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マクロ

統計力学とは何か？？

熱力学：時間発展はない
流体力学：保存量密度の時間発展

スケール
ミクロ

平衡統計力学

非平衡統計力学}

現象論

問題
ミクロな記述とマクロな記述をどう関係させられるか？

古典論：古典力学
量子論：量子力学・場の量子論 (例. QCD)

}

基礎理論

A. マクロな現象論とミクロな理論の関係づけを行いたい！
(有効理論)(基礎理論)

n4®T%◆ 非相対論的Navier-Stokes方程式 ＝ 非線形！
✓
◆
✓
◆
@
1
⌘
⇣
1⌘
+v·r v =rp +v++
r(r · v) + K
@t
⇢
⇢
⇢ 3⇢
<latexit sha1_base64="K79jLcePhnf2jdfAGCqj+y1Ilqk=">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</latexit>(1) 通常の流体力学{- 運動方程式 ＝ 非線形な決定論的偏微分方程式

場の理論で言うと
古典Yang-Mills理論- 保存量密度の平均値のみのダイナミクスを記述の運動方程式は非線形！

(2) 揺らぐ流体力学

Yang-Mills理論の{- 運動方程式 ＝ 非線形な確率論的偏微分方程式- 保存量密度のn点関数のダイナミクスを記述(非線形な流体揺らぎ)

非線形相互作用項から生じる量子揺らぎを考慮！

KwáT%°eª¢Part 0. 非平衡現象論としての流体力学 [Talk by 藤井]
Part 1. 通常の流体力学

Part 1.5. 流体力学中の線形揺らぎ [Ref. Mori (1965), Minami-Hidaka (2013)]
森の射影演算子法 (1965) ＝ カレント代数の線形非平衡系への拡張

Part 2. 揺らぐ流体力学

KwáT%°eª¢Part 0. 非平衡現象論としての流体力学 [Talk by 藤井]
Part 1. 通常の流体力学[Ref. Hayata-Hidaka-MH-Noumi, PRD(2015),
MH Ann. Phys.(2017)]

Part 1.5. 流体力学中の線形揺らぎ [Ref. Mori (1965), Minami-Hidaka (2013)]
森の射影演算子法 (1965) ＝ カレント代数の線形非平衡系への拡張

Part 2. 揺らぐ流体力学[Ref. Crossley-Glorioso-Liu, JHEP(2017)]

KwáT%°eª¢Part 0. 非平衡現象論としての流体力学 [Talk by 藤井]
Part 1. 通常の流体力学[Ref. Hayata-Hidaka-MH-Noumi, PRD(2015),
MH Ann. Phys.(2017)]

Part 1.5. 流体力学中の線形揺らぎ [Ref. Mori (1965), Minami-Hidaka (2013)]
森の射影演算子法 (1965) ＝ カレント代数の線形非平衡系への拡張

Part 2. 揺らぐ流体力学[Ref. Crossley-Glorioso-Liu, JHEP(2017)]

Part 1. 出題編

そもそも流体力学とは？

­#/ÒRAf/©®Ô­#/ÒRAf/©®Ô
保存則
µ T̂

µ

(x) = 0,µˆ
µ J (x) = 0

これだけでは解けない！

i«Ò:>Af/
保存則

n
+
t

·j =0

j=

D n

構成方程式

物性情報

拡散係数 D の値は？

n
t

D

2

n=0

­#/ÒRAf/©®Ô
与 保存則
µ T̂

µ

µ
ˆ
µ J (x) = 0(x) = 0,構成方程式
保存量に関するカレントが保存量の情報から決まる

T̂µ(x) = T

µ

[T

0

0

,J ] = T

µ

µ
µ
0
0
µ
ˆ
J (x) = J [T , J ] = J [

[

, ]
, ]

物性情報
状態方程式(静的性質)：

p = p[T0輸送係数(動的性質)： Li = Li [T0, J ] = p[00

, J ] = Li [

, ]
, ]

pDÒR!­&
ミクロ

マクロ

場の量子論
(QCD/QED/…)

流体力学
よく使われる方法

弱結合系にしか正当化できない

Kinetic theory
ボルツマン方程式

局所熱平衡分布のまわりで展開pDÒR!­&
ミクロ
場の量子論
(QCD/QED/…)

？

マクロ
流体力学

pDÒR!­&ミクロ

Nakajima (1957), Mori (1958), McLennan (1960)
Zubarev et al. (1979), Becattini et al. (2015)
Hayata-Hidaka-MH-Noumi (2015), MH (2017)

場の量子論(QCD/QED/…)局所熱平衡状態+ 小さなズレ
強結合系にも
適用が可能

マクロ
流体力学

Part 1. 解答編

流体力学の場の理論的導出[参考文献.
Hayata-Hidaka-MH-Noumi, PRD(2015), MH Ann. Phys.(2017)]&­¥±­¼¿Á¸¿Æ[Sasa PRL (2014),

Hayata, Hidaka, MH, Noumi PRD (2015)]

非平衡からの教訓：初期についてだけ適切な分布を入れる！
- 初期状態は局所熱平衡分布： ˆLG = e

d3 x µ (x)p̂µ (x)[ µ]- パラメーター( 温度・速度場 )の時空間変調はゆるやか

t

hT̂

µ⌫

i =?
t̄0
x
µ

(x) で特徴付けられる分布

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局所版への一般化◆ 局所熱平衡状態
局所ギブス分布：⇢ˆLG = e

K̂

K̂ =

d3 x

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ˆLG [t0 ;a
t0 ]e Ŝ[ t0 ] = e Ŝ[ t ]+Ŝ[ t ] Ŝ[ t0 ] = e Ŝ[ t ] Û (t)- =

with Û (t)

