Spin hydrodynamics
inultra-relativistic heavy-ion collisions
Hidetoshi TAYA
(Fudan University)
Ref: Hattori, Hongo, Huang, Matsuo, HT, Phys. Lett. B795, 100 (2019) [arXiv:1901.06615]

Spin hydrodynamics
in①
ultra-relativistic heavy-ion collisions
Hidetoshi TAYA
(Fudan University)
Ref: Hattori, Hongo, Huang, Matsuo, HT, Phys. Lett. B795, 100 (2019) [arXiv:1901.06615]

②
Spin hydrodynamics
in①
ultra-relativistic heavy-ion collisions
Hidetoshi TAYA
(Fudan University)
Ref: Hattori, Hongo, Huang, Matsuo, HT, Phys. Lett. B795, 100 (2019) [arXiv:1901.06615]

Ultra-relativistic heavy ion collisions (HIC)

cartoon stolen from Internet

Purpose: Study quark-gluon plasma (QGP)(⇒ early Universe, origin of mass, QCD … )Lesson: QGP behaves like a perfect liquidand hydrodynamics works very well

Strong vorticity in HIC (1/2)
L&ω

cartoon stolen from Internet

Huge angular momentum: 𝑳𝑳 ∼ 𝒑𝒑 × 𝒙𝒙 ∼ 105 ℏ⇒ QGP should be “rotating quickly”⇒ QGP should have strong vorticity 𝝎𝝎 = rot 𝒗𝒗 ≫ 1 (? )Strong vorticity in HIC (2/2)
Galaxies

TyphoonWashing
machineHelium 4(strongest in lab.)

QGP in HIC[Gomez et al. (2014)]~ 10−15 Hz< 10−1 Hz< 10+3 Hz

10+7 Hz

10+20 Hz= 1 ∼ 10 MeV[Jiang, Lin, Liao (2017)] [Li et al. (2017)][Xia et al (2018)] [Wei, Deng, Huang (2019)] …

Strong vorticity in HIC (2/2)
Galaxies

TyphoonWashing
machineHelium 4(strongest in lab.)

QGP in HIC[Gomez et al. (2014)]~ 10−15 Hz< 10−1 Hz< 10+3 Hz

10+7 Hz

10+20 Hz= 1 ∼ 10 MeV[Jiang, Lin, Liao (2017)] [Li et al. (2017)][Xia et al (2018)] [Wei, Deng, Huang (2019)] …

The STRONGEST vorticity ever in our Universe !
 Very NEW & UNIQUE opportunity to study phys. of strong vorticity
 Vorticity appears in various systems ⇒ interdisciplinary interest

Strong vorticity in HIC (2/2)
Galaxies

TyphoonWashing
machineHelium 4

QGP in HIC(strongest in lab.)[Gomez et al. (2014)]~ 10−15 Hz< 10−1 Hz< 10+3 Hz

10+7 Hz

10+20 Hz= 1 ∼ 10 MeV[Jiang, Lin, Liao (2017)] [Li et al. (2017)][Xia et al (2018)] [Wei, Deng, Huang (2019)] …

The STRONGEST vorticity ever in our Universe !
 Very NEW & UNIQUE opportunity to study phys. of strong vorticity
 Vorticity appears in various systems ⇒ interdisciplinary interest

First step: What happens to QGP ?
cf. talk by Yang, Yamamoto

Expectation: QGP is polarized along 𝝎𝝎[Liang, Wang (2005)] [Voloshin (2004)]

Expectation: QGP is polarized along 𝝎𝝎[Liang, Wang (2005)] [Voloshin (2004)]

because of spin-vorticity coupling 𝐸𝐸 → 𝐸𝐸 − 𝜔𝜔 ⋅ 𝑠𝑠(~ LS force 𝛿𝛿𝛿𝛿 ∝ 𝐿𝐿 ⋅ 𝑠𝑠)[Vilenkin (1980)] [Hehl, Ni (1990)][Matsuo et al. (2011)] [Beccatini (2012)] [Beccatini et al. (2013)] [Hattori et al . (2019)] …

Expectation: QGP is polarized along 𝝎𝝎[Liang, Wang (2005)] [Voloshin (2004)]

because of spin-vorticity coupling 𝐸𝐸 → 𝐸𝐸 − 𝜔𝜔 ⋅ 𝑠𝑠(~ LS force 𝛿𝛿𝛿𝛿 ∝ 𝐿𝐿 ⋅ 𝑠𝑠)[Vilenkin (1980)] [Hehl, Ni (1990)][Matsuo et al. (2011)] [Beccatini (2012)] [Beccatini et al. (2013)] [Hattori et al . (2019)] …
[Takahashi et al (2015)]

cf. talk by Matsuo

 such a coupling is already confirmed in spintronics

Experimental fact ➡ Observed[STAR, PRC 98, 014910 (2018)]

central𝜔𝜔~0non-central𝜔𝜔 ≫1Exp. suggest 𝜔𝜔 = 𝑂𝑂(1020 Hz) = 𝑂𝑂(1 ∼ 10 MeV) consistent w/ theor. estimates

How about theory?
Hydro for spin polarized QGP?

