Van der Waals equation on a nuclear scale

Volodymyr Vovchenko

Acknowledgements toD. Anchishkin, M. Gorenstein, R. Poberezhnyuk, and H. Stoecker

FIGSS Seminar

FIAS, Frankfurt25 April 2016

HGS-HIReHelmholtz Graduate School for Hadron and Ion Research

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Outline

1 Introduction

2 Generalizations of Van der Waals equationGrand Canonical EnsembleQuantum statistics

3 ApplicationsCritical fluctuationsNuclear matter as VDW gas of nucleonsInteracting pion gas with van der Waals equationVan der Waals interactions in Hadron Resonance Gas

4 Summary

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Strongly interacting matter

Protons and neutrons interact via strong forceNot elementary, composed of quarks, interaction mediated by gluonsTheory: Quantum Chromodynamics (QCD)

Tempe

rature

net-baryon density0

Tc

nuclei

hadron gas

Quark Gluon Plasma

Big bang

neutron stars

nucleon gas

Heavy Ion Collisions

ScalesLength: 1 fm = 10−15 m = 10−5 AEnergy: 100 MeV = 1010 times room T

Where is it relevant?Early universeNeutron starsHeavy-ion collisions (laboratory!)

First principles of QCD are rather established,but direct calculations are problematic

Phenomenological tools are very useful3 / 25

Thermodynamics

Put a lot of particles in a box and wait...

System will reach state of thermodynamic equilibrium,characterized by few macroscopic state variables

Equation of State – relation between different state variables

Ideal gas law

P(T ,V ,N) =N TV

= n T

Good starting point for applying thermodynamic approachSimple mathematical propertiesWith proper input (atoms, molecules, hadrons, ...) can describe wellcertain regions of corresponding phase diagramCannot be used to describe phase transitions (no liquid-vapor)

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Van der Waals equationVan der Waals equation

P(T ,V ,N) =NT

V − bN− a

N2

V 2

Formulated in 1873.Nobel Prize in 1910.

Two ingredients:1) Short-range repulsion: particles are hard spheres,

V → V − bN, b = 44πr3

32) Attractive interactions in mean-field approximation,P → P − a n2

Later derived from statistical physics5 / 25

Van der Waals equation

VDW isotherms show irregular behavior below certain temperature TC

Interpreted as the appearance of phase transitionBelow TC isotherms are corrected by Maxwell’s rule of equal areas

1 2 3 4-0.5

0.0

0.5

1.0

1.5

2.0

T=0.8

T=0.9

T=1

P

v

T=1.1

mixed phase

liquid

Tn

gas

Critical point∂p∂v = 0, ∂2p

∂v2 = 0, v = V/N

pC = a27b2 , nC = 1

3b , TC = 8a27b

Reduced variables

p =p

pC, n =

nnC, T =

TTC

6 / 25

Van der Waals equation

VDW equation is quite successful in describing qualitative features ofliquid-vapour phase transition in classical substances

But can it provide insight on phase transitions in QCD?

7 / 25

Van der Waals equation

VDW equation is quite successful in describing qualitative features ofliquid-vapour phase transition in classical substances

Tempe

rature

net-baryon density0

Tc

nuclei

hadron gas

Quark Gluon Plasma

Big bang

neutron stars

nucleon gas

Heavy Ion Collisions

But can it provide insight on phase transitions in QCD?

7 / 25

Statistical ensembles

VDW equation originally formulated in canonical ensemble

Canonical ensembleSystem of N particles in fixed volume V exchanges energy with largereservoir (heat bath)State variables: T , V , NThermodynamic potential – free energy F (T ,V ,N)

All other quantities determined from F (T ,V ,N)

Grand canonical ensembleSystem of particles in fixed volume V exchanges both energy andparticles with large reservoir (heat bath)State variables: T , V , µN no longer conserved. Chemical potential µ regulates 〈N〉Pressure P(T , µ) as function of T and µ contains complete information

GCE is more natural for systems with variable number of particlesGCE formulation opens possibilities for new applications in nuclear physics

8 / 25

How to transform CE pressure P(T ,n) into GCE pressure P(T , µ)?Calculate µ(T ,V ,N) from standard TD relationsInvert the relation to get N(T ,V , µ) and put it back into P(T ,V ,N)

Consistency due to thermodynamic equivalence of ensembles

Result: transcendental equation for n(T , µ)

NV≡ n(T , µ) =

nid(T , µ∗)

1 + b nid(T , µ∗), µ∗ = µ − b

n T1− b n

+ 2a n

Implicit equation in GCE, in CE it as explicitMay have multiple solutions below TC

Choose one with largest pressure – equivalent to Maxwell rule in CE

Advantages of the GCE formulation1) Hadronic physics applications: number of hadrons usually not conserved.2) CE cannot describe particle number fluctuations. N-fluctuations in asmall (V � V0) subsystem follow GCE results.3) Good starting point to include effects of quantum statistics.