T e

1
d
0

a
Ŝ [ at ] Ŝ [ t ]0LG
任意の演算子に関する恒等式： hÔ(x)i = hÛ Ô(x)it<latexit sha1_base64="X9Qybdn2aFazdiFUgaxV2ocHXN8=">AAACx3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQu4xuQk5qXnpCrEZCSWVPvXalRoxhRBRGwVUORCa2FqFBCK4kviqn3ca+MFlA30LC3MjEzMFAz0DAzMDY0MQQwjcxNjEwVDoAgIKDNAQUC+wDaGGIYUhnyGZIZShlyGVIY8hhIgO4chkaEYCKMZDBkMGAqAYrEM1UCxIiArEyyfylDLwAXUWwpUlQpUkQgUzQaS6UBeNFQ0D8gHmVkM1p0MtCUHiIuAOhUYVA2uGqw0+GxwwmC1wUuDPzjNqgabAXJLJZBOguhNLYjn75II/k5QVy6QLmHIQOjC6+YShjQGC7BbM4FuLwCLgHyRDNFfVjX9c7BVkGq1msEig9dA9y80uGlwGOiDvLIvyUsDU4Nm43FPEtAtoOiBxYECbkaokZ6lnmGgibKDEzSeOBikGZQYNICRYc7gwODBEMAQCjR8NcNFhlsMt5m8mAqYypgqIEqZGKF6hBlQAFMjAPtPq4o=</latexit>

ÂÊÈÏ¾·#±µKÒF¶¥=[x©N¬¡¨FAC·]µÒ

Fastest Apparent Convergence： Û Jˆ0a (x) LG=0t

Jˆ0a (x) = Jˆ0a (x) LGtÀÛÌÚÐÝ]9¬¶¸¡°(\◆ 任意の演算子に関する恒等式 = ゆらぎの定理
hÔ(x)i = hÛ Ô(x)iLG
t ,

with

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ˆ t0 ; ] ⌘⌃[t,<latexit sha1_base64="1D5lplLfyW6+TmW7GceJBB7JK1o=">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</latexit>=<latexit sha1_base64="Q/K93xQIucNGTCl7l53IPDjUI54=">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</latexit>Û (t) ⌘ T⌧ e<latexit sha1_base64="JsFqLy3RzsHc0YV66EakbuARP3o=">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</latexit>

Z t

t0
Z t
t0

ds
ds

Z

Z

R1

a
a
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Ŝ
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0

d3 xN Jˆµa (x)rµ a (x)µaˆd xN J a (x)rµ (x)3aˆ⌃[t, t0 ; ] はパラメータの微分 (= rµ ) に比例！
<latexit sha1_base64="H5yAvuoY0ed+E1ni4B0rclMPADw=">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</latexit><latexit sha1_base64="/n0VC7Pccsoh7yeG6xUpY9qON/o=">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</latexit>

ˆ t0 ; ] に関する展開は微分展開になっている！！
⌃[t,<latexit sha1_base64="/n0VC7Pccsoh7yeG6xUpY9qON/o=">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</latexit>

参考. 第二法則を導出するゆらぎの定理： PF ( ; j] = e PB (<latexit sha1_base64="kl/sB69A/Htu+jf+2pNNJWIDwag=">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</latexit>
sha1_base64="YjF9HAGYaUYQxGevBp7HQIKkJtE=">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</latexit>
sha1_base64="uINm0A0ez612+x/MfARssvoABb8=">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</latexit>
sha1_base64="3RRELZDBqXkHJnKshs8aEK7zKvY=">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</latexit>; j]6-¯·ºL9Bg2°+
◆ 構成方程式のエントロピーによる展開 ＝ 散逸微分補正
hT̂ µ⌫ (x)i = hÛ T̂ µ⌫ (x)iLGt<latexit sha1_base64="6ohPvKUgZJT5F2yNa8ED/4Sgvtg=">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</latexit>

= hT̂

µ⌫

LG
(x)it +

<latexit sha1_base64="hFFHAsMDCjqI5tLM7cmNyW49eVE=">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</latexit>

hJˆµ (x)i = hÛ Jˆµ (x)iLGt<latexit sha1_base64="LgVCUGQ7EiRqh18uPRIb2RNWmZc=">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</latexit>µ
LGˆ= hJ (x)it +<latexit sha1_base64="t5HVMo6dN7mgYUn7Ky8aKXLgbDI=">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</latexit>

を代入すると， h T̂<latexit sha1_base64="dMFgu0AVG6qAA8blkZCrQf052UA=">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</latexit>

µ⌫

0

µ⌫
LG
ˆ
d⌧ hT⌧ ⌃⌧ [t, t0 ]T̂ (x)it + · · ·⌘ h T̂

µ⌫(x)i(1)<latexit sha1_base64="M4ivP4NQoAmcIgBr4DQvqXzyA1o=">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</latexit>

Z 1

ˆ t0 ; ] =エントロピー生成： ⌃[t,<latexit sha1_base64="QZvdYL4lLuq1HzHuhmjgc3gKjDg=">AAADLnichVHLahRBFL3dvuL4yEQ3gpvCIRLRDNUaiCjCqBtxlWQySWBqpqnurukpUv1Id/VgbNoP8AdcuFJwIdm48gfc+AMKWYhbxWUENy6809OgMRhvU31Pnbrn1imuEyuZakp3DfPI0WPHT0ydrJ06febsdH3m3FoaZYkrOm6komTD4alQMhQdLbUSG3EieOAose5s3h+fr49EksooXNXbsegF3A/lQLpcI2XXd9iQ65y1pR/wgnT1NW3T20xhA4/3yB0yT5gMtZ0jXfQ18VLi9W88IjWWbiU6n/cLhEoM9BxhnlCak7LfatFnQWbnT1iYFYSF3FHcRoYwR2jeR5bUCLm6T/Ow1PxZjDjDMsyJ9If6il1v0CYtgxwEVgUaUMVSVH8LDDyIwIUMAhAQgkasgEOKXxcsoBAj14McuQSRLM8FFFBDbYZVAis4spv493HXrdgQ9+Oeaal28RaFK0ElgVn6gb6me/Q93aFf6c9/9srLHmMv25idiVbE9vTTC+0f/1UFmDUMf6sO9axhADdLrxK9xyUzfoU70Y8eP9tr31qZzS/Tl/Qb+n9Bd+k7fEE4+u6+WhYrzw/x46CXAsdj/T2Mg2DtetOiTWt5odG6Vw1qCi7CJZjDaSxCCx7AEnTANWaMRaNl3DXfmB/NT+bnSalpVJrzsC/ML78AMUPNXg==</latexit>

Z 1

0

µ
LG
ˆ
ˆ
d⌧ hT⌧ ⌃⌧ [t, t0 ]J (x)it + · · ·Z tt0⌘ h Jˆµ (x)i(1)

