19-24 May, 2014

Tetsufumi Hirano Sophia Univ.

In collaboration with R. Kurita, K. Murase and K. Nagai 1  Introduction  Fluctuating Hydrodynamics in Bjorken Expansion  Fluctuation theorem  Summary

2 • Conventional hydrodynamics  Space-time evolution of (coarse- grained) thermodynamic quantities • Microscopic information  Lost through coarse-graining process • Does the lost information play an important role in dynamics in small system and/or on an e-by-e basis?  Thermal (Hydrodynamic) fluctuation!

Calzetta, Kapusta, Muller, Stephanov, Moore, Murase, Young, … 3 Thermal equilibrium state

Entropy = Maximum entropy state

4 dissipation Thermal equilibrium state

Entropy = Maximum entropy state

Balance between fluctuation and dissipation fluctuation  Stability of thermal equilibrium state

5 dissipation Thermal equilibrium state

Entropy = Maximum entropy state

Balance between fluctuation and dissipation fluctuation  Stability of thermal equilibrium state 훿Π(푥)훿Π(푥′) = 푇퐺∗(푥, 푥′) 퐺∗ : Symmetrized correlation function

훿Π : Thermal (hydrodynamic) fluctuation 6 7 Equation of motion 푒 : Energy density 푑푒 푒 + 푃 푒 휋 − Π 푃 : Pressure = − 1 − 푑𝜏 𝜏 푠푇 푠 : Entropy density 휋 Constitutive equations : Shear stress Π : Bulk pressure 푑휋 4휂 𝜏 + 휋 = + 휉 Shear: 휋 푑𝜏 3𝜏 휋 휂 : Shear viscosity 휁 : Bulk viscosity 푑Π 휁 Bulk: 𝜏Π + Π = − + 휉Π 𝜏휋, 𝜏Π : Relaxation time 푑𝜏 𝜏

8 Equation of motion 푒 : Energy density 푑푒 푒 + 푃 푒 휋 − Π 푃 : Pressure = − 1 − 푑𝜏 𝜏 푠푇 푠 : Entropy density 휋 Constitutive equations : Shear stress Π : Bulk pressure 푑휋 4휂 𝜏 + 휋 = + 휉 Shear: 휋 푑𝜏 3𝜏 휋 휂 : Shear viscosity 휁 : Bulk viscosity 푑Π 휁 Bulk: 𝜏Π + Π = − + 휉Π 𝜏휋, 𝜏Π : Relaxation time 푑𝜏 𝜏 휉: Gaussian white noise* Hydrodynamic *K. Murase, TH, arXiv:1304.3243; noises! K. Murase, Poster H-21 9 푇 − 푇 푇 − 푇 1 − tanh 푐 1 + tanh 푐 푑 = 푑 푑 + 푑 푑

eff 퐻 2 푄 2 . . 2 푑푄 = 37 4휋 3 d.o.f 푠 푇 = 푑 푇 eff 90 푇

푇푐 = 170 MeV 푝 푇 = 푠 푇′ 푑푇′ Effective Effective 푑퐻 = 3 0 푑 = 푇푐/50 푒 = 푇푠 − 푝 푇 (MeV)

D.H.Rischke, nucl-th/9809044 10 Shear viscosity 휂 1 휂 = 푠 4휋 푠 P.Kovtun et al., PRL94, 111601 (2005) 휁 Bulk viscosity 2 푠 휁 1 휂 = 15 − 푐2 푠 3 푠 푠 S.Weinberg, Astrophys.J.168, 175 (1971) 3휂 Relaxation time 𝜏 = 𝜏 = 휋 Π 2푝

Caveat: Just for demonstration! 11 Initial conditions Fluctuating hydro 𝜏0 = 1 fm 푇 = 0.22 GeV Viscous fluid 0 휋0 = Π0 = 0

Perfect fluid

12 Fluctuating hydro

Viscous fluid

Perfect fluid

Fixed initial condition  Final entropy fluctuation due to hydrodynamic noises 13

Perfect fluid Viscous fluid The number number The events of

Fluctuating entropy around mean value 14 15 D.J.Evans et al., PRL71,2401(1993); G.Gallavotti and E.G.D.Cohen, PRL74, 2694(1995); D.J.Evans and D.J.Searles, PRE52, 5839(1995).

