Kinetic regime of hydrodynamic
fluctuations in the Bjorken expansion
=!  gBh/05k Derek Teaney, Aleksas Mazeliauskas (Stony Brook)

Td\cDU^bXe>TVd81
Bjorken7lj&Hubble7

τ
Bjorken7i(PE(PELdamping rate = γk2
6(L7* = 1/τ2016/8/22-24)L?<JHL,.'@ae] k*>>1/τ
à Kinetic theory

Kinetic regime of hydrodynamic
fluctuations in the Bjorken expansion
9"=!  gBh/05k Derek Teaney, Aleksas Mazeliauskas (Stony Brook)(PELLA-S:2g3&Ih!
!*δ g(t, k)⋅ δ g (t, k)

Out ofequilibrium

Quasi-equilibrium?
Flow scale

2016/8/22-24k*)L?<JHL,l−1
mfpk

Kinetic regime of hydrodynamic
fluctuations in the Bjorken expansion
=!  gBh/05k Derek Teaney, Aleksas Mazeliauskas (Stony Brook)

Td\cDU^bXe>TVd81
Bjorken7lj&Hubble7

τ
Bjorken7i(PE(PELdamping rate = γk2
6(L7* = 1/τ
2016/8/22-24)L?<JHL,.'@ae] k*>>1/τ
à Kinetic theory

Kinetic regime of hydrodynamic
fluctuations in the Bjorken expansion
9"=!  gBh/05k Derek Teaney, Aleksas Mazeliauskas (Stony Brook)(PELLA-S:2g3&Ih!
!*δ g(t, k)⋅ δ g (t, k)

Out ofequilibrium

Quasi-equilibrium?
Flow scale

2016/8/22-24k*)L?<JHL,l−1mfpk

t moments
nsverse mo-(see Eq. (66a)). This asymptotic form is responsible forthe renormalization of the pressure and shear viscosity.
For wavenumbers of order k⇤ the hydrodynamic fluctuations are not in equilibrium at all, but reach a nonequilibrium steady state at late times. A graph of thisnon-equilibrium
is given in Fig. 2.
=!steadystategBh

Kinetic regime of hydrodynamic
fluctuations in the Bjorken expansion(88)

TheDerek
deviationof hydrodynamicfluctuations (Stonyfrom equi/05k
Teaney,Aleksas Mazeliauskas
Brook)

librium has consequences for the evolution of the system.
stant shift
4#Indeed, the longitudinal pressure ⌧ 2 T ⌘⌘ receives a corescribed atg
rection from the unequilibrated modes
scales •as (PELW[\V_JFQGN0• @A$CIL(PEKORLong-Time TailL+e first termal thermal• %
L  ⌦ 2 ⌘⌘ ↵ h⌧ Tp4 ⌘1.08318=+ghe+pe+p 3 ⌧s (4⇡ ⌘ ⌧ )3/2i(⌘⌧)
81⇡ON+, (91)2e + p 9⌧

c equations;3M`YZeIf
mic fluctua& arXiv:1606.07742 [nucl-th]these equawhere we have repeated Eq. (74a) for convenience. The)L?<JHL,tion to2016/8/22-24
thecorrection to the pressure⇠T /( ⌘ ⌧ )3/2 is of order ⇠T k⇤3 ,t moments
nsverse mo-(see Eq. (66a)). This asymptotic form is responsible forthe renormalization of the pressure and shear viscosity.
For wavenumbers of order k⇤ the hydrodynamic fluctuations are not in equilibrium at all, but reach a nonequilibrium steady state at late times. A graph of thisnon-equilibrium
is given in Fig. 2.
=!steadystategBh

Kinetic regime of hydrodynamic
fluctuations in the Bjorken expansion(88)

TheDerek
deviationof hydrodynamicfluctuations (Stonyfrom equi/05k
Teaney,Aleksas Mazeliauskas
Brook)

librium has consequences for the evolution of the system.
stant shift4#Indeed, the longitudinal pressure ⌧ 2 T ⌘⌘ receives a corescribed atg
rection from the unequilibrated modes
scales •as (PELW[\V_JFQGN0• @A$CIL(PEKORLong-Time TailL+e first term
al thermal• %
L  ⌦ 2 ⌘⌘ ↵ h⌧ Tp4 ⌘1.08318=+ghe+pe+p 3 ⌧s (4⇡ ⌘ ⌧ )3/2i(⌘⌧)
81⇡
ON+, (91)2e + p 9⌧

c equations;3M`YZeIf
mic fluctua& arXiv:1606.07742 [nucl-th]these equawhere we have repeated Eq. (74a) for convenience. The)L?<JHL,tion to2016/8/22-24
thecorrection to the pressure⇠T /( ⌘ ⌧ )3/2 is of order ⇠T k⇤3 ,