⇣
⌘
p
3
µ
⌫
µ
ˆ
dsd x
g T̂ ⌫ rµ + J rµ ⌫<latexit sha1_base64="F+seVB2zMBJeCu0snv0rv/i32IA=">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</latexit>µˆ(x)i(1) , h J (x)i(1) を評価することが可能！

L9Bg2¬◆ 構成方程式の散逸補正⇣ µ⌫ ⇢µ⌫
h T̂ (x)i(1) =h h r⇢<latexit sha1_base64="5uRtZ9NlXrRPkT4ZrXhbyNX98mw=">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</latexit>µˆh J (x)i(1) =<latexit sha1_base64="XlKrJDwpSnLaLeu54SX0Un3XCaE=">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</latexit>

2
⌘

µ⇢ ⌫

2 h h rh⇢iµ⌫h r⌫ ⌫◆ 輸送係数の久保公式⇣ = (x)<latexit sha1_base64="Fhl+KlO3fPyl0wUxQxC5RuVJUOM=">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</latexit>

Z t

1

dd x0

Z 1
0

Z t

d⌧ heK̂⌧ ˜p̂(x0 )e K̂⌧ ˜p̂(x)iLG
t
Z 1

(x)
⌘=
dd x0
d⌧ heK̂⌧ ˜⇡
ˆ µ⌫ (x0 )e K̂⌧ ˜⇡
ˆ ⇢ (x)iLGt hµ⇢ (x)h⌫ (x)(d + 1)(d 2) 1
0
Z t
Z 1
(x)
=
dd x0
d⌧ heK̂⌧ ˜Jˆµ (x0 )e K̂⌧ ˜Jˆ⌫ (x)iLGt hµ⌫ (x)d 1 10<latexit sha1_base64="bvsSgY5EugVPrN6Gjf35mzALT68=">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</latexit><latexit sha1_base64="tiu7m2sLBvCvkMTZkfXy2vYC1dI=">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</latexit>¹O¥ã
◆ 構成方程式のエントロピーによる展開 ＝ 散逸微分補正
hT̂ µ⌫ (x)i = hÛ T̂ µ⌫ (x)iLGt<latexit sha1_base64="6ohPvKUgZJT5F2yNa8ED/4Sgvtg=">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</latexit>

= hT̂

µ⌫

LG
(x)it +？

<latexit sha1_base64="hFFHAsMDCjqI5tLM7cmNyW49eVE=">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</latexit>

hJˆµ (x)i = hÛ Jˆµ (x)iLGt<latexit sha1_base64="LgVCUGQ7EiRqh18uPRIb2RNWmZc=">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</latexit>µ
LGˆ= hJ (x)it +？

<latexit sha1_base64="t5HVMo6dN7mgYUn7Ky8aKXLgbDI=">AAADJHiclVFLaxRBEK6ZqInxkTVeBC+DSySiWWpETAgIQQ+KeEiy2SSwsxl6eju7TXoezPQsxmH+gH/Ag6cICuLdP+DFgzfJIRfv4jGCFw/WPEQ0GLGa7q76qr7qrykvUjLRiAeGOXbi5KnxidOTZ86eOz/VuDC9noRpzEWHhyqMNz2WCCUD0dFSK7EZxYL5nhIb3s69Ir8xEnEiw2BN70ai57NBILclZ5ogt7F3x3IUCwZKWM6Q6exhvpU5fprPPr5mOXGZKACmh7GfPbqf526mc+u65chAu7hlW31Hs/RnizW3iKpOTlsOfJZXSFffoPLef77B+6FO3EYTW1iaddSxa6cJtS2HjbfgQB9C4JCCDwIC0OQrYJDQ6oINCBFhPcgIi8mTZV5ADpPETalKUAUjdIfOAUXdGg0oLnomJZvTK4p2TEwLZnAfX+Mhvsc3+Bm//7VXVvYotOzS7VVcEblTTy+1v/2T5dOtYfiLdaxmDduwUGqVpD0qkeIXvOKPnjw7bC+uzmRX8QV+If17eIDv6AfB6Ct/uSJWnx+jxyMtOY3H/nMYR531my0bW/bKrebS3XpQE3AZrsAsTWMeluABLEMHuDFuzBm3jXnzlfnB/GjuV6WmUXMuwm9mfvoBD3HNww==</latexit>

を代入すると， h T̂<latexit sha1_base64="dMFgu0AVG6qAA8blkZCrQf052UA=">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</latexit>

µ⌫

0

µ⌫
LG
ˆ
d⌧ hT⌧ ⌃⌧ [t, t0 ]T̂ (x)it + · · ·⌘ h T̂

µ⌫(x)i(1)<latexit sha1_base64="M4ivP4NQoAmcIgBr4DQvqXzyA1o=">AAAC43ichVHLThRBFL00PhAfjLgxYVNxAhkSM7k9Thh0RWTDktcAydQ4VjfFUKH6YXf1ROz0D+iOmLggLDRxYdz7AybGH3DBJxjdobhx4e2emRgX4K1U1a1T95w6leuEWsUG8XjEGr1w8dLlsSvjV69dvzFRujm5EQdJ5MqmG+gg2nJELLXyZdMoo+VWGEnhOVpuOnuL+f1mT0axCvx1sx/Ktie6vtpRrjAEdUqPU16ItKKu006xen9+rlafu4tVxIZds/Ok1qjfq2dcPklUj3Et/K6WjG9LbQTju8Kk69mjlHsJ95Os8nSW8ago6aQVezbLOqXyUJUNVdlQldmE5FGGQSwHpQ/AYRsCcCEBDyT4YCjXICCm0QIbEELC2pASFlGminsJGYwTN6EqSRWC0D1au3RqDVCfzrlmXLBdekXTjIjJYBq/4Ds8wc/4Hr/i7zO10kIj97JPu9PnyrAz8fz22q//sjzaDez+ZZ3r2cAOzBdeFXkPCyT/hdvn9569Oll7sDqdzuAb/Eb+X+MxfqQf+L0f7tsVuXp4jh+HvOTtGfaAnZ1s1Ko2Vu2Vennh4aBRYzAFd6BC3WjAAizBMjRJ/RN8h59waknrhXVgveyXWiMDzi34J6yjPxBms2c=</latexit>