푃 𝜎 = 퐴 = 푒퐴푡 푃 𝜎 : Probability distribution 푃 𝜎 = −퐴 of entropy production rate 1 푡 𝜎 푡 = 𝜎 푠 푑푠 Entropy production rate 푡 0 averaged over 0 < 푠 < 푡

푃 𝜎 Probability of negative entropy production 푃 𝜎 = 퐴 푃 𝜎 = −퐴  Quantified through fluctuation theorem! −퐴 0 퐴 𝜎

*Milestone in non-equilibrium statistical physics since linear response theory 16 Consequence of fluctuation theorem in Bjorken expansion*

∆푆 2 ∆ 𝜏푠 1 fin = 푆fin 𝜏0푠0 + ∆ 𝜏푠 ∆휂푠Δ푥Δ푦 Δ 𝜏푠 : Average entropy production per volume of local thermal system

Δ푉 = 𝜏Δ휂푠Δ푥Δ푦: Volume of local thermal system

*For brief derivation, see backup slide 17 The number of independent local thermal system: 푆 푏 Δ푥 푁 = Δ푥Δ푦 Δ푦 Theorem: Upper bound of total entropy fluctuation in Bjorken expansion Δ푆 1 ∆푆 2 ∆ 𝜏푠 1 tot = fin = 푆 푆 𝜏 푠 + ∆ 𝜏푠 tot 푁 fin 0 0 ∆휂푠푆 푏 1 1 1 ≤ = 2𝜏0푠0 ∆휂푠푆 푏 2푆ini 18 The number of independent local thermal system: 푆 푏 Δ푥 푁 = Δ푥Δ푦 Δ푦 Theorem: Upper bound of total entropy fluctuation in Bjorken expansion Δ푆 1 ∆푆 2 ∆ 𝜏푠 1 tot = fin = 푆 푆 𝜏 푠 + ∆ 𝜏푠 tot 푁 fin 0 0 ∆휂푠푆 푏 1 1 1 ≤ = 2𝜏0푠0 ∆휂푠푆 푏 2푆ini 19 푆

푏 p-p collision 𝜎in = 61 mb at LHC

) 푅 = 6 fm 𝜎 = 42 mb at RHIC

1 in −

fm 1 ( ~0.4 − 0.5 𝜎

푆 in 1 Enhancement of 푏 (fm) fluctuation effects in small system 20 • Hydrodynamic fluctuation in 1D Bjorken expansion • Final entropy (multiplicity) fluctuation due to hydro noises • Fluctuation theorem in 1D Bjorken expansion  Upper bound of fluctuation • More quantitative analysis needed • Fluctuation of flow velocity? • Higher harmonics in 2D expansion?

21 Finite number of

hadrons time Propagation of jets

Hydrodynamic fluctuation collision axis 0 Initial state fluctuation

Fluctuations appear everywhere! 22 • Hydro at work to describe elliptic flow (~ 2001) 푑 ≲ 5 fm

• E-by-e hydro at work to describe higher harmonics 푑 ≲ 1 fm (~ 2010)

• Hydro at work in p-p 푑 ≲ 1 fm? and/or p-A??? (2012-) Is fluctuation important

in such a small system? 23 1 휂 = lim lim 푑푡푑푥 푒푖(휔푡−푞푥) 휔→0푞→0 2휔

× 푇푥푦 푡, 푥 , 푇푥푦(0,0) Slow dynamics  How slow? Macroscopic time scale ~ 1 휔  푡"macro" Microscopic time scale ~ 𝜏 cf.) Long tail problem (liquid in 2D, glassy system, super-cooling, etc. ) 24 Constitutive equations Instantaneous response at Navier-Stokes level violates causality  Critical issue in 휋휇휈 = 2휂휕<휇푢휈>, relativistic theory 휇 Π = −𝜍휕휇푢 ,  Relaxation plays an … essential role

휅 thermodynamics force

/ 푅

or Realistic response

퐹 푡 푡0 𝜏 25 Linear response to thermodynamic force ′ ′ ′ Π 푡 = 푑푡 퐺푅 푡, 푡 퐹 푡 Retarded Green function (as an example) 휅 푡 − 푡′ 퐺 푡, 푡′ = exp − 휃(푡 − 푡′) 푅 𝜏 𝜏 Differential form Π 푡 − 휅퐹 푡 휅 Π 푡 = − , 푣 = < 푐 𝜏 signal 𝜏 Maxwell-Cattaneo Eq. (simplified Israel-Stewart Eq.) 26 퐺 퐺 푅 Non-Markovian 푅 Markovian 푡 휅 − coarse 퐺 = 푒 휏 퐺푅 ≈ 휅훿 푡′ 푅 𝜏 graining Maxwell-Cattaneo ??? Navier Stokes  causality 푡 → 푡′

푡 푡′ Existence of upper bound in coarse-graining time (or lower bound of frequency) in relativistic theory???

