The spinodal instability near a critical point in a droplet

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The spinodal instability near a critical point in a droplet

Maximilian Attems (IGFAE)

arXiv:1603.01254 arXiv:1703.09681 arXiv:1807.05175

arXiv:1809.XXXXX

Collaborators:

Yago Bea (UB), Jorge Casalderrey-Solana (Oxford, UB), David Mateos (UB), Miguel Zilhao (CENTRA)

Benasque COST workshop on collectivity in small systems

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Motivation I

Phase transition:

Vaporization

Gregory-Laflamme Instability:

Black ring pinching off gauge/gravity correspondence:

bridge between physical phenomena in gauge theories and gravity.

Maximilian Attems, UB The spinodal instability near a critical point in a droplet 2/22

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Motivation II

sQGP

uSC dSC CFL 2SC

Critical Point

Quarkyonic Matter Quark-Gluon Plasma

Hadronic Phase

Color Superconductors

?

Temperature T

Baryon Chemical Potential mB Inhom

ogen eous

ScB Liquid-Gas

Nuclear Superfluid CFL-K , Crystalline CSC Meson supercurrent Gluonic phase, Mixed phase

0

QCD phase diagram[Fukushima, Hatsudo 2010]

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Motivation III

Black-hole engineered critical point w QCD lattice match[Critelli, Noronha, Noronha-Hostler, Portillo, Ratti, Rougemont 2017]

●●

CEP

Black Hole Engineering Lattice QCD

Finite-Size Scaling

Unlikely due to transition line curvature

Allowed Region

0 200 400 600 800

0 50 100 150 200

μ_B[MeV]

T[MeV]

Baryon susceptibilityχ₂ extended in the full T −µ_B plane and critical point located atT_CEP = 89 MeV andµ^CEP_B = 724 MeV

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Outline

Excitation and saturation of the spinodal instability Non-conformal General Relativity model Non-conformal thermodynamics

Spinodal instability Inhomogeneous Horizon

Hydrostatic + Hydrodynamic evolution Phase separation

Unseeded spinodal instability Pressure evolution

Preferred final state Phase mergers

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Introduction gauge gravity duality

Quantum gravity ind+ 1 dimension AdS ↔QFT in d dimension

IIB string theory onAdS₅×S₅ ↔ N = 4 Super-Yang-Mills [Maldacena 1998, Witten 1998]

shear visocity over entropy density ratio ^η_s = _4π¹ ≈0.08 [Policastro, Son, Starinets 2001; Kovtun, Son, Starinets 2005]

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Introduction gauge gravity duality

Holographic dictionary relates:

Black hole gµν

⇐⇒

Equilibrium state with temperature

Tµν

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Introduction gauge gravity duality

Use of the duality:

To solve complicated dynamical problems in non-abelian theories.

As a source of new modeling ideas for strongly coupled QGP.

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Non-conformal General Relativity model

Dual field theory: ‘mimics’a deformation of N=4 SYM with a dimension 3 operatorO and Λ as ‘mass’

SGaugeTheory =S_conformal+ Z

d⁴xΛO

First order phase transition at the critical temperature seen in the the multivalued plot of the energy density in function of the temperature

0.24 0.25 0.26 0.27

-4 -3 -2 -1

T/Λ Log10(ℰ/Λ4)

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Non-conformal thermodynamics

Einstein-Hilbert action coupled to a scalar with non-trivial potential (single parameterφM) in five-dimensional bottom-up model:

S = 2 κ²₅

Z d⁵x√

−g 1

4R −1

2(∇φ)²−V(φ)

Holographic renormalization[Bianchi, Freedman, Skenderis 2002]

V(φ) =− 1

12φ⁴_Mφ⁸+ 1

2φ⁴_M∓ 1 3φ²_M

φ⁶−1

3φ³−3

2φ²−3. Small IR modification of the model leads to rich phase structure

0.1 0.2 0.3 0.4 0.5 0.6

0.0 0.2 0.4 0.6 0.8 1.0

T

ξ/η

Bulk/Shear in terms of temperature(ϕ_M=2.3i)

0.0 0.1 0.2 0.3 0.4 0.5

-0.1 0.0 0.1 0.2 0.3

T cs2

cs2against T(ϕM=2.3i)

Deconfined strong non-conformality with 1st order phase transition

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Spinodal instability

Homogeneous periodic box in thermodynamical unstable region:

Energy density versus temperature for the gauge theory:

0.24 0.25 0.26 0.27 10^-4

0.001 0.010 0.100

T/Λ

ℰ/Λ4

0.24 0.25 0.26 -0.05

0.00 0.05 0.10

T/Λ cs2

The dashed red curve is locally unstable, the dotted green curve metastable.