Z 1

ˆ t0 ; ] =エントロピー生成： ⌃[t,<latexit sha1_base64="QZvdYL4lLuq1HzHuhmjgc3gKjDg=">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</latexit>

Z 1

0

µ
LG
ˆ
ˆ
d⌧ hT⌧ ⌃⌧ [t, t0 ]J (x)it + · · ·Z tt0⌘ h Jˆµ (x)i(1)

⇣
⌘
p
3
µ
⌫
µ
ˆ
dsd x
g T̂ ⌫ rµ + J rµ ⌫<latexit sha1_base64="F+seVB2zMBJeCu0snv0rv/i32IA=">AAAC4HichVE9bxNBEB1f+AjmIyY0SDSrWEGOhKw5Y8WBKoIGUdlJnETyGWvvsnFW2fvgbs8inK5HNChVpKQCiQLR8wfS5A+kyE+I6DASDQVzZ1uIImFOezv7Zt7btxo7UDLSiGcFY+rK1WvXp28Ub966fWemdHd2PfLj0BFtx1d+uGnzSCjpibaWWonNIBTctZXYsHefZ/WNgQgj6Xtrei8QXZf3PbktHa4J6pW6iZWLdMK+3U2w+mRpsVZffIRVxIZZM7Ok1qg/rqeWeB3LAbMU9/pKMGtLKM2ZtcN18jJ9lVhunFbeLDArzOu9pGIupGmvVJ5Isokkm0gyk5AsyjCOpl/6BhZsgQ8OxOCCAA805Qo4RPR1wASEgLAuJISFlMm8LiCFInFj6hLUwQndpX+fTp0x6tE504xytkO3KFohMRnM4yl+wSGe4Fc8x98XaiW5RuZlj3Z7xBVBb+b9/dVf/2W5tGvY+cu61LOGbVjKvUryHuRI9gpnxB+8PRiuPl2ZTx7iJ/xO/j/iGR7TC7zBT+dzS6wcXeLHJi/ZeCYzYBcn67WqiVWzVS8vPxsPahoewBxUaBoNWIYX0IQ2qR/DOfyAoWEb74wPxv6o1SiMOffgnzAO/wCCprIA</latexit>µˆ(x)i(1) , h J (x)i(1) を評価することが可能！

'9V+rj­V!ÇÀÎºÉË
局所ギブス分布 w/tdµ= { µ, }µ t̄t̄t̄(x) = const.

x◆ マシュー・プランク汎関数[t̄; ]

log Tr exp= log Tr exp

d

t̄

d3 x̄

µ

(x)T̂ µ (x) + (x)Jˆ (x)gµ̄(x̄)T̂ 0̄µ̄ (x̄) + (x̄)Jˆ0̄ (x̄)

'9V+rj­/[ Banerjee et al.(2012), Jensen et al.(2012) , Haehl et al. (2015), MH(2017)]◆ 変分公式 in “hydrostatic gauge”
hT̂

µ⌫

2
LG
(x)it̄ = pg gµ⌫ (x)1µLGˆ[t̄; ], hJ (x)it̄ = p

g Aµ (x)[t̄; ]

'9V+rj­V!ÇÀÎºÉËt̄dµ

局所ギブス分布 w/

t
µ t̄

tµ

= { µ, }

µ
x
t̄

t̄
t̄(x) = const.

x◆ マシュー・プランク汎関数[t̄; ]

log Tr exp= log Tr exp

d

t̄

d3 x̄

µ

(x)T̂ µ (x) + (x)Jˆ (x)gµ̄(x̄)T̂ 0̄µ̄ (x̄) + (x̄)Jˆ0̄ (x̄)

Hydrostatic gauge fixing
Picture before gauge fixing
Future time
directionµPicture in hydrostatic gauge(t̄2 , x̄)tµt̄(x) = t̄2µ(t̄1 , x̄)

tµ

t̄
Gauge
fixingtµ = e u µ(e ⌘ / 0 )t̄(x) = t̄1

t x

j

xi

µ̄

x̄i = const.

(x) =

t̄2
x̄j
t̄1

x̄i

µ
µ
t
(x)
⌘
@
x
局所的に時間方向を定めることができる：
t̄

Hydrostatic gauge conditionµt (x) =µ

(x)/ 0 , A0̄ (x) = ⌫(x)

µ̄
0 0̄

'9V+rj­/[ Banerjee et al.(2012), Jensen et al.(2012) , Haehl et al. (2015), MH(2017)]◆ 変分公式 in “hydrostatic gauge”
hT̂

µ⌫

2
LG
(x)it̄ = p

証明.
@t̄

g gµ⌫ (x)1µLGˆ[t̄; ], hJ (x)it̄ = p

g Aµ (x)[t̄; ][ ] の時間微分を考えると，

Z

⇣
⌘
p
⌫
µ
ˆ
[t̄; ] = dd 1 x̄g rµ ⌫ hT̂ µ⌫ iLG
+
(r
⌫
+
F
)h
J
it̄
µ
⌫µ
t̄
✓
◆
Z
1
d 1 p
⌫
⌫
ˆµ it̄
= d x̄
g(rµ ⌫ + r⌫ µ )hT̂ µ⌫ iLG
+
(
r
A
+
A
r
)h
J
⌫
µ
⌫
µ
t̄
2
✓
◆
Z
1
d 1 p
ˆµ it̄
= d x̄
g£ gµ⌫ hT̂ µ⌫ iLG
+
£
A
h
J
µ
t̄
2

一方で，tµ =@t̄ [t̄; ] =Zµだったので，左辺は次のようにも表すことができる．
✓

dd 1 x̄ £ gµ⌫gµ⌫+ £ AµAµ◆上の2通りの表式を比較することで，変分公式が得られる．

log Z ·ªvg¢µÔ'V+rj­­[x­3o有限温度の場の量子論 = 松原形式

大局熱平衡

[Matsubara, 1955]

サイズ

T = const.