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Scaled variance in VDW equation

New application from GCE formualtion: particle number fluctuations

Scaled variance is an intensive measure of N-fluctuations

σ2

N= ω[N] ≡ 〈N

2〉 − 〈N〉2

〈N〉=

Tn

(∂n∂µ

)T

=Tn

(∂2P∂µ2

)T

In ideal Boltzmann gas fluctuations are Poissonian and ωid [N] = 1.

ω[N] in VDW gas (pure phases)

ω[N] =

[1

(1− bn)2 −2anT

]−1

Repulsive interactions suppress N-fluctuationsAttractive interactions enhance N-fluctuations

N-fluctuations are useful because theyCarry information about finer details of EoS, e.g. phase transitionsMeasurable experimentally

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Phase transition signatures

Imagine we are measuring distribution of particle number

Ideal gas

0 1 2 30

1

2

3

T/Tc

n / n c

n / n C

0 . 0 0

1 . 0 0

2 . 0 0

3 . 0 0

VDW gas

0 1 2 30

1

2

3

m i x e d p h a s e

l i q u i d

T/Tc

n / n c

g a s

n / n C

0 . 0 0

1 . 0 0

2 . 0 0

3 . 0 0

Measuring average 〈N〉 (or average energy) gives no information aboutphase transitions

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Phase transition signatures

What about fluctuations?

Ideal gas

0 1 2 30

1

2

3

T/Tc

n / n c

ω[ N ]

0 . 0 10 . 1 0

1 . 0 0

1 0 . 0 0

VDW gas

0 1 2 30

1

2

3

1 02

1 0 . 5

m i x e d p h a s e

l i q u i d

T/Tc

n / n c

g a s

0 . 1 ω[ N ]

0 . 0 10 . 1 0

1 . 0 0

1 0 . 0 0

Deviations from unity signal interaction effectsFluctuations grow rapidly near critical point

V. Vovchenko et al., J. Phys. A 305001, 48 (2015)

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Skewness

Higher-order (non-gaussian) fluctuations are even more sensitive

Skewness: Sσ =〈(∆N)3〉σ2 = ω[N] +

Tω[N]

(∂ω[N]

∂µ

)T

asymmetry

0 1 2 30

1

2

3

metastablemetas

table

10

1

0

-1

unstable

liquid

T

n

gas-10

S

-40.00

-10.00

-1.000.001.00

10.00

40.00

Skewness isPositive (right-tailed) in gaseous phaseNegative (left-tailed) in liquid phase

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Kurtosis

Kurtosis: κσ2 =〈(∆N)4〉 − 3 〈(∆N)2〉2

σ2 peakedness

0 1 2 30

1

2

3

100100-100 10

1

0-1

metastablemetas

table

10

10 -1

unstable

liquid

T

n

gas

-10

-40.00

-10.00

-1.000.001.00

10.00

40.00

Kurtosis is negative (flat) above critical point (crossover), positive (peaked)elsewhere and very sensitive to the proximity of the critical point

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VDW equation with quantum statistics

Nucleon-nucleon potential:Repulsive core at small distancesAttraction at intermediate distancesSuggestive similarity to VDWinteractionsCould nuclear matter described by VDWequation?

