EPJ manuscript No.(will be inserted by the editor)

Kinetics of the chiral phase transition in a linear σ model

Christian Wesp1a, Hendrik van Hees1,2b, Alex Meistrenko1c, and Carsten Greiner1d

1 Institut fur theoretische Physik, Goethe-Universitat Frankfurt am Main, Max-von-Laue-Straße 1, 60438 Frankfurt, Germany2 Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, 60438 Frankfurt, Germany

the date of receipt and acceptance should be inserted later

Abstract. We study the dynamics of the chiral phase transition in a linear quark-meson σ model usinga novel approach based on semiclassical wave-particle duality. The quarks are treated as test particlesin a Monte-Carlo simulation of elastic collisions and the coupling to the σ meson, which is treated asa classical field, via a kinetic approach motivated by wave-particle duality. The exchange of energy andmomentum between particles and fields is described in terms of appropriate Gaussian wave packets. It hasbeen demonstrated that energy-momentum conservation and the principle of detailed balance are fulfilled,and that the dynamics leads to the correct equilibrium limit. First schematic studies of the dynamics ofmatter produced in heavy-ion collisions are presented.

PACS. XX.XX.XX No PACS code given

1 Introduction

The exploration of the phase diagram of strongly inter-acting matter [1] is among the prime goals of on-goingheavy-ion research at the Relativistic Heavy Ion collider(Beam-Energy Scan at RHIC) at the Brookhaven NationalLaboratory (BNL) and one of the prime motivations forthe construction of the Facility for Anti-Proton and IonResearch (FAIR) in Darmstadt and the Nuclotron-basedIon Collider Facility (NICA) in Dubna.

The low-energy regime of QCD is governed by approx-imate chiral symmetry in the light-quark sector, which isspontaneously broken in the vacuum via the formation of aquark condensate, 〈qq〉 6= 0. At high temperatures and/ordensities one expects the restoration of chiral symmetry,i.e., a cross-over or phase transition.

One problem for theoretical heavy-ion physics is to findpossible signatures of such phase transitions for the highlydynamical medium created in heavy-ion collisions. Fromlattice-QCD (lQCD) calculations it is known that at van-ishing net-baryon number, i.e., µB = 0, the chiral tran-sition is a rapid cross-over transition at a temperatureof Tpc ' 160 MeV, while for µB 6= 0 lQCD is plaguedby the sign problem. Various effective chiral models likequark-meson σ- or Nambu-Jona-Lasinio (NJL) models (orextensions of these with Polyakov loops to implement glu-onic degrees of freedom) indicate that at µB 6= 0 the phasetransition becomes of first order, and the first-order phase-transition line in the phase diagram ends in a critical

a e-mail: cwesp@th.physik.uni-frankfurt.deb e-mail: hees@th.physik.uni-frankfurt.dec e-mail: meistrenko@th.physik.uni-frankfurt.ded e-mail: carsten.greiner@th.physik.uni-frankfurt.de

point, where the phase transition becomes of second order[2,3,4,5,6,7,8]. Possible indications for this critical pointis the divergence of susceptibilities of conserved quantitieslike the baryon number or electric charge and the corre-sponding grand-canonical fluctuations of these quantities[9,10,11,12,13,14,15].

However, the medium created in a relativistic heavy-ion collision consists of a rapidly expanding and coolingfireball. Thus to find possible experimental signatures forthe different kind of phase transitions or even to localizethe critical point in the QCD phase diagram a dynamicalnon-equilibrium treatment is necessary.

On the one hand, at higher collision energies, as achievedat RHIC and the Large Hadron Collider (LHC) at CERN,the bulk properties of the medium are astonishingly welldescribed using (viscous) hydrodynamics, i.e., the fireballis close to local thermal equilibrium. On the other handto study fluctuations a kinetic approach is more appro-priate. One intermediate level is to use hydrodynamicsand impose fluctuations in the sense of a Langevin treat-ment on top [16,17,18]. Here, however, usually one as-sumes Gaussian, white noise for the fluctuations, and thespecial statistical properties of a medium close to a phasetransition, particularly around the critical point, wherethe phase transition becomes of 2nd order and the corre-lation lengths are expected to extent over the entire sys-tem, a microscopic treatment in terms of a Boltzmann-liketransport equation is desirable.