How about theory?
Hydro for spin polarized QGP?

Far from complete∵ Relativistic spin hydro has NOT even been formulated !

Current status of formulation of spin hydro Non-relativistic case

[Eringen (1998)] [Lukaszewicz (1999)]

Already established (e.g. micropolar fluid)

spintronics:- applied to pheno. and is successful e.g. [Takahashi
et al. (2015)]- spin must be dissipative because of mutual
conversion b/w spin and orbital angular momentum

 Relativistic caseCurrent status of formulation of spin hydro Non-relativistic case

[Eringen (1998)] [Lukaszewicz (1999)]

Already established (e.g. micropolar fluid)

spintronics:- applied to pheno. and is successful e.g. [Takahashi
et al. (2015)]- spin must be dissipative because of mutual
conversion b/w spin and orbital angular momentum

 Relativistic case

[Florkowski, Ryblewski, Kumar, …]Some preceding works do exist, but- only for “ideal” fluid (no dissipative corrections)- some claim spin should be conserved (!?)

Purpose of this talk
 Formulate relativistic spin hydro with 1storder dissipative corrections for the first time Clarify spin must be dissipativePurpose of this talk
 Formulate relativistic spin hydro with 1storder dissipative corrections for the first time Clarify spin must be dissipative
1. Introduction

Outline2. Formulation based on entropy-current analysis
3. Linear mode analysis
4. Summary

Outline
1. Introduction
2. Formulation based on entropy-current analysis
3. Linear mode analysis
4. Summary

Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)

Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz

Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇
Step 2:4 eqs

Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

4 eqs

Step 2: Express 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o hydro variables (constitutive relation)--

Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

4 eqs

Step 2: Express 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o hydro variables (constitutive relation)𝜇𝜇- define hydro variables: {𝛽𝛽, 𝑢𝑢 }--𝑢𝑢𝜇𝜇 = 𝛾𝛾(1, 𝒗𝒗/𝑐𝑐)
𝑢𝑢2 = −1

1 + (4-1) = 4 DoGs
“chemical potential” for 𝑃𝑃𝜇𝜇

Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

4 eqs

Step 2: Express 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o hydro variables (constitutive relation)𝜇𝜇

𝑢𝑢𝜇𝜇 = 𝛾𝛾(1, 𝒗𝒗/𝑐𝑐)
𝑢𝑢2 = −1

1 + (4-1) = 4 DoGs

- define hydro variables: {𝛽𝛽, 𝑢𝑢 }“chemical potential” for 𝑃𝑃𝜇𝜇
- write down all the possible tensor structures of 𝑇𝑇𝜇𝜇𝜇𝜇-

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑓𝑓1 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑓𝑓2 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈
+𝑓𝑓3 𝛽𝛽 𝜖𝜖 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜎𝜎 + 𝑓𝑓4 𝛽𝛽 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑓𝑓5 𝛽𝛽 𝜕𝜕 𝜈𝜈 𝑢𝑢𝜇𝜇+𝑓𝑓6 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓7 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓8 𝛽𝛽 𝑢𝑢𝜇𝜇 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + ⋯ + 𝑂𝑂(𝜕𝜕 2 )Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

4 eqs

Step 2: Express 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o hydro variables (constitutive relation)𝜇𝜇

𝑢𝑢𝜇𝜇 = 𝛾𝛾(1, 𝒗𝒗/𝑐𝑐)
𝑢𝑢2 = −1

1 + (4-1) = 4 DoGs

- define hydro variables: {𝛽𝛽, 𝑢𝑢 }“chemical potential” for 𝑃𝑃𝜇𝜇
- write down all the possible tensor structures of 𝑇𝑇𝜇𝜇𝜇𝜇
𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑓𝑓1 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑓𝑓2 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈
+𝑓𝑓3 𝛽𝛽 𝜖𝜖 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜎𝜎 + 𝑓𝑓4 𝛽𝛽 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑓𝑓5 𝛽𝛽 𝜕𝜕 𝜈𝜈 𝑢𝑢𝜇𝜇+𝑓𝑓6 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓7 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓8 𝛽𝛽 𝑢𝑢𝜇𝜇 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + ⋯ + 𝑂𝑂(𝜕𝜕 2 )- simplify the tensor structures by (assumptions in hydro)(1) symmetry(2) power counting ➡ gradient expansion(3) other physical requirements ➡ thermodynamics (see next slide)Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

4 eqs

Step 2: Express 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o hydro variables (constitutive relation)𝜇𝜇

𝑢𝑢𝜇𝜇 = 𝛾𝛾(1, 𝒗𝒗/𝑐𝑐)
𝑢𝑢2 = −1

1 + (4-1) = 4 DoGs

- define hydro variables: {𝛽𝛽, 𝑢𝑢 }“chemical potential” for 𝑃𝑃𝜇𝜇
- write down all the possible tensor structures of 𝑇𝑇𝜇𝜇𝜇𝜇
𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑓𝑓1 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑓𝑓2 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈
+𝑓𝑓3 𝛽𝛽 𝜖𝜖 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜎𝜎 + 𝑓𝑓4 𝛽𝛽 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑓𝑓5 𝛽𝛽 𝜕𝜕 𝜈𝜈 𝑢𝑢𝜇𝜇+𝑓𝑓6 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓7 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓8 𝛽𝛽 𝑢𝑢𝜇𝜇 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + ⋯ + 𝑂𝑂(𝜕𝜕 2 )- simplify the tensor structures by (assumptions in hydro)(1) symmetry(2) power counting ➡ gradient expansion(3) other physical requirements ➡ thermodynamics (see next slide)Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

