Critical fluctuations in models with van der Waalsinteractions∗

V. Vovchenkoa,b,c, D.V. Anchishkinc,d, M.I. Gorensteina,d,R.V. Poberezhnyukd, H. Stoeckera,b,e

aFrankfurt Institute for Advanced Studies, Goethe Universitat Frankfurt,D-60438 Frankfurt am Main, Germany

bInstitut fur Theoretische Physik, Goethe Universitat Frankfurt, D-60438Frankfurt am Main, Germany

cTaras Shevchenko National University of Kiev, 03022 Kiev, UkrainedBogolyubov Institute for Theoretical Physics, 03680 Kiev, Ukraine

eGSI Helmholtzzentrum fur Schwerionenforschung GmbH, D-64291 Darmstadt,

Germany

Particle number fluctuations are considered within the van der Waals(VDW) equation, which contains both attractive (mean-field) and repulsive(eigenvolume) interactions. The VDW equation is used to calculate thescaled variance of particle number fluctuations in generic Boltzmann VDWsystem and in nuclear matter. The strongly intensive measures ∆[E∗, N ]and Σ[E∗, N ] of the particle number and excitation energy fluctuationsare also considered, and, similarly, show singular behavior near the criticalpoint. The ∆[E∗, N ] measure is shown to attain both positive and negativevalues in the vicinity of critical point. Based on universality argument,similar behavior is expected to occur in the vicinity of the QCD criticalpoint.

PACS numbers: 21.65.-f, 21.65.Mn, 05.70.Jk

1. Introduction

The study of event-by-event fluctuations in high-energy nucleus-nucleuscollisions is presently an essential task in the search of phase transitions andcritical point (CP) in QCD (see, e.g., Refs. [1, 2] and references therein).The van der Waals (VDW) equation of state is a simple and popular modelto describe repulsive and attractive interactions between particles [3] and

∗ Presented at CPOD 2016, Wroclaw, Poland, May 30 – June 4, 2016

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2 CPOD16˙Vovchenko printed on September 12, 2018

is suitable for studying the qualitative features of the critical behavior offluctuations in the vicinity of the CP. Two extensions of the classical VDWequation are considered: (i) the grand canonical ensemble (GCE) formula-tion, and (ii) the inclusion of the quantum statistics.

2. Extensions of the Van der Waals equation

In the classical VDW equation the pressure is expressed as a function oftemperature T , volume V , and particle number N as

p(T, V,N) =NT

V − bN− a

N2

V 2, (1)

where a and b are, respectively, the attraction and repulsion VDW param-eters. Eq. (1) determines the pressure in the canonical ensemble (CE).

However, the CE pressure does not give a complete thermodynamicaldescription of the system. This is because parameters V , T , and N are notthe natural variables for the pressure. The thermodynamical potential inthe CE is the free energy F (T, V,N). Recalling the thermodynamic identityp = −(∂F/∂V )T,N , and by requiring that for a = 0 and b = 0 the systemreduces to the ideal gas, one obtains the free energy of the VDW gas

F (T, V,N) = Fid(T, V − bN,N) − aN2

V, (2)

where Fid is the free energy of the corresponding ideal gas. The Fid(T, V,N)contains additional information which is missing in the standard VDW equa-tion (1) such as particle’s mass m and degeneracy d. The F (T, V,N) func-tion itself contains complete thermodynamic information about the system.

2.1. Quantum statistics

Relation (2) is rigorously formulated for the case of Boltzmann statistics.In the following we postulate that it is also valid for quantum statistics, byassuming that Fid(T, V,N) is the free energy of the corresponding idealquantum gas. In such a way we obtain the VDW equation which includesthe effects of quantum statistics. The CE pressure reads

p(T, V,N) = −(∂F/∂V )T,N = pid(T, V − bN,N)− a N2

V 2, (3)

where pid(T, V,N) is the CE pressure of the ideal quantum gas. For Boltz-mann statistics pid(T, V − bN,N) = NT/(V − bN), and in this case Eq. (3)coincides with Eq. (1).