Original VDW equation is for Boltzmann statisticsNucleons are fermions, obey Pauli exclusion principleUnlike for classical fluids, uncertainty principle is important

Requirements for VDW equation with quantum statistics

1) Reduce to ideal quantum gas at a = 0 and b = 02) Reduce to classical VDW when quantum statistics are negligible3) s ≥ 0 and s → 0 as T → 0

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VDW equation with quantum statistics in GCE

Ansatz: Take pressure in the following form

p(T , µ) = pid(T , µ∗)− an2, µ∗ = µ− b p − a b n2 + 2an

where pid(T , µ∗) is pressure of ideal quantum gas.

n(T , µ) =

(∂p∂µ

)T

=nid(T , µ∗)

1 + b nid(T , µ∗)

s(T , µ) =

(∂p∂T

)µ

=sid(T , µ∗)

1 + b nid(T , µ∗)

ε(T , µ) = Ts + µn − p = [εid(T , µ∗)− an] n

This formulation explicitly satisfies requirements 1-3

Algorithm for GCE

1) Solve system of eqs. for p and n at given (T , µ) (there may be multiplesolutions)2) Choose the solution with largest pressure

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Nuclear matter as a VDW gas of nucleons

Nuclear matter is known to have a liquid-gas phase transition atT ≤ 20 MeV and exhibit VDW-like behaviorUsually studied analyzing nuclear fragment distributionHas long history...

Theory:Csernai, Kapusta, Phys. Rept. (1986)Stoecker, Greiner, Phys. Rept. (1986)Serot, Walecka, Adv. Nucl. Phys. (1986)Bondorf, Botvina, Ilinov, Mishustin, Sneppen, Phys. Rept. (1995)

Experiment:Pochodzalla et al., Phys. Rev. Lett. (1995)Natowitz et al., Phys. Rev. Lett. (2002)Karnaukhov et al., Phys. Rev. C (2003)

Our description: Nuclear matter as a system of nucleons (d = 4,m = 938 MeV) described by VDW equation with Fermi statistics. Pions,resonances and nuclear fragments are neglected

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VDW gas of nucleons: zero temperature

How to fix a and b? For classical fluid usually tied to CP location.Different approach: Reproduce saturation density and binding energy

From EB ∼= −16 MeV and n = n0 ∼= 0.16 fm−3 at T = p = 0 we obtain:

a ∼= 329 MeV fm3 and b ∼= 3.42 fm3 (r ∼= 0.59 fm)

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28-20

-10

0

10

20

30

40(b)

E B (M

eV)

n (fm-3)

Mixed phase at T = 0 is special:A mix of vacuum (n = 0) and liquidat n = n0

1 0 - 2 1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 8 1 0 91 0 - 1 71 0 - 1 61 0 - 1 51 0 - 1 41 0 - 1 31 0 - 1 21 0 - 1 11 0 - 1 01 0 - 91 0 - 8

A r g o nC O 2

r (fm)

a / b ( e V )

N u c l e a r m a t t e r

W a t e r

VDW eq. now at very different scale!

18 / 25

VDW gas of nucleons: pressure isotherms

CE pressure

p = pid[T , µid

( n1− bn

,T)]− a n2

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.280.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b)

T=15 MeV

T=18 MeV

T=19.7 MeV

P (M

eV/fm

3 )

n (fm-3)

T=21 MeV

5 10 15 20 25 30 35 40 45 50 55 60 65 70-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(a)

T=15 MeV

T=18 MeV

T=19.7 MeV

P (M

eV/fm

3 )v (fm3)

T=21 MeV

Behavior qualitatively same as for Boltzmann caseMixed phase results from Maxwell construction

Critical point at Tc ∼= 19.7 MeV and nc ∼= 0.07 fm−3

Experimental estimate1: Tc = 17.9± 0.4 MeV, nc = 0.06± 0.01 fm−3

1J.B. Elliot, P.T. Lake, L.G. Moretto, L. Phair, Phys. Rev. C 87, 054622 (2013)19 / 25

VDW gas of nucleons: (T , µ) plane

Density in (T , µ) plane2

880 890 900 910 920 9300

5

10

15

20

25

30

35

liquid

T (M

eV)

(MeV)

gas

n (fm-3)

0.00

0.02

0.04

0.07

0.09

0.11

0.13

0.16

0.18

Crossover region at µ < µC∼= 908 MeV is clearly seen

1V. Vovchenko et al., Phys. Rev. C 91, 064314 (2015) 20 / 25

VDW gas of nucleons: (T , µ) plane

Density in (T , µ) plane2

880 890 900 910 920 9300

5

10

15

20

25

30

35

liquid

T (M

eV)

(MeV)

gas

n (fm-3)

0.00

0.02

0.04

0.07

0.09

0.11

0.13

0.16

0.18

Boltzmann CP

Boltzmann: TC = 28.5 MeV. Fermi statistics important at CP1V. Vovchenko et al., Phys. Rev. C 91, 064314 (2015) 20 / 25