平らな”時空”上

d

経路積分

の

の場の理論

x
ギブス分布: ˆG =e

(Ĥ µN̂ )Z=e(Ĥ µN̂ )[ , ]◆ 熱力学関数 with Euclidean作用[ , ] = log Tr e

(Ĥ µN̂ )= log( )=± (0)

= log

d

± |e

D e+SE [ ] , SE [ ] =(Ĥ µN̂ )d0|d3 x LE ( , µ )'9V+rj­­
局所熱平衡 { (x), v(x)}

経路積分

x

局所熱平衡系の場の量子論

事例研究：スカラー場
g µ̄¯
2

L=

µ̄

T̂

[

µ

µ

V( )
2

µ

S
=
g gµ

µˆ

3

µ̄

] = log Tr exp
= log

S[ ,

¯

0

]=

d

d3 x̄

D exp (SE [ ,ge u0̄

0
0

=

d
0

d3 x̄

d x̄

g̃

e 2
˙ )2
(i
2u0̄ u0̄g̃ µ̄¯
2

µ̄

¯

g
µ

ˆ + g µ L( ˆ,ˆ)0̄(x̄)T̂ µ̄ (x̄)]) = log

D exp (SE [ , g̃])e uī ˙(i ) ī
0̄
u u0̄

1
2

V( )

(e (x̄)

ī j̄
u
u
īj̄
+ 0̄
u u0̄

ī

(x̄)/ 0 )

j̄

V( )

V_v¬²µV!?­w|
Z
D exp (SE [ , ; g̃])[t̄; ] = log

Thermal metricg̃µ̄¯ =
(e

e2
e uī

(x̄)

Inverse thermal metrice uj̄g̃ µ̄¯ =īj̄(x̄)/ 0 )e 2
u0̄ u0̄ī
e u
u0̄ u0̄e uj̄
u0̄ u0̄
ī j̄
uu
īj̄
+ 0̄
u u0̄

◆ 得られた結果の意味するところ[t̄; ] は「曲がった時空」中の経路積分で記述される
2

ds̃ =

2

ī 2

e (dt̃ + aī dx ) +

(aī

e

uī ,

īj̄

0
ī
j̄
īj̄ dx dxīj̄ + uī uj̄ ,

dt̃ =

id )

'V+r©'9V+r­O{
有限温度の場の量子論 = 松原形式

大局熱平衡

[Matsubara, 1955]

サイズ

T = const.

平らな”時空”上

d

経路積分

の

の場の理論

x
局所熱平衡系の場の量子論

局所熱平衡 { (x), v(x)}[Hayata, Hidaka, MH, Noumi (2015), MH (2017)]

“線素”が
2

経路積分

d

(x)

2

ds̃ = ds̃ ( , v)
で与えられる“曲がった時空”上の場の量子論x⌘ log Z を構築する2つの方法<latexit sha1_base64="ki7zn0mtd4wRwCRXRAtluSG1S5E=">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</latexit>

1. 一般座標不変性とゲージ不変性を使う！

{- Ψは {g̃µ⌫ , Ãµ } で表されるd<latexit sha1_base64="WDRXOkHYFfPr8LlRksPGZDoBq1M=">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</latexit>(x)- Ψは一般座標変換不変+ゲージ不変
ΨのBuilding blocks：=<latexit sha1_base64="OnRpdRoxQy6wAm8H/0xYPs2XjGQ=">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</latexit>

I

ds̃,

µ=

I

x
Ã, R̃, F̃µ⌫<latexit sha1_base64="dCRw/yRkgUVO6h5EaiG5FKnFSzQ=">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</latexit>

2. 虚時間形式に起因する時空対称性を使う！

{- Ψは空間座標の貼り替えに対して不変- ΨはKaluza-Kleinゲージ変換に対して不変<latexit sha1_base64="ki7zn0mtd4wRwCRXRAtluSG1S5E=">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</latexit>⌘ log Z はこれらの対称性をrespectする！！

[cf. Hydrostatic partition function method Banerjee et al.(2012), Jensen et al.(2012)]

⌘ log Z を構築する2つの方法<latexit sha1_base64="ki7zn0mtd4wRwCRXRAtluSG1S5E=">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</latexit>

1. 一般座標不変性とゲージ不変性を使う！

{- Ψは {g̃µ⌫ , Ãµ } で表されるd<latexit sha1_base64="WDRXOkHYFfPr8LlRksPGZDoBq1M=">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</latexit>(x)- Ψは一般座標変換不変+ゲージ不変
ΨのBuilding blocks：=<latexit sha1_base64="OnRpdRoxQy6wAm8H/0xYPs2XjGQ=">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</latexit>

I

ds̃,

µ=

I

x
Ã, R̃, F̃µ⌫<latexit sha1_base64="dCRw/yRkgUVO6h5EaiG5FKnFSzQ=">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</latexit>

2. 虚時間形式に起因する時空対称性を使う！

{- Ψは空間座標の貼り替えに対して不変- ΨはKaluza-Kleinゲージ変換に対して不変<latexit sha1_base64="ki7zn0mtd4wRwCRXRAtluSG1S5E=">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</latexit>⌘ log Z はこれらの対称性をrespectする！！

[cf. Hydrostatic partition function method Banerjee et al.(2012), Jensen et al.(2012)]

“時空”の対称性 2: Kaluza-Klein対称性
2

ds =

2

ī 2

e (dt̃ + aī dx̄ ) +ī

īj̄ dx̄ dx̄

j̄(dt̃ =

パラメータ

id⌧ )

は虚時間

に

依存しない！
Kaluza-Kleinゲージ変換：(x)d[ ] = log(fīj̄Zx
D ¯D eī aj̄

S[ , ¯,ẽ]j̄ aī )t̃t̃ + (x̄)
x̄x̄aī (x̄)aī (x̄)ī

f īj̄ fīj̄ , · · ·aī , aī aī , · · ·(x̄)

'9V+rj­­

x

局所熱平衡系の場の量子論

局所熱平衡 { (x), v(x)}[Hayata, Hidaka, MH, Noumi (2015), MH (2017)]

“線素”が
経路積分

[t̄; ]①②

log Tr expddds̃2 = ds̃2 ( , v)

(x)

t̄

µ

で与えられる“曲がった時空”上の場の量子論x(x)T̂ µ (x) + (x)Jˆ (x)[ ] は局所熱平衡状態の生成汎関数： hT̂

µ⌫

(x)i

LG

2
=p

g gµ⌫ (x)[ ][ ] は次のような「曲がった時空」中の経路積分公式で与えられる

ds̃2 =e2 (dt̃ + aī dxī )2 +

0
ī
j̄
dx
dx
īj̄

対称性：空間座標の張り替え + Kaluza-Kleinゲージ対称性

log Z ·ªvg¢µÔ

u%d6¬§¨Ñ4(Ð

'9V!ÇÀÎºÉË­ÌºÃ◆ マシュー・プランク汎関数[ ] = log

D e

- Building blocks：

S[ ,g̃] =[See also Banerjee et al.(2012), Jensen et al.(2012)]

(0)

(1)

[ ]+
0

[ , @] + O(@ 2 )
1

O(p )

O(p )

= {e , aī , µ, Aī }- 対称性：Spatial diﬀeo，Kaluza-Klein，ゲージ対称性
Aī ：not Kaluza-Klein inv.