27 Figures from J.B.Bell et al., ESAIM 44-5 (2010)1085

Ex.) Seeds for instabilities

Rayleigh-Taylor instability Kelvin-Helmhortz instability Non-linearity, instability, dynamic critical phenomena,… 28 29 Linear response to thermodynamic force

′ ′ ′ Π 푡 = 푑푡 퐺푅 푡, 푡 퐹 푡

Dissipative current 횷 Thermodynamic force 푭 Shear Shear stress tensor Gradient of flow Bulk Bulk pressure Divergence of flow Diffusion Diffusion current Gradient of chemical potential

퐺푅 Transport coefficient, relaxation time, … 30 • Generalized Langevin Eq. for currents

4 ′ ′ ′ Π 푥 = 푑 푥 퐺푅 푥, 푥 퐹 푥 + 훿Π(푥)

• Fluctuation-Dissipation Relation (F.D.R.) 훿Π(푥)훿Π(푥′) = 푇퐺∗(푥, 푥′) 퐺∗: Symmetrized correlation function 훿Π : Hydrodynamic fluctuation K. Murase and TH, arXiv:1304.3243[nucl-th] 31 휅 푡 − 푡′ 퐺 푡, 푡′ = exp − 휃(푡 − 푡′) 푅 𝜏 𝜏

2휋 4훿(휔 − 휔′) 훿 3 (풌 − 풌′) 훿Π∗ 훿Π = 2휅 휔,풌 휔′,풌′ 1 + 휔2𝜏2

 Colored noise!  (Indirect) consequence of causality

K. Murase and TH, arXiv:1304.3243[nucl-th] 32 Integral form 4 ′ ′ ′ Π 푥 = 푑 푥 퐺푅 푥, 푥 퐹 푥 + 훿Π(푥)

Differential form 푑Π 푥 𝜏 + Π 푥 = 퐹 푥 + 휉 푥 푑푡 휉 = ℒ 푑 푑푡 훿Π: white noise  No memory effect nor colored noise  Practically convenient K. Murase and TH, arXiv:1304.3243[nucl-th] 33 푑휋 4휂 𝜏 + 휋 = + 휉 휋 푑𝜏 3𝜏 휋

Discretized 휋 4휂 휉 휋 𝜏 + Δ𝜏 = 휋 𝜏 + − + + 휋 Δ𝜏 𝜏휋 3𝜏휋𝜏 𝜏휋

34 F.D.R. for shear viscosity

휇휈 훼훽 휇휈훼훽 (4) ′ 휉휋 (푥)휉휋 (푥′) = 4푇휂Δ 훿 푥 − 푥 1 1 Δ휇휈훼훽 = Δ휇훼Δ휈훽 + Δ휇훽Δ휈훼 − Δ휇휈Δ훼훽 2 3 Δ휇휈 = 푔휇휈 − 푢휇푢휈 휇 Bjorken expansion case 푢 = (cosh 휂푠 , 0, 0, sinh 휂푠) 00 33 00 33 휋 = 휋 − 휋 ⟹ 휉휋 = 휉휋 − 휉휋

35 00 33 Need F.D.R. for 휉휋 = 휉휋 − 휉휋

00 00 00 33 33 33 휉휋 푥 휉휋 푥′ = 휉휋 휉휋 − 2 휉휋 휉휋 + 휉휋 휉휋

Magnitude of fluctuation in discretized space-time 4푇휂 8푇휂 ⟹ Δ0000 − 2Δ0033 + Δ3333 = Δ𝜏Δ푉 3Δ𝜏Δ푉 Width of Gaussian white noise

Δ푉 = 𝜏Δ휂푠Δ푥Δ푦, Δ휂푠 = 1, Δ푥 = Δ푦 = 1 (fm) 36 Our approach Conventional method 푑 푠𝜏 휋 − Π 푑 푠𝜏 𝜏 3휋2 Π2 = ≶ 0 = + ≥ 0 푑𝜏 푇 푑𝜏 푇 4휂 휁 Entropy production  Not positive definite Constitutive equations due to hydrodynamic noises designed to obey 2nd law  2nd law of thermodynamics of thermodynamics on (ensemble) average

37 Suppose entropy production rate obeys Gaussian 1 𝜎 − 𝜎 2 푃 𝜎 = exp − 2휋푎2 2푎2 From fluctuation theorem, 2 𝜎 = 푡 ⟹ 푡푎 = 2 𝜎 푡 푎2

2 푆fin − 푆fin 푃 푆fin ∝ exp − 2 2 2 𝜎 푡

38 39