Quark Matter 2015 September 29 2015 Kobe Japan

Quark Matter 2015 September 29, 2015 Kobe, Japan Causal hydrodynamic fluctuation in Bjorken expansion Kenichi Nagai 1) In collaboration with Ryuichi Kurita 2) , Koichi Murase 2) and Tetsufumi Hirano 1) 1) Department of Physics, Sophia Univ, Japan. 2) Department of Physics, The University of Tokyo, Japan.

Contents 1. Introduction 2. Causal hydrodynamic fluctuation in Bjorken expansion 3. Results 4. Summary 2

Contents 1. Introduction 2. Causal hydrodynamic fluctuation in Bjorken expansion 3. Results 4. Summary 3

Introduction time Hydrodynamic fluctuation! Hadron gas QGP fluid collision axis 0 Initial collision 4

Entropy Introduction (contd. ) Thermal equilibrium state = Maximum entropy state dissipation fluctuation Fluctuation dissipation relation The effect of causal hydrodynamic fluctuation on entropy production on an event by event basis.

Contents 1. Introduction 2. Causal hydrodynamic fluctuation in Bjorken expansion 3. Results 4. Summary 6

Hydrodynamic equations in Bjorken expansion Bjorken’s equation of energy density evolution Correction from viscosities 7

Constitutive equation First order constitutive equations for 3 D and Bjorken expansion 3 -dimensional expansion Bjorken expansion

Stochastic constitutive equations For shear stress • For bulk pressure First order theory Hydrodynamic fluctuation Relaxation term 9

Fluctuation dissipation relation Fluctuation and dissipation are related with each other! 10

Fluctuation dissipation relation in Bjorken expansion Fluctuation dissipation relation Bulk pressure: Shear stress: Other properties:

Model for Eo. S Equation of state with crossover M. Asakawa and T. Hatsuda, Phys. Rev. D 55, 4488 (1997).

Models for transport coefficients Temperature dependence of transport coefficients P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005). S. Weinberg, Astrophys. J. 168, 175 (1971) 13

Contents 1. Introduction 2. Causal hydrodynamic fluctuation in Bjorken expansion 3. Results 4. Summary 14

Time evolution of temperature Perfect Viscous

Time evolution of entropy Viscous Perfect Hydrodynamic fluctuation⇒final entropy fluctuation Entropy production rate: 16

Time evolution of shear stress and bulk pressure.

Final entropy distribution ideal • Final entropy in hydrodynamic fluctuation is distributed around that of the viscous fluid. • Final entropy can be smaller than initial entropy in a small number of events. • The smaller volume, the larger variance of entropy. 18

Contents 1. Introduction 2. Causal hydrodynamic fluctuation in Bjorken expansion 3. Results 4. Summary 19

Summary • Causal hydrodynamic fluctuation in Bjorken expansion model • Final entropy fluctuates around the result of viscous fluid. • Fluctuation effect would be important in small systems. 20

Backup

Bjorken expansion Flow velocity: 22

Parameter dependence of final entropy *This simulation was done without bulk pressure Relaxation time dependence**

Introduction Thermal equilibrium state = Maximum entropy state Dissipation Entropy Fluctuation the effect of causal hydrodynamic fluctuation on entropy production on an event by event basis. State