µaī

0

- Power counting scheme：fīj̄ ⌘ @ī aj̄Āī ⌘ Aī

1@j̄ aī = O(p )= O(p )

2f f = O(p )Ψ(0)：Order

0

O(p )

◆ マシュー・プランク汎関数[ ] = log

D e

S[ ,g̃] =- Building blocks：
(0)

(0)

[ ]+
0

(1)

[ , @] + O(@ 2 )
1

O(p )

O(p )

= {e , aī , µ, Aī }
0

[ ]=

d

3

d x̄

0

e p( , µ)

完全流体
µ
LG
µ
µ
T̂ (x) t̄ = (e + p)u u + pµJˆµ (x) LG=nut̄結果のまとめ: Navier-Stokesの導出
与 保存則
µ T̂

µ

µ
ˆ
µ J (x) = 0(x) = 0,

構成方程式 (1st order)
T̂

µ

µ

(x) = (e + p)u u + pgµµ
ˆJ (x) = nu

h

µ

µ

h h

2

µ

µ

物性情報
状態方程式(静的性質)：
輸送係数(動的性質)：

D i eSE [ i ;g̃] =[ ] = log
t̄

= (x)

1

d4 x
0

d3 x̄

p( , µ)d eK̂ ˜p̂(x )e K̂ ˜p̂(x) LGt̄ etc.

KwáT%°eª¢Part 0. 非平衡現象論としての流体力学 [Talk by 藤井]
Part 1. 通常の流体力学{[Ref. Hayata-Hidaka-MH-Noumi, PRD(2015),
MH Ann. Phys.(2017)]- 流体力学 ＝ 実時間発展に関する最適化された摂動展開- 虚時間形式の拡張：局所熱平衡系 ＝「曲がった時空中」の量子論

Part 1.5. 流体力学中の線形揺らぎ [Ref. Mori (1965), Minami-Hidaka (2013)]
森の射影演算子法 (1965) ＝ カレント代数の線形非平衡系への拡張

Part 2. 揺らぐ流体力学[Ref. Crossley-Glorioso-Liu, JHEP(2017)]

KwáT%°eª¢Part 0. 非平衡現象論としての流体力学 [Talk by 藤井]
Part 1. 通常の流体力学{[Ref. Hayata-Hidaka-MH-Noumi, PRD(2015),
MH Ann. Phys.(2017)]- 流体力学 ＝ 実時間発展に関する最適化された摂動展開- 虚時間形式の拡張：局所熱平衡系 ＝「曲がった時空中」の量子論

Part 1.5. 流体力学中の線形揺らぎ [Ref. Mori (1965), Minami-Hidaka (2013)]
森の射影演算子法 (1965) ＝ カレント代数の線形非平衡系への拡張

Part 2. 揺らぐ流体力学[Ref. Crossley-Glorioso-Liu, JHEP(2017)]

Part 2. 出題編
有効場の理論(EFT)とは？

[lj$Y\°WáMUN
Simplified ver.

⇠ 100 GeV ⇠ 10

Full ver.

18

m まで正しいが，標準模型に基づいた記述は難しすぎる…!!

ÀÍØÂÝÇÅÝØ°.L
E⇤GUT ⇠ 10

15

GeV?

⇤EW ⇠ 100 GeV⇤QCD ⇠ 1 GeV⇤ PT ⇠ 100 MeV⇤Nucl ⇠ 10 MeV

MUNÁÎÝ¦ºÀÍØÂÝ
Ñ¿ØÔ°3\}
^q®
Á½×Ø>}
^q®…

⇤Atom ⇠ 10 eV…

ÀÍØÂÝ°Y\¯°´Sc¥·ÞÀÍØÂÝF\}EFT¬±ã◆ h±?°ãRayleigh@

自由度： 原子
⇣
†
LLO
=
i@t
e↵
散乱断面積

, 光子 Aµ ＋ 対称性：U(1)対称性，ゲージ対称性
p2 ⌘+ a30 † cE E 2 + cB B 22m⇠ a60 ! 4 より，青色の散乱が大！

◆ 有効場の理論の基本思想
p ⌧ ⇤U V という「低エネルギー」領域を考える

E

⇤U V
p

興味のある
エネルギー領域{

1. 問題になる自由度と対称性を定め，
2. p/⇤U V ⌧ 1 に基づいて，微分展開として理論を構築する！

凝縮系物理¯¢ºáT◆ 超流動の低エネルギー有効理論

保存量

NG mode

基礎理論/モデルÀÍØÂÝF\}

Lennard-Jones Liquid

T°F\}

Gross-Pietaevski(二流体モデル)

長波長で残る(低エネルギーの)自由度のみ
残してあとは積分する！

https://en.wikipedia.org/wiki/Superfluidity#/
media/File:Liquid̲helium̲Rollin̲film.jpg◆ 長波長で残る自由度たち ( = ゼロモード )保存量

NG modeゲージ場(磁場)QCD¯¢ºáÁ½×Ø>}◆ QCDの低エネルギー有効理論

NG mode

基礎理論/モデル

低エネルギー有効理論

量子色力学(QCD)

Pionの有効理論

NJL model(Chiral Lagrangian)

長波長で残る(低エネルギーの)自由度のみ
残してあとは積分する！

◆ 長波長で残る自由度たち ( = ゼロモード )保存量

NG modeゲージ場(磁場)ÀÍØÂÝF\}dh°ÙÆÐ
Step 1.p«º°%bs_1¬*f7¼(¦ºPion: ⇠ = ei⇡+ Chiral symmetry breaking