4 eqs

Step 2: Express 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o hydro variables (constitutive relation)𝜇𝜇

𝑢𝑢𝜇𝜇 = 𝛾𝛾(1, 𝒗𝒗/𝑐𝑐)
𝑢𝑢2 = −1

1 + (4-1) = 4 DoGs

- define hydro variables: {𝛽𝛽, 𝑢𝑢 }“chemical potential” for 𝑃𝑃𝜇𝜇
- write down all the possible tensor structures of 𝑇𝑇𝜇𝜇𝜇𝜇
𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑓𝑓1 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑓𝑓2 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈
+𝑓𝑓3 𝛽𝛽 𝜖𝜖 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜎𝜎 + 𝑓𝑓4 𝛽𝛽 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑓𝑓5 𝛽𝛽 𝜕𝜕 𝜈𝜈 𝑢𝑢𝜇𝜇+𝑓𝑓6 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓7 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓8 𝛽𝛽 𝑢𝑢𝜇𝜇 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + ⋯ + 𝑂𝑂(𝜕𝜕 2 )- simplify the tensor structures by (assumptions in hydro)(1) symmetry(2) power counting ➡ gradient expansion(3) other physical requirements ➡ thermodynamics (see next slide)Introduction to rela. hydro w/o spin (1/3)
An effective theory that describes spacetime
Hydro = evolution of Iong-wavelength modes (hydro modes)
See textbook

 Phenomenological formulation by Landau & Lifshitz
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

4 eqs

Step 2: Express 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o hydro variables (constitutive relation)𝜇𝜇

𝑢𝑢𝜇𝜇 = 𝛾𝛾(1, 𝒗𝒗/𝑐𝑐)
𝑢𝑢2 = −1

1 + (4-1) = 4 DoGs

- define hydro variables: {𝛽𝛽, 𝑢𝑢 }“chemical potential” for 𝑃𝑃𝜇𝜇
- write down all the possible tensor structures of 𝑇𝑇𝜇𝜇𝜇𝜇
𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑓𝑓1 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑓𝑓2 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈
+𝑓𝑓3 𝛽𝛽 𝜖𝜖 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜎𝜎 + 𝑓𝑓4 𝛽𝛽 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑓𝑓5 𝛽𝛽 𝜕𝜕 𝜈𝜈 𝑢𝑢𝜇𝜇+𝑓𝑓6 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓7 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓8 𝛽𝛽 𝑢𝑢𝜇𝜇 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + ⋯ + 𝑂𝑂(𝜕𝜕 2 )- simplify the tensor structures by (assumptions in hydro)(1) symmetry(2) power counting ➡ gradient expansion(3) other physical requirements ➡ thermodynamics (see next slide) Hydrodynamic eq. = conservation law + constitutive relation

Introduction to rela. hydro w/o spin (2/3)
 Constraints by thermodynamics

Introduction to rela. hydro w/o spin (2/3)
 Constraints by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o derivatives
𝜇𝜇𝜇𝜇

𝜇𝜇𝜈𝜈

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0) + 𝑇𝑇(1) + 𝑂𝑂(𝜕𝜕 2 ) where

because 𝑇𝑇𝜇𝜇𝜇𝜇

static eq.

𝜇𝜇𝜇𝜇

𝑇𝑇(0) = (𝑒𝑒, 𝑝𝑝, 𝑝𝑝, 𝑝𝑝)𝜇𝜇𝜇𝜇

𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )Introduction to rela. hydro w/o spin (2/3)
 Constraints by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o derivatives
𝜇𝜇𝜇𝜇

𝜇𝜇𝜈𝜈

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0) + 𝑇𝑇(1) + 𝑂𝑂(𝜕𝜕 2 ) where

because 𝑇𝑇𝜇𝜇𝜇𝜇

static eq.

𝜇𝜇𝜇𝜇

𝑇𝑇(0) = (𝑒𝑒, 𝑝𝑝, 𝑝𝑝, 𝑝𝑝)𝜇𝜇𝜇𝜇

𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )1st law of thermodynamics says
𝑑𝑑𝑑𝑑 = 𝛽𝛽𝛽𝛽𝛽𝛽, 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝)

With EoM 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇 , div. of entropy current 𝑆𝑆𝜇𝜇 = 𝑠𝑠𝑢𝑢𝜇𝜇 + 𝑂𝑂(𝜕𝜕) can be evaluated as𝜇𝜇𝜇𝜇

𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = −𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 + 𝑂𝑂(𝜕𝜕 3 )Introduction to rela. hydro w/o spin (2/3)
 Constraints by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o derivatives
𝜇𝜇𝜇𝜇

𝜇𝜇𝜈𝜈

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0) + 𝑇𝑇(1) + 𝑂𝑂(𝜕𝜕 2 ) where

because 𝑇𝑇𝜇𝜇𝜇𝜇

static eq.