VDW gas of nucleons: scaled variance

Scaled variance in quantum VDW:

ω[N] = ωid(T , µ∗)

[1

(1− bn)2 −2anT

ωid(T , µ∗)

]−1

880 890 900 910 920 9300

5

10

15

20

25

30

35

10

2

1 0.5

liquid

T (M

eV)

(MeV)

gas

0.1

0.01

0.10

1.00

10.00

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VDW gas of nucleons: skewness and kurtosis

Skewness

Sσ = ω[N] +Tω[N]

(∂ω[N]

∂µ

)T

880 890 900 910 920 9300

5

10

15

20

25

30

35-1

10

1

0 -1

liquid

T (M

eV)

(MeV)

gas

-10

S

-40.00

-10.00

-1.000.001.00

10.00

40.00

Kurtosis

κσ2 = (Sσ)2 + T(∂[Sσ]

∂µ

)T

880 890 900 910 920 9300

5

10

15

20

25

30

35

-100100

10010

1

10

10-1

liquid

T (M

eV)

(MeV)

gas

-10

-40.00

-10.00

-1.000.001.00

10.00

40.00

V. Vovchenko et al., Phys. Rev. C 92, 054901 (2015)

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Pion gas with van der Waals equationInteracting pion gas as a VDW gas with Bose statistics

VDW parameters: r = 0.3 fm and a/b = 500 MeV

No conserved charges (µ = 0), only temperature dependence

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 00

2

4

6

8

1 0 V a n d e r W a a l s g a s o f p i o n s P i o n s + H a g e d o r n s p e c t r u m i d e a l g a s

C [fm

-3 ]

T [ M e V ]

x

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 00 . 0

0 . 1

0 . 2

0 . 3

0 . 4 V a n d e r W a a l s g a s o f p i o n s P i o n s + H a g e d o r n s p e c t r u m i d e a l g a s

xc2 s

T [ M e V ]

At some T > T0 there are no solutions!Van der Waals attraction leads to emergence of limiting temperatureConsequence of GCE and Bose statisticsSuggestive similarity to Hagedorn mass spectrumHint of phase transition to new state of matter?

R. Poberezhnyuk et al., arXiv:1508.0458523 / 25

VDW interactions in hadron resonance gas

Hadron resonance gas – successful model for low density part of QCD

Usually modeled as non-interacting gas of hadrons and resonances

Add repulsive VDW interactions

Better agreement with lattice

0 200 400 600 800 1000 12000

100

200

300

400

500

200

GeV

Lines: S/B const.

17.3 GeV12.3 GeV

8.8 GeV

7.6 GeV

Circles: rp = 0.5 fm

"Crossterms" EV, rp = 0.5 fm

T (M

eV)

B (MeV)

Contours show valleys of 2 minima

Diamonds: rp = 0.0 fm

6.3 GeV

Big change in fits to exp. data

Eigenvolume interactions had largely been overlooked in past!

V. Vovchenko, H. Stoecker, arXiv:1512.08046

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Summary

1 Classical VDW equation is transformed to GCE and generalized toinclude effects of quantum statistics.

2 Scaled variance, skewness, and kurtosis of particle number fluctuationsare calculated for VDW equation. Role of repulsive ans attractiveinteractions is clarified.

3 VDW equation with Fermi statistics for nucleons is able to describeproperties of symmetric nuclear matter. VDW equation with Bosestatistics for pions shows limiting temperature. Strong effect of VDWinteractions in HRG

4 Fluctuations are very sensitive to the proximity of the critical point.Gaseous phase is characterized by positive skewness while liquid phasecorresponds to negative skewness. The crossover region is clearlycharacterized by negative kurtosis in VDW model.

Thanks for your attention!

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Summary

1 Classical VDW equation is transformed to GCE and generalized toinclude effects of quantum statistics.

2 Scaled variance, skewness, and kurtosis of particle number fluctuationsare calculated for VDW equation. Role of repulsive ans attractiveinteractions is clarified.

3 VDW equation with Fermi statistics for nucleons is able to describeproperties of symmetric nuclear matter. VDW equation with Bosestatistics for pions shows limiting temperature. Strong effect of VDWinteractions in HRG

4 Fluctuations are very sensitive to the proximity of the critical point.Gaseous phase is characterized by positive skewness while liquid phasecorresponds to negative skewness. The crossover region is clearlycharacterized by negative kurtosis in VDW model.

Thanks for your attention!

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