In this paper we employ a novel kind of method toderive a numerical realization of such a kinetic process,based on a linear quark-meson σ model [5,6,7]. The parti-cle content of the corresponding quantum field theory areσ mesons, pions, and (constituent) quarks. By adjusting

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2 Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model

the coupling constants and evaluating the model at fi-nite temperature and density the different kinds of phasetransitions (cross-over, 1st, and 2nd order) can be realized.E.g., at a given value of g with increasing baryochemicalpotential, µB the order of the transition (as a function ofT ) can be changed from crossover to first order through asecond-order point. Since the σ field is the order param-eter for the chiral phase transition one has to simulateparticles and (mean) fields in a consistent way.

In our approach (Dynamical Solution of a Linear σModel, DSLAM) we treat the σ field as a mean field andthe quarks in a Monte-Carlo test-particle approach to sim-ulate a Vlasov-type of equation with elastic two-body col-lisions for the quarks and anti-quarks.

To also implement the “chemical processes” of σ-mesondecay to a quark-anti-quark pair and the corresponding re-verse process of quark-anti-quark annihilation, which turnout to be crucial for the correct realization of the vari-ous kinds of phase transitions in the limit of thermal andchemical equilibrium, we employ the “wave-particle dual-ity” of “old quantum mechanics”. To this end we realizethe mean field on a spatial grid, which is used to numeri-cally solve for the mean-field equations. The annihilationprocess qq → σ is realized by the test-particle approachemploying the corresponding cross section from the quan-tum field theory. Since the σ field is realized as a meanfield but not in a particle picture, the energy and momen-tum of the qq pair is transferred to the mean field in termsof a relativistic Gaussian wave packet, guaranteeing accu-rate energy-momentum conservation. To realize the backreaction, i.e., the decay σ → qq we use the coarse-grainingapproach, i.e., for each grid cell the energy and momen-tum of the σ field is determined and from this interpretedas a phase-space distribution of σ mesons in local thermalequilibrium employing the equation of state determinedfrom the quantum-field theoretical model in thermal equi-librium on the mean-field level. Then in a Monte-Carlostep, σ mesons are decayed to quark-anti-quark pairs, us-ing the corresponding decay width, consistent with thecross section used for the pair-annihilation process. In thisway again energy-momentum conservation as well as theprinciple of detailed balance for the reactions σ ↔ qq isrealizable with high numerical accuracy.

The remainder of the paper is organized as follows. InSect. 2 we present the DSLAM model for the kinetic sim-ulation in detail (Sect. 2.1) and demonstrate the validityof energy-momentum conservation and the stability of theexpected equilibrium solution (Sect. 2.2) and the correctapproach of a simple off-equilibrium initial state (“thermalquench”) to thermal and chemical equilibration, particu-larly the fulfillment of the principle of detailed balance,in “box calculations” (Sect. 2.3). Finally we simulate anexpanding fireball as a simple model for the kinetics ofthe medium as created in heavy-ion collisions (Sect. 3)followed by brief conclusions and outlook in Sect. 4.

2 Non-equilibrium simulation of the linear σmodel

2.1 Particle-field dynamics

In the following we simulate the dynamics of the chiralphase transition using the linear quark-meson σ modelbased on the chiral symmetry group SU(2)L × SU(2)R,acting separately on the left- and right-handed parts ofthe two-flavor quark field ψ = (u, d) and real-valued scalarfields (σ,π), which transform under the SO(4) represen-tation of the chiral group. The Lagrangian is

L =ψ[i/∂ − g(σ + iγ5π · τ )]ψ

+1

2(∂µσ∂

µσ + ∂µπ · ∂µπ)− U(σ,π).(1)

The meson-field potential includes explicit chiral symme-try breaking and is given by

U(σ,π) =λ2

4(σ2 + π2 − ν2)2 − fπm2

πσ − U0. (2)