𝜇𝜇𝜇𝜇

𝑇𝑇(0) = (𝑒𝑒, 𝑝𝑝, 𝑝𝑝, 𝑝𝑝)𝜇𝜇𝜇𝜇

𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )1st law of thermodynamics says
𝑑𝑑𝑑𝑑 = 𝛽𝛽𝛽𝛽𝛽𝛽, 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝)

With EoM 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇 , div. of entropy current 𝑆𝑆𝜇𝜇 = 𝑠𝑠𝑢𝑢𝜇𝜇 + 𝑂𝑂(𝜕𝜕) can be evaluated as𝜇𝜇𝜇𝜇

𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = −𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 + 𝑂𝑂(𝜕𝜕 3 )2st law of thermodynamics says 𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 ≥ 0, which is guaranteed
if RHS is expressed as a semi-positive bilinear as
𝜇𝜇𝜇𝜇

𝜇𝜇𝜇𝜇

−𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 = ∑𝑋𝑋 ∈ 𝑇𝑇 𝜆𝜆𝑖𝑖 𝑋𝑋𝑖𝑖 𝑋𝑋𝑖𝑖 𝜈𝜈𝜈𝜈 ≥ 0 with 𝜆𝜆𝑖𝑖 ≥ 0
𝑖𝑖1(strong constraint !!)Introduction to rela. hydro w/o spin (2/3)
 Constraints by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o derivatives
𝜇𝜇𝜇𝜇

𝜇𝜇𝜈𝜈

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0) + 𝑇𝑇(1) + 𝑂𝑂(𝜕𝜕 2 ) where

because 𝑇𝑇𝜇𝜇𝜇𝜇

static eq.

𝜇𝜇𝜇𝜇

𝑇𝑇(0) = (𝑒𝑒, 𝑝𝑝, 𝑝𝑝, 𝑝𝑝)𝜇𝜇𝜇𝜇

𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )1st law of thermodynamics says
𝑑𝑑𝑑𝑑 = 𝛽𝛽𝛽𝛽𝛽𝛽, 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝)

With EoM 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇 , div. of entropy current 𝑆𝑆𝜇𝜇 = 𝑠𝑠𝑢𝑢𝜇𝜇 + 𝑂𝑂(𝜕𝜕) can be evaluated as𝜇𝜇𝜇𝜇

𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = −𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 + 𝑂𝑂(𝜕𝜕 3 )2st law of thermodynamics says 𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 ≥ 0, which is guaranteed
if RHS is expressed as a semi-positive bilinear as
𝜇𝜇𝜇𝜇

𝜇𝜇𝜇𝜇

−𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 = ∑𝑋𝑋 ∈ 𝑇𝑇 𝜆𝜆𝑖𝑖 𝑋𝑋𝑖𝑖 𝑋𝑋𝑖𝑖 𝜈𝜈𝜈𝜈 ≥ 0 with 𝜆𝜆𝑖𝑖 ≥ 0ex) heat current:𝑖𝑖

1(strong constraint !!)𝜇𝜇𝜇𝜇2ℎ(𝜇𝜇 𝑢𝑢𝜈𝜈) ≡ ℎ𝜇𝜇 𝑢𝑢𝜈𝜈 + ℎ𝜈𝜈 𝑢𝑢𝜇𝜇 ∈ 𝑇𝑇 1 (𝑢𝑢𝜇𝜇 ℎ𝜇𝜇 = 0)𝜇𝜇𝜇𝜇

⇒ 𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 = −𝛽𝛽ℎ𝜇𝜇 𝛽𝛽𝜕𝜕⊥𝜇𝜇 𝛽𝛽 −1 + 𝑢𝑢𝜈𝜈 𝜕𝜕𝜈𝜈 𝑢𝑢𝜇𝜇 ≥ 0⇒ ℎ𝜇𝜇 = −𝜅𝜅 𝛽𝛽𝜕𝜕⊥𝜇𝜇 𝛽𝛽−1 + 𝑢𝑢𝜈𝜈 𝜕𝜕𝜈𝜈 𝑢𝑢𝜇𝜇 with 𝜅𝜅 ≥ 0

Introduction to rela. hydro w/o spin (3/3)
 Constitutive relation up to 1st order w/o spin𝜇𝜇𝜇𝜇

𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )
𝜇𝜇𝜇𝜇

(𝜇𝜇

<𝜇𝜇

𝑇𝑇(1) = −2𝜅𝜅 𝐷𝐷𝑢𝑢(𝜇𝜇 + 𝛽𝛽𝜕𝜕⊥ 𝛽𝛽−1 𝑢𝑢𝜈𝜈) − 2𝜂𝜂𝜕𝜕⊥ 𝑢𝑢𝜈𝜈> − 𝜁𝜁 𝜕𝜕𝜇𝜇 𝑢𝑢𝜇𝜇 Δ𝜇𝜇𝜇𝜇heat current

shear viscosity

bulk viscosityIntroduction to rela. hydro w/o spin (3/3)
 Constitutive relation up to 1st order w/o spin𝜇𝜇𝜇𝜇

𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )
(𝜇𝜇

𝜇𝜇𝜇𝜇

<𝜇𝜇

𝑇𝑇(1) = −2𝜅𝜅 𝐷𝐷𝑢𝑢(𝜇𝜇 + 𝛽𝛽𝜕𝜕⊥ 𝛽𝛽−1 𝑢𝑢𝜈𝜈) − 2𝜂𝜂𝜕𝜕⊥ 𝑢𝑢𝜈𝜈> − 𝜁𝜁 𝜕𝜕𝜇𝜇 𝑢𝑢𝜇𝜇 Δ𝜇𝜇𝜇𝜇heat current

shear viscosity

bulk viscosity Relativistic hydrodynamic eq. w/o spin
Hydrodynamic eq. = conservation law + constitutive relation

Euler eq.
Navier-Stokes eq.
⋮

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇⋮

𝜇𝜇𝜇𝜇

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0)

𝜇𝜇𝜇𝜇

𝜇𝜇𝜇𝜇

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0) + 𝑇𝑇(1)⋮(*) Correctly reproduce the non-relativistic eqs. w/ 𝑢𝑢 → 1, 𝒗𝒗/𝑐𝑐 + 𝑂𝑂( 𝑣𝑣⁄𝑐𝑐 2 )Formulation of rela. hydro with spin (1/4)
 Strategy is the same
 Phenomenological formulation
Step 1: Write down the conservation law
Step 2: Construct a constitutive relation- define hydro variables- write down all the possible tensor structures- simplify the tensor structures by e.g. thermodynamics

Formulation of rela. hydro with spin (2/4)
Step 1: Write down the conservation law(1) energy conservation

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇(canonical)

Formulation of rela. hydro with spin (2/4)
Step 1: Write down the conservation laws
(1) energy conservation

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇(canonical)(2) total angular momentum conservation

0 = 𝜕𝜕𝜇𝜇 𝑀𝑀𝜇𝜇,𝛼𝛼𝛼𝛼Formulation of rela. hydro with spin (2/4)
Step 1: Write down the conservation laws
(1) energy conservation

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇(canonical)(2) total angular momentum conservation

0 = 𝜕𝜕𝜇𝜇 𝑀𝑀𝜇𝜇,𝛼𝛼𝛼𝛼= 𝜕𝜕𝜇𝜇 (𝐿𝐿𝜇𝜇,𝛼𝛼𝛼𝛼 +Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )𝜓𝜓(𝑥𝑥) → 𝑆𝑆 Λ 𝜓𝜓(Λ−1 𝑥𝑥)= 𝜕𝜕𝜇𝜇 (𝑥𝑥 𝛼𝛼 𝑇𝑇𝜇𝜇𝜇𝜇 − 𝑥𝑥 𝛽𝛽 𝑇𝑇𝜇𝜇𝜇𝜇 + Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )∴ 𝜕𝜕𝜇𝜇 Σ𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑇𝑇 𝛼𝛼𝛼𝛼 − 𝑇𝑇𝛽𝛽𝛽𝛽

Formulation of rela. hydro with spin (2/4)
Step 1: Write down the conservation laws
(1) energy conservation

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇(canonical)(2) total angular momentum conservation

0 = 𝜕𝜕𝜇𝜇 𝑀𝑀𝜇𝜇,𝛼𝛼𝛼𝛼= 𝜕𝜕𝜇𝜇 (𝐿𝐿𝜇𝜇,𝛼𝛼𝛼𝛼 +Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )𝜓𝜓(𝑥𝑥) → 𝑆𝑆 Λ 𝜓𝜓(Λ−1 𝑥𝑥)= 𝜕𝜕𝜇𝜇 (𝑥𝑥 𝛼𝛼 𝑇𝑇𝜇𝜇𝜇𝜇 − 𝑥𝑥 𝛽𝛽 𝑇𝑇𝜇𝜇𝜇𝜇 + Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )∴ 𝜕𝜕𝜇𝜇 Σ𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑇𝑇 𝛼𝛼𝛼𝛼 − 𝑇𝑇𝛽𝛽𝛽𝛽𝜇𝜇𝜇𝜇 Spin is not conserved if (canonical) 𝑇𝑇𝜇𝜇𝜇𝜇 has anti-symmetric part 𝑇𝑇(a) There’s no a priori reason (canonical) 𝑇𝑇𝜇𝜇𝜇𝜇 must be symmetricFormulation of rela. hydro with spin (2/4)
Step 1: Write down the conservation laws
(1) energy conservation

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇(canonical)(2) total angular momentum conservation