Step 2.Power counting scheme ¼Q(¦º0
2⇠ = O(p ), @µ = O(p), ⌘ 2BM = O(p )Le↵
Step 3.6-¼^«ß{¤»ºtb®¼D
¦
2
2
f
f
(2)
⇡
⇡
† µ
L PT =
tr @µ U @ U +
tr
4
4

†

U +

†

U

Le↵
Step 4. ¯³»ºÀÍØÂÝ
A¼matching¯·¹(µºf⇡ = 130 MeV, m⇡ = 140 MeV, · · ·Goal. @</®­°Y\¼yi¦ºÞpDÒR!­&
ミクロ
場の量子論
(QCD/QED/…)

？

マクロ
流体力学

T%âÀÍØÂÝF\}◆ 流体力学の場合

保存量

ミクロ：基礎理論

マクロ：有効理論

Any theory(QCD, NJL, Scaler …)

相対論的流体力学(揺らぐ流体)長波長で残る(低エネルギーの)自由度のみ
残してあとは積分する！

not ゼロ温度, not 平衡系, but with 散逸！

Schwinger-Keldysh形式！

Part 2. 解答編

非平衡系の有効場の理論[参考文献. Crossley-Glorioso-Liu, JHEP(2017)]

Non-equilibrium
E ffective Field Theory in a Nutshell
1. Basic formulation
2. Application to fluctuating hydroNon-equilibrium
E ffective Field Theory in a Nutshell
1. Basic formulation
2. Application to fluctuating hydroSchwinger-Keldysh42in a Nutshell½ÍS.­­xqµW[Òe;*

Schwinger-Keldysh1/
qµW[ÒI2

X

hout|Ô|iniinS ↵ = h |Ŝ|↵i

Z[J] = hvac|ÛJ |vaci
J(t)

h'i
ˆ = Tr (⇢ˆ0 ')ˆlzCTP)\8P?

\8P?

ti

hin|Ô|ini

Z[J1 , J2 ] = Tr

Re t†
⇢ˆ0 ÛJ2 ÛJ1

Im t

ti

tf

⇣

J1 (t)⇢0J2 (t)tf
Re t⌘

CTP]9PA°7
◆ CTP\8P?
⇣⌘†Z[J1 , J2 ] = Tr ⇢ˆ0 ÛJ2 ÛJ1

Im t

ti

J1 (t)

⇢0

tf
Re t

J2 (t)◆ CTP生成汎関数の基本的性質① Doubled symmetry：Z[j1 , j2 ] = Z[j1 +② Unitarity condition：

Z[j1 = j, j2 = j] = 1

1

j 1 , j2 +

2

j2 ]

③ Reflectivity condition： Z[j1 , j2 ] = Z[j2 , j1 ]⇤④ KMS conditionG)1Vi$X0uZ8°¬ áZ[j1 (x), j2 (x)] = Z[✏P Tj j1 ( x), ✏P Tj j2 ( x

i )]

F
Z[j1 , j2 ] ==ZZ^°ÙÆÐw|

d'd'F

Z

D 1D 2eD'1 D'2 eiSeff [

i(S['1 ;j1 ] S['2 ,j2 ])h'|⇢ˆ0 |'F i

1 , 2 ;j1 ,j2 ]◆ 有効作用 Se↵ [h1 , h2 ] の性質②’ Unitarity condition： Se↵ [

1 =

,

2 =

]=0

③’ Reflectivity + Positivity condition：Se↵ [ 1 ,

2]

⇤

=

Se↵ [ 2 ,④’ Local KMS condition： Se↵ [

1 ],
1,

Im Se↵ [ 1 ,2]0

2 ; j1 , j2 ] = Ss [j1 , j2 ] + · · ·Ss [j1 (x), j2 (x)] = Ss [j1 ( x), j2 ( xi )],:KMSI¬bKMSI
Z[j1 (x), j2 (x)] = Z[✏P Tj j1 ( x), ✏P Tj j2 ( x④’ Local KMS condition： Se↵ [

1,

i )]

2 ; j1 , j2 ] = Ss [j1 , j2 ] + · · ·Ss [j1 (x), j2 (x)] = Ss [j1 ( x), j2 ( xi )]④’’ Dynamical KMS condition：Se↵ [ R , jR ;
~!0 で

(

A , jA ] = Se↵ [ ˜R , j̃R ; ˜A , j̃A ]j̃R (x) = jR ( x), j̃A (x) = jA ( x) + i @ t jR ( x),˜R (x) = R ( x), ˜A (x) = A ( x) + i @ t R ( x),ÀÍØÂÝF\}dh°ÙÆÐ
Step 1.p«º°%bs_1¬*f7¼(¦ºPion: ⇠ = ei⇡+ Chiral symmetry breaking

Step 2.Power counting scheme ¼Q(¦º0
2⇠ = O(p ), @µ = O(p), ⌘ 2BM = O(p )Le↵
Step 3.6-¼^«ß{¤»ºtb®¼D
¦
2
2
f
f
(2)
⇡
⇡
† µ
L PT =
tr @µ U @ U +
tr
4
4

†

U +

†

U

Le↵
Step 4. ¯³»ºÀÍØÂÝ
A¼matching¯·¹(µºf⇡ = 130 MeV, m⇡ = 140 MeV, · · ·Goal. @</®­°Y\¼yi¦ºÞÀÍØÂÝ0uF\}°ÙÆÐ
Step 1.p«º°%bs_1¬*f7¼(¦ºStep 2.Power counting scheme¼Q(¦º

Le↵
Step 3.6-¼^«ß{¤»ºtb®¼D
¦Le↵
Step 4. ¯³»ºÀÍØÂÝ
A¼matching¯·¹(µºGoal. Y\

°Ca-®­¼yi¦ºÞNon-equilibrium
E ffective Field Theory in a Nutshell
1. Basic formulation
2. Application to fluctuating hydroStep 1-0. NG mode °¬ °5o◆ 超流動/超伝導(ゼロ温度)の復習
Z[Aµ ] =Z