Energy density evolution of black branes afflicted by the Gregory-Laflamme instability:

excited unstable mode growth until non-linear saturation

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Inhomogeneous Horizon

Final entropy density

s_hor s_EoS

-20 -10 0 10 20 0

4 8 12

zΛ

Final entropy density extracted from the area of the horizon and estimated from the equation of state

Evolution of Fourier modes of the local energy density

n=3 n=6

n=9

n=1 n=4 n=2

0 200 400

-4 -7 -10 -13

tΛ Log10|ℰ/Λ4 |

momentum dependent growth rate Γ(k)' |c_s|k− 1

2T 4

3 η s + ζ

s

k².

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Hydrostatic + Hydrodynamic evolution

Hydrodynamics description with transport coefficientsc_L/T,f_L/T: P_L/T^hyd =P_eq(E) +c_L/T(E)(∂_zE)²+f_L/T(E)(∂_z²E)

PL

PT PL

hyd

PT

Peq hyd

0

-8 -4 4 8

0 -2 -4

-6

zΛ

Pressures agree with hydrodynamic prediction for a different state

Pressures predicted by hydro match:

Peq

PL PT

PL hyd

PT hyd

PL hyd(1)

PT hyd(1)

100 200 300 400

2 0 -2 -4 -6

tΛ

Early time behaviour with exponential decay of quasi-normal modes

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Stripped phases I

Bigger periodic box with single excited mode:

tΛ=1000 tΛ=1400 tΛ=1500

-60 -40 -20 0 20 40 60

0.00 0.01 0.02 0.03 0.04 0.05 0.06

zΛ

ϵ/Λ4

Cooled and hot regions are on the respective stable phase[Janik, Jankowski, Soltanpanahi 2017]

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Stripped phases II

Final static state:

tΛ=5450.5

-60 -40 -20 0 20 40 60

10^-4 0.001 0.010 0.100

zΛ

ϵ/Λ4

mode analysis

n₌1 n₌2 n₌3 n₌4 n₌5

0 1000 2000 3000 4000

10-9 10-7 10-5 0.001 0.100

tΛ

Initialn = 1 excited Phase separated final state

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Spinodal instability triggered by numerical noise I

Unexcited initial state:

Minimizing free energy to have only a single blob

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Spinodal instability triggered by numerical noise II

Final static state:

tΛ=5478.5

-40 -20 0 20 40

10^-4 0.001 0.010 0.100

zΛ

ϵ/Λ4

Phase separated final state

Unstable mode growth

n₌1 n₌2 n₌3 n₌4 n₌5

0 1000 2000 3000 4000

10-16 10-12 10-8 10-4 1

tΛ

n= 4 unstable mode pushing four initial peaks

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Pressure evolution

Pressures at midpoint:

PL/Λ⁴ PT/Λ⁴

0 1000 2000 3000 4000 5000 6000

-0.0010 -0.0005 0.0000 0.0005

tΛ

Pressures at the ”wall”:

PL/Λ⁴ PT/Λ⁴

0 1000 2000 3000 4000 5000 6000

-0.0005 0.0000 0.0005

tΛ

Longitudinal merger snapshots:

tΛ=3770 tΛ=4300

-40 -20 0 20 40

-0.0004 -0.0002 0.0000 0.0002 0.0004 0.0006

zΛ PT/Λ4

Final equilibrated state:

PT/Λ⁴ PL/Λ⁴

-40 -20 0 20 40

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005

zΛ

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Preferred final state

Large boxes with different extent and initial conditions:

L=6 L=5 L=4 L=4, noise

-50 0 50

1.×10^-4 5.×10^-4 0.001 0.005 0.010 0.050

zΛ

ϵ/Λ4

L=6 L=5 L=4 L=4, noise

-50 0 50

1.×10^-4 5.×10^-4 0.001 0.005 0.010 0.050

zΛ

ϵ/Λ4

Universal ”domain-wall” type phase separation.

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Phase mergers I

Different phase separated region:

Merger to preferred state.

Snapshot at phase separation:

tΛ=1950

-50 0 50

10^-4 0.001 0.010 0.100

zΛ

ϵ/Λ4

Early time evolution:

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Phase mergers II

Premerger time snapshot: Ongoing evolution:

Separated phases over longer times merge to preferred state

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One more thing: Collisions near a critical point

1st order: 2nd order: crossover:

Different thermodynamics with similar off-equilibrium blob formation

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One more thing: Collisions near a critical point

M¨uller-Israel-Stewart-type hydrodynamics fails to describe the pressure evolution at mid rapidity in the formed blob:

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Well described by the constitutive relations of second-order hydrodynamics that include all spatial second-order gradients.

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Conclusions

First simulation of a holographic spinodal instability Excitation of the Gregory-Laflamme instability Final set of static inhomogeneous black branes

Spinodal instabilities lead to phase separation and mergers The system settles (!?) into a preferred inhomogenous state (!) The final state is also described by hydrodynamics!

New example of theapplicability of hydrodynamics to systems with large gradients in energy densities - even in non-trivial phase structure - both for the time evolution of the spinodal instability and the static final states

More studies are on the way

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The spinodal instability near a critical point in a droplet