0 = 𝜕𝜕𝜇𝜇 𝑀𝑀𝜇𝜇,𝛼𝛼𝛼𝛼= 𝜕𝜕𝜇𝜇 (𝐿𝐿𝜇𝜇,𝛼𝛼𝛼𝛼 +Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )𝜓𝜓(𝑥𝑥) → 𝑆𝑆 Λ 𝜓𝜓(Λ−1 𝑥𝑥)= 𝜕𝜕𝜇𝜇 (𝑥𝑥 𝛼𝛼 𝑇𝑇𝜇𝜇𝜇𝜇 − 𝑥𝑥 𝛽𝛽 𝑇𝑇𝜇𝜇𝜇𝜇 + Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )∴ 𝜕𝜕𝜇𝜇 Σ𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑇𝑇 𝛼𝛼𝛼𝛼 − 𝑇𝑇𝛽𝛽𝛽𝛽𝜇𝜇𝜇𝜇 Spin is not conserved if (canonical) 𝑇𝑇𝜇𝜇𝜇𝜇 has anti-symmetric part 𝑇𝑇(a) There’s no a priori reason (canonical) 𝑇𝑇𝜇𝜇𝜇𝜇 must be symmetricConsequence(1) Spin must not be a hydro mode in a strict sense𝜇𝜇𝜇𝜇(2) Nevertheless, it behaves like a hydro mode if 𝑇𝑇(𝑎𝑎) ≪ 1➡ inclusion of dissipative nature is crucially importantFormulation of rela. hydro with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇 , 𝛴𝛴𝜇𝜇,𝛼𝛼𝛼𝛼(1) define hydro variables

4 DoGs𝛽𝛽, 𝑢𝑢𝜇𝜇(2) simplify the tensor structure by thermodynamics

Formulation of rela. hydro with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇 , 𝛴𝛴𝜇𝜇,𝛼𝛼𝛼𝛼(1) define hydro variables

Introduce spin chemical potential𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝝎𝝎𝝁𝝁𝝁𝝁

4 + 6 = 10 DoGs = # of EoMs

with 𝜔𝜔 𝜇𝜇𝜇𝜇 = −𝜔𝜔 𝜈𝜈𝜈𝜈𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 are independent w/ each other at this stage (𝜔𝜔 𝜇𝜇𝜇𝜇 ≠ thermal vorticity)(2) simplify the tensor structure by thermodynamics

Formulation of rela. hydro with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇 , 𝛴𝛴𝜇𝜇,𝛼𝛼𝛼𝛼(1) define hydro variables

Introduce spin chemical potential𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝝎𝝎𝝁𝝁𝝁𝝁

4 + 6 = 10 DoGs = # of EoMs

with 𝜔𝜔 𝜇𝜇𝜇𝜇 = −𝜔𝜔 𝜈𝜈𝜈𝜈𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 are independent w/ each other at this stage (𝜔𝜔 𝜇𝜇𝜇𝜇 ≠ thermal vorticity)(2) simplify the tensor structure by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 , Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 , i.t.o derivatives𝜇𝜇𝜈𝜈𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑇𝑇 1 + 𝑂𝑂 𝜕𝜕 2 ,where I defined spin density 𝜎𝜎 𝛼𝛼𝛼𝛼

Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑢𝑢𝜇𝜇 𝜎𝜎 𝛼𝛼𝛼𝛼 + 𝑂𝑂(𝜕𝜕1 )Formulation of rela. hydro with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇 , 𝛴𝛴𝜇𝜇,𝛼𝛼𝛼𝛼(1) define hydro variables

Introduce spin chemical potential𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝝎𝝎𝝁𝝁𝝁𝝁

4 + 6 = 10 DoGs = # of EoMs

with 𝜔𝜔 𝜇𝜇𝜇𝜇 = −𝜔𝜔 𝜈𝜈𝜈𝜈𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 are independent w/ each other at this stage (𝜔𝜔 𝜇𝜇𝜇𝜇 ≠ thermal vorticity)(2) simplify the tensor structure by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 , Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 , i.t.o derivatives𝜇𝜇𝜈𝜈𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑇𝑇 1 + 𝑂𝑂 𝜕𝜕 2 ,where I defined spin density 𝜎𝜎 𝛼𝛼𝛼𝛼

Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑢𝑢𝜇𝜇 𝜎𝜎 𝛼𝛼𝛼𝛼 + 𝑂𝑂(𝜕𝜕1 )Generalize 1st law of thermodynamics with spin as
𝑑𝑑𝑑𝑑 = 𝛽𝛽(𝑑𝑑𝑑𝑑 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝑑𝑑𝑑𝑑𝜇𝜇𝜇𝜇 ), 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝜎𝜎 𝜇𝜇𝜇𝜇 )Formulation of rela. hydro with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇 , 𝛴𝛴𝜇𝜇,𝛼𝛼𝛼𝛼(1) define hydro variables

Introduce spin chemical potential𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝝎𝝎𝝁𝝁𝝁𝝁

4 + 6 = 10 DoGs = # of EoMs

with 𝜔𝜔 𝜇𝜇𝜇𝜇 = −𝜔𝜔 𝜈𝜈𝜈𝜈𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 are independent w/ each other at this stage (𝜔𝜔 𝜇𝜇𝜇𝜇 ≠ thermal vorticity)(2) simplify the tensor structure by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 , Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 , i.t.o derivatives𝜇𝜇𝜈𝜈𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑇𝑇 1 + 𝑂𝑂 𝜕𝜕 2 ,where I defined spin density 𝜎𝜎 𝛼𝛼𝛼𝛼

Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑢𝑢𝜇𝜇 𝜎𝜎 𝛼𝛼𝛼𝛼 + 𝑂𝑂(𝜕𝜕1 )Generalize 1st law of thermodynamics with spin as
𝑑𝑑𝑑𝑑 = 𝛽𝛽(𝑑𝑑𝑑𝑑 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝑑𝑑𝑑𝑑𝜇𝜇𝜇𝜇 ), 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝜎𝜎 𝜇𝜇𝜇𝜇 )With EoMs, div. of entropy current 𝑆𝑆𝜇𝜇 = 𝑠𝑠𝑢𝑢𝜇𝜇 + 𝑂𝑂(𝜕𝜕) can be evaluated as𝜇𝜇𝜇𝜇
𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = −𝑇𝑇 1s

𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 + 𝜕𝜕𝜈𝜈 𝛽𝛽𝛽𝛽𝜇𝜇𝜇𝜇𝜇𝜇− 𝑇𝑇 1a2𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 − 𝜕𝜕𝜈𝜈 𝛽𝛽𝛽𝛽𝜇𝜇− 2𝛽𝛽𝜔𝜔𝜇𝜇𝜇𝜇 + 𝑂𝑂(𝜕𝜕 3 )2

Formulation of rela. hydro with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇 , 𝛴𝛴𝜇𝜇,𝛼𝛼𝛼𝛼(1) define hydro variables

Introduce spin chemical potential𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝝎𝝎𝝁𝝁𝝁𝝁

4 + 6 = 10 DoGs = # of EoMs

with 𝜔𝜔 𝜇𝜇𝜇𝜇 = −𝜔𝜔 𝜈𝜈𝜈𝜈𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 are independent w/ each other at this stage (𝜔𝜔 𝜇𝜇𝜇𝜇 ≠ thermal vorticity)(2) simplify the tensor structure by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 , Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 , i.t.o derivatives𝜇𝜇𝜈𝜈𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑇𝑇 1 + 𝑂𝑂 𝜕𝜕 2 ,where I defined spin density 𝜎𝜎 𝛼𝛼𝛼𝛼

Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑢𝑢𝜇𝜇 𝜎𝜎 𝛼𝛼𝛼𝛼 + 𝑂𝑂(𝜕𝜕1 )Generalize 1st law of thermodynamics with spin as
𝑑𝑑𝑑𝑑 = 𝛽𝛽(𝑑𝑑𝑑𝑑 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝑑𝑑𝑑𝑑𝜇𝜇𝜇𝜇 ), 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝜎𝜎 𝜇𝜇𝜇𝜇 )With EoMs, div. of entropy current 𝑆𝑆𝜇𝜇 = 𝑠𝑠𝑢𝑢𝜇𝜇 + 𝑂𝑂(𝜕𝜕) can be evaluated as𝜇𝜇𝜇𝜇
𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = −𝑇𝑇 1s

𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 + 𝜕𝜕𝜈𝜈 𝛽𝛽𝛽𝛽𝜇𝜇𝜇𝜇𝜇𝜇− 𝑇𝑇 1a2𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 − 𝜕𝜕𝜈𝜈 𝛽𝛽𝛽𝛽𝜇𝜇− 2𝛽𝛽𝜔𝜔𝜇𝜇𝜇𝜇 + 𝑂𝑂(𝜕𝜕 3 )2𝜇𝜇𝜇𝜇 2nd law of thermodynamics 𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 ≥ 0 gives strong constraint on 𝑇𝑇 1
 In global equil. 𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = 0, so 𝜔𝜔 = vorticity (spin-vorticity coup.).

Formulation of rela. hydro with spin (4/4)
 Constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇

𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )heat current
(𝜇𝜇

𝜇𝜇𝜇𝜇

𝜇𝜇𝜇𝜇

up to 1st order with spin

shear viscosity<𝜇𝜇bulk viscosity

𝑇𝑇(1) = −2𝜅𝜅 𝐷𝐷𝑢𝑢(𝜇𝜇 + 𝛽𝛽𝜕𝜕⊥ 𝛽𝛽−1 𝑢𝑢𝜈𝜈) − 2𝜂𝜂𝜕𝜕⊥ 𝑢𝑢𝜈𝜈> − 𝜁𝜁 𝜕𝜕𝜇𝜇 𝑢𝑢𝜇𝜇 Δ𝜇𝜇𝜇𝜇
[𝜇𝜇

[𝜇𝜇

𝜇𝜇

−2𝜆𝜆 −𝐷𝐷𝑢𝑢[𝜇𝜇 + 𝛽𝛽𝜕𝜕⊥ 𝛽𝛽−1 + 4𝑢𝑢𝜌𝜌 𝜔𝜔 𝜌𝜌[𝜇𝜇 𝑢𝑢𝜈𝜈] − 2𝛾𝛾 𝜕𝜕⊥ 𝑢𝑢𝜈𝜈] − 2Δ𝜌𝜌 Δ𝜆𝜆𝜈𝜈 𝜔𝜔 𝜌𝜌𝜌𝜌NEW !