The total entropy S = −(∂F/∂T )V,N of the VDW gas reads S(T, V,N) =Sid(T, V − bN,N). One can easily see that entropy is always positive andthat S → 0 with T → 0, in accordance with the 3rd law of thermodynamics.

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2.2. Grand canonical ensemble

In the GCE the pressure p(T, µ) given as a function of its natural vari-ables T and µ contains complete information about the system. In orderto transform the VDW equation to GCE we calculate the CE chemicalpotential as µ(T, V,N) = (∂F/∂N)T,V , and, denoting µ ≡ µ(T, V,N) andN/V ≡ n, solve resulting equation to obtain GCE particle density n(T, µ) [4]

n(T, µ) =nid(T, µ∗)

1 + b nid(T, µ∗), (4)

where µ∗ = µ−b p−a b n2+2 an. The GCE pressure p(T, µ) reads p(T, µ) =pid(T, µ∗) − a [n(T, µ)]2. At any given T and µ equations for p(T, µ) andn(T, µ) should be solved simultaneously. It is possible that there is morethan a single solution at a given T and µ. In that case the solution with thelargest pressure should be chosen, in accordance with the Gibbs criteria.

3. Calculation results

3.1. Particle number fluctuations

The scaled variance of the total particle number fluctuations in the GCEcan be calculated as [4]

ω[N ] =T

n

(∂n

∂µ

)T

= ωid(T, µ∗)

[1

(1− bn)2− 2an

Tωid(T, µ∗)

]−1

, (5)

where ωid corresponds to the scaled variance of particle number fluctuationsin the ideal gas (in the Boltzmann approximation it is reduced to ωid = 1).

In our calculations we consider two representative VDW systems: (1)the generic VDW system obeying Boltzmann statistics and (2) the nuclearmatter described as VDW equation with Fermi statistics for nucleons. Inthe first case all quantities of interest do not explicitly depend on VDWparameters a and b. For the second case, following Ref. [4], we take a =329 MeV fm3 and b = 3.42 fm3 for nucleons. Results of the calculationsof ω[N ] for the Boltzmann VDW gas and for nuclear matter are shown inFig. 1. The ω[N ] has a very similar qualitative behavior in both cases: itis close unity at small densities in the gaseous phase, close to zero at largedensities in the liquid phase, and diverges at the CP.

3.2. Strongly intensive quantities

The results for ω[N ] demonstrate a strong increase of the particle num-ber fluctuations in a vicinity of the CP. However, the potential observation of

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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

1

1 0

8 8 0 8 9 0 9 0 0 9 1 0 9 2 0 9 3 005

1 01 52 02 53 03 54 0

1 0

2

1 0 . 5

l i q u i d

T (Me

V)

µ ( M e V )

g a s0 . 1

ω[ N ]

0 . 0 10 . 1

1

1 0

Fig. 1. The density plots of scaled variance ω[N ] of particle number fluctuations

on the phase diagram of the classical VDW system (left panel) and of the nuclear

matter (right panel).

these fluctuations can be masked by the fluctuations of the system volume,which cannot be completely avoided in the heavy-ion collision experiments.

The strongly intensive measures of the fluctuations defined in terms oftwo extensive quantities A and B were suggested in Ref. [5], and they areinsensitive to the trivial fluctuations of the system volume. Supposedly,these fluctuation measures also show critical behavior in the vicinity of CP.For example, they are used by the NA61/SHINE collaboration as probesfor the QCD CP [2]. However, there is no proper model calculations whichwould confirm the presence of such critical behavior. Thus, we consider thestrongly intensive measures of the fluctuations of excitation energy E∗ =E−mN N and nucleon number N in VDW model. The general expressionsfor these quantities read