D eiS[ ;Aµ ] =Z

D| |D⇡ eiS[| |,⇡;Aµ ] =Se↵ [⇡; Aµ ] = Se↵ [@µ ⇡F2Aµ ] =(@µ ⇡2

ZD⇡ eiSeff [⇡;Aµ ]Aµ ) 2 + · · ·◆ ゲージ変換による方法
Z

Z[Aµ ] = Z[Aµ + @µ ✓] / D✓Z[Aµ + @µ ✓]ZZ2
2i F2 (@µ ⇡ @µ ✓ Aµ )2 +···i F2 (@µ ✓ Aµ )2 +···
= D✓D⇡ e= D✓D⇡ eZF2i 2 (@µ ✓ Aµ )2 +···/ D✓ e

教訓：有効ラグランジアンはゲージ自由度で記述！

Step 1-0.*f7"=°Ï×ÕÝÉã◆ CTP生成汎関数の展開：

iW [j1 , j2 ] = log Z[j1 , j2 ] =◆ Key pointZ

A

jA G + jA G

1. 感受率は 有限の値を持つ : GABABjB + · · ·6= 0

2. CTP生成汎関数はゲージ不変 :W [j1 , j2 ] = W [j1 +1

j 1 , j2 +

2

j2 ]

iW [j1 , j2 ] は非局所 ← ゼロモードを積分したため！

ゼロモード(流体モード)をどう取り扱うべきか？

Step 1-1. Goldstone-like®!
◆ 流体力学に関わるゲージ変換 = 一般座標変換
無限小変換の場合：
0gµ⌫ ! gµ⌫
= gµ⌫ + X gµ⌫ ,cf.

J µ $ AµT µ⌫ $ gµ⌫

X gµ⌫ = rµ X⌫ + r⌫ Xµ

流体のラグランジュ記述を取ると，X = 流体粒子の位置！

Step 1-1. 流体力学に付随する対称性◆ 流体の対称性 (＝“流体の定義”)
流体のEFTに対して次の2つの対称性を要請する:(1) 流体粒子ごとの時間張り替え : t̄ ! t̄0 = f (t̄, x̄),(2) 流体粒子のラベルの張り替え：t̄ ! t̄,ix̄ ! x̄

x̄i ! x̄0i (x̄i )i

Step 2. Power counting schemeµ0@0 Xr = O(p ),@µ = O(p),µµXa ⌧ Xr

◆ A場に関する展開
Se↵
A
A
Se↵ [QR , g ] = Se↵ [g = 0] + A
gµ⌫

A
gµ⌫
+
g A =02

Se↵
Ag⇢A gµ⌫g A =0A
A A= hTRµ⌫ [Q]igµ⌫+ h TRµ⌫ [Q] TR⇢ [Q]igµ⌫g⇢ + · · ·◆ 微分展開µ⌫µ⌫hTRµ⌫ [Q]i = T(0)[Q] + T(1)[Q, @Q] + · · · : 構成方程式

h TRµ⌫ [Q] TR⇢ [Q]i = · · ·: 熱揺らぎの性質

A A
gµ⌫g⇢ + · · ·Step 3-1. Leading-order＝完全流体◆ A場に関するLeading-orderの展開

withZ[j1 , j2 ] ==Z

Z

DXrµ DXa⌫ e
DXrµ

i

R

µ⌫
dd xT̂hydro@µ Xa⌫µ⌫@µ T̂hydro: 完全流体の方程式！

Step 3-2. NLO＝揺らぐ流体力学◆ A場に関するNext-Leading-orderの展開

揺らぐ流体力学とは？
[See e.g. Kovtun et al., arXiv:1405.3967]◆ 保存則:
@µ T

µ⌫

= 0,

µ

@µ J = 0◆ 構成方程式:◆ ノイズの性質 = 揺動散逸関係
with確率過程に対する経路積分形式◆ Martin-Siggia-Rose-Jansen-DeDominics(MSRJD)形式
Step 1: ランジュバン方程式を書く： EoM = (q̇ + q
Step 2: ノイズ)=0に対する１つの解を次式で表す：q⇠ (t1 )q⇠ (t2 ) · · · =Z

Dq (EoM )q(t1 )q(t2 ) · · ·Step 3: ノイズ平均を次のように表す：
··· =

D e

W[ ]

(· · · ),with

Step 4: デルタ関数の積分表示： (EoM ) =Step 5: ノイズ

1
W[ ] =
2C
Dpei

dt (t )2

dtp(t)(EoM )(t) についての積分を行う！

確率過程に関するMSRJDの有効作用(経路積分形式)！

MSRJDの有効作用 for 揺らぐ流体[See e.g. Kovtun et al., arXiv:1405.3967]◆ 揺らぐ流体に関するMSRJDの有効作用

NLOの有効作用として書き下したものに他ならない！

Step 3-2. NLO: 揺らぐ流体力学◆ A場に関するNext-Leading-orderの展開◆ 揺動散逸関係は局所KMS条件から従う！

Step 4. 基礎理論とのMatching◆ A場に関するNext-Leading-orderの展開◆ Matching condition = グリーン・久保公式！

³¬µáT%°eª¢Part 0. 非平衡現象論としての流体力学 [Talk by 藤井]
Part 1. 通常の流体力学{[Ref. Hayata-Hidaka-MH-Noumi, PRD(2015),
MH Ann. Phys.(2017)]- 流体力学 ＝ 実時間発展に関する最適化された摂動展開- 虚時間形式の拡張：局所熱平衡系 ＝「曲がった時空中」の量子論

Part 1.5. 流体力学中の線形揺らぎ [Ref. Mori (1965), Minami-Hidaka (2013)]
森の射影演算子法 (1965) ＝ カレント代数の線形非平衡系への拡張

Part 2. 揺らぐ流体力学{[Ref. Crossley-Glorioso-Liu, JHEP(2017)]- 実時間形式 ＝ 非平衡系の量子論(Schwinger-Keldysh形式)の性質- 揺らぐ散逸流体力学の有効ラグランジアン・有効場の理論(QCDにおけるカイラル摂動論)

今後の課題？
◆ 局所熱力学汎関数の格子シミュレーション◆ 微分展開に関する非摂動効果とResurgence構造?
◆ 非平衡量子開放系の有効場の理論(EFT)◆ ブラックホール熱力学へのEFT的アプローチ

次回(?)※ 画像は開発中のものです．