“boost heat current”“rotational (spinning) viscosity
e.g. Eringen (1998); Lukaszewicz (1999)

 Relativistic generalization of a non-relativistic micropolar fluid
 “boost heat current” is a relativistic effect Relativistic hydrodynamic eq. up to 1st order with spin
𝜇𝜇𝜇𝜇

𝜇𝜇𝜇𝜇

0 = 𝜕𝜕𝜇𝜇 (𝑇𝑇 0 + 𝑇𝑇 1 + 𝑂𝑂(𝜕𝜕 2 ))

𝛼𝛼𝛼𝛼

𝛽𝛽𝛽𝛽

𝜕𝜕𝜇𝜇 (𝑢𝑢𝜇𝜇 𝜎𝜎 𝛼𝛼𝛼𝛼 ) = 𝑇𝑇(1) − 𝑇𝑇 1 + 𝑂𝑂(𝜕𝜕 2 )Outline
1. Introduction
2. Formulation based on entropy-current analysis
3. Linear mode analysis
4. SummaryLinear mode analysis (1/2)
Setup: small perturbations on top of global therm. equilibrium𝑖𝑖𝜕𝜕𝑡𝑡

𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿
𝜇𝜇
= 𝑀𝑀 𝛿𝛿𝛿𝛿𝜇𝜇 + 𝑂𝑂(𝛿𝛿 2 )𝛿𝛿𝛿𝛿𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇hydro evolution

𝛽𝛽 = 𝛽𝛽0

𝛽𝛽 = 𝛽𝛽0 + 𝛿𝛿𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇 = 0𝜔𝜔 𝜇𝜇𝜇𝜇 = 0 + 𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇

𝑢𝑢𝜇𝜇 = 1, 𝟎𝟎

𝑢𝑢𝜇𝜇 = 1, 𝟎𝟎 + 𝛿𝛿𝑢𝑢𝜇𝜇

propagate if Im 𝜔𝜔 = 0
dissipate if Im 𝜔𝜔 < 0unstable if Im 𝜔𝜔 > 0

Linear mode analysis (2/2)

 Hydro w/o spin {𝛽𝛽, 𝑢𝑢𝜇𝜇 }4 gapless modes

2 sound modes 𝜔𝜔 = ±𝑐𝑐s 𝑘𝑘 + 𝑂𝑂(𝑘𝑘 2 )2 shear modes 𝜔𝜔 = 0 + 𝑂𝑂(𝑘𝑘 2 )where 𝑐𝑐s2 ≡ 𝜕𝜕𝜕𝜕/𝜕𝜕𝜕𝜕 Hydro with spin {𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 }Linear mode analysis (2/2)

 Hydro w/o spin {𝛽𝛽, 𝑢𝑢𝜇𝜇 }4 gapless modes

2 sound modes 𝜔𝜔 = ±𝑐𝑐s 𝑘𝑘 + 𝑂𝑂(𝑘𝑘 2 )2 shear modes 𝜔𝜔 = 0 + 𝑂𝑂(𝑘𝑘 2 )where 𝑐𝑐s2 ≡ 𝜕𝜕𝜕𝜕/𝜕𝜕𝜕𝜕 Hydro with spin {𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 }4 gapless modes

2 sound modes 𝜔𝜔 = ±𝑐𝑐s 𝑘𝑘 + 𝑂𝑂(𝑘𝑘 2 )2 shear modes 𝜔𝜔 = 0 + 𝑂𝑂(𝑘𝑘 2 )+ 6 dissipative gapped modes

3 “boost” modes 𝜔𝜔 = −2𝑖𝑖𝜏𝜏b−1 + 𝑂𝑂(𝑘𝑘 2 )3 “spin” modes𝜔𝜔 = −2𝑖𝑖𝜏𝜏s−1 + 𝑂𝑂(𝑘𝑘 2 )𝜕𝜕𝜎𝜎𝑖𝑖𝑖𝑖 ⁄𝜕𝜕𝜔𝜔𝑖𝑖𝑖𝑖
𝜕𝜕𝜎𝜎𝑖𝑖0 ⁄𝜕𝜕𝜔𝜔𝑖𝑖0
where 𝜏𝜏s ≡, 𝜏𝜏b ≡4𝛾𝛾
4𝜆𝜆

 We explicitly confirmed that spin is dissipative Time-scale of the dissipation is controlled by the new viscous constants 𝛾𝛾, 𝜆𝜆Outline
1. Introduction
2. Formulation based on entropy-current analysis
3. Linear mode analysis
4. Summary

Summary Spin polarization in QGP is one of the hottest topics in HIC.
But, its theory, in particular hydrodynamic framework, is still
under construction
 Relativistic spin hydrodynamics with 1st orderdissipative corrections is formulated for the first time
based on the phenomenological entropy-current analysis
 Spin must be dissipative because of the mutual conversion
between the orbital angular momentum and spin Linear mode analysis of the spin hydrodynamic equation
also suggests that spin must be dissipative, whose time-scale
is controlled by the new viscous constants 𝛾𝛾, 𝜆𝜆Outlook: extension to 2nd order, Kubo formula, MHD,
application to cond-mat, numerical simulations

BACKUPLinearlized hydro eq.
whereDispersion relations

Further simplification by EoM
The 1st order constitutive relation reads

By using LO hydro eq.,
we can further simplify h,q as