∆[E∗, N ] = C−1∆

[〈N〉ω[E∗]− 〈E∗〉ω[N ]

], (6)

Σ[E∗, N ] = C−1Σ

[〈N〉ω[E∗] + 〈E∗〉ω[N ]− 2

(〈E∗N〉 − 〈E∗〉 〈N〉

)],(7)

where ω[E∗] is scaled variance of the excitation energy fluctuations, andC−1

∆ and C−1Σ are the normalization factors that have been suggested in

the following form: C∆ = CΣ = 〈N〉ω[ε∗] with ω[ε∗] being the scaledvariance of a single-particle excitation (kinetic) energy distribution in theVDW system [5]. The details of the calculations of ∆[E∗, N ] and Σ[E∗, N ]for classical VDW systems are described in Ref. [4]. For Fermi statisticsthe procedure is essentially the same. Results of calculations for the nuclearmatter are shown in Fig. 2. Both the ∆[E∗, N ] and Σ[E∗, N ] diverge atthe CP, and, thus, appear to be suitable probes for the search of criticalbehavior. It is notable that ∆ measure has a richer structure: for instance,it can take negative values in the vicinity of CP.

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8 8 0 8 9 0 9 0 0 9 1 0 9 2 0 9 3 005

1 01 52 02 53 03 54 0

g a s

0 . 5

0 . 50 . 5

0

1- 1

l i q u i d

T (Me

V)

µ ( M e V )

∆ [ E * , N ]

- 5

- 10

1

2

8 8 0 8 9 0 9 0 0 9 1 0 9 2 0 9 3 005

1 01 52 02 53 03 54 0

1

210 . 5

l i q u i d

T (Me

V)

µ ( M e V )

g a s 0 . 1

Σ[ E * , N ]

0 . 1

0 . 51

2

1 0

Fig. 2. The density plots of the strongly intensive measures ∆[E∗, N ] (left panel)

and Σ[E∗, N ] (right panel) on the phase diagram of the nuclear matter.

4. Summary

The critical behavior of different measures of fluctuations in systems withvan der Waals interactions in the vicinity of the CP is demonstrated. Inparticular, model calculations confirm that the strongly intensive measuresof energy and particle number fluctuations are suitable probes of the criticalbehavior. The VDW interactions should also play a significant role in thefull hadron resonance gas, in the regions of phase diagram away from nuclearmatter, for instance where the chemical freeze-out in heavy-ion collisions canbe expected to occur. This has recently been demonstrated for the crossoverregion at µB = 0 in Ref. [7]. Further studies in this direction can shed newlight on the role of VDW interactions in the confined phase of QCD.

REFERENCES

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[2] M. Stephanov, K. Rajagopal, and E. Shuryak, Phys. Rev. Lett. 81, 4816 (1998);Phys. Rev. D 60, 114028 (1999); M. Gazdzicki and P. Seyboth, Acta Phys.Polon. B 47, 1201 (2016).

[3] L. D. Landau and E. M. Lifshitz, Statistical Physics (Oxford: Pergamon) 1975.

[4] V. Vovchenko, D. V. Anchishkin, and M. I. Gorenstein, J. Phys. A48,305001 (2015); Phys. Rev. C 91, 14 (2015); V. Vovchenko, D. V. Anchishkin,M. I. Gorenstein, and R. V. Poberezhnyuk, Phys. Rev. C 92, 054901 (2015).

[5] M. I. Gorenstein, M. Gazdzicki, Phys. Rev. C 84, 014904 (2011); M. Gazdzicki,M. I. Gorenstein, M. Mackowiak-Pawlowska, Phys. Rev. C 88, 024907 (2013).

[6] V. Vovchenko, R. V. Poberezhnyuk, D. V. Anchishkin, and M. I. Gorenstein,J. Phys. A49, 015003 (2016).

[7] V. Vovchenko, M. I. Gorenstein, and H. Stoecker, arXiv:1609.03975 [hep-ph].