With ν2 = f2π − m2

π/λ the minimum of this potentialis given by the vacuum expectation value σ0 = fπ, i.e.,the approximate chiral symmetry is broken to the SU(2)V

isospin symmetry, and the mass of the σ mesons is given

by m(0)σ = 2λ2f2

π +m2π. Choosing λ ' 20 leads to m

(0)σ '

600 MeV. Through the Yukawa couplings of the σ field tothe quark field in (1) the quarks acquire the constituent-quark mass mq = g2σ2

0 , and the variation of g ∈ (3.3, 5.5)leads to a cross-over (g = 3.3), a second-order (g = 3.63),or first-order phase transition at finite temperature andvanishing baryochemical potential, as determined from thegrand-canonical potential in the mean-field approximationfor the mesons,

Ω(T, µ) = U(σ,π) +Ωψψ (3)

with the ideal-gas grand-canonical quark potential

Ωψψ = −dn∫

d3p

(2π)3[T ln(1 + exp(−β(E − µ)))

+ T ln(1 + exp(−β(E + µ)))]

(4)

with E =√p2 + g2(σ2 + π2) and the degeneracy factor of

the quarks dn = 2NfNc = 12. The mean fields in thermalequilibrium are determined by the equilibrium condition∂Ω/∂σ = ∂Ω/∂π = 0. In this paper we only consider thecase 〈π〉 = 0. In Fig. 1 the results for the mean field 〈σ〉 forvanishing baryochemical potential µ = 0 and the result-ing σ-meson mass, mσ = ∂2Ω/∂σ2, are shown. To simu-late non-equilibrium simulations we employ our recentlydeveloped semiclassical Monte-Carlo algorithm DSLAM(Dynamical Simulation of a Linear Sigma Model) [6] toevaluate the dynamics of the σ mean field on the one handand the quarks as particles in a Boltzmann-transport ap-proach on the other hand. The challenge here is that toachieve both thermal and chemical equilibrium besides theusual collision terms for qq and qq elastic scattering one

Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08 0.1

T [G

eV

]

σ [GeV]

CrossoverSecond Order

First OrderParametrization

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

m [

GeV

]

T [GeV]

Crossover (g=3.30)Second Order (g=3.63)

First Order (g=5.50)

Fig. 1: The phase diagram for the linear σ model in mean-field approximation at µB = 0, 〈π〉 = 0.Left: the order parameter 〈σ〉 and the effective σ mass mσ = ∂2Ω/∂σ2.

also has to take “chemical” processes like qq ↔ σ intoaccount. Here, energy-momentum conservation as well asthe principle of detailed balance have to be ensured toguarantee the correct equilibrium limit of the simulation.

Schematically our approach can be summarized in astochastic mean-field and Boltzmann-Vlasov transportequation,

σ + λ(σ2 − ν2)σ − fπm2π + g

⟨ψψ⟩

= I(σ ↔ qq), (5)[∂t +

p

Eq·∇x −∇xEψ(t,x,p) ·∇p)

]fq(t,x,p)

= C(ψψ → ψψ, σ ↔ qq),

(6)

where I and C denote collision terms contributing in astochastic way to the time evolution of the mean field σand the quark phase-space distribution function f , respec-tively.

This scheme is numerically realized on a space-timegrid, and the elastic binary collisions of the quarks aresimulated with the usual stochastic Monte-Carlo methodassuming a constant cross section of σelastic = 15 mb.

The decay process σ → qq and the reverse recombina-tion process qq → σ are defined by the quantum field the-ory as processes involving the “particle picture”, leadingto (perturbative) decay rates and cross sections in termsof the corresponding S-matrix elements. The correspond-ing “interaction processes” are considered as local, quasi-discrete, events in comparison to the involved macroscopicscales of the spatial and temporal changes of the meanfield. Thus, within our mean-field approach to describethe mesons we need to define, how to transfer energy andmomentum from and to the mean field corresponding tothe stochastic and discrete decay and recombination pro-cesses.

For the recombination process, the initial state is al-ready given in terms of the test quarks and anti-quarks,and the cross section is determined by the decay width ofthe σ meson via the Breit-Wigner cross section

σqq→σ(s) =Γ 2

(√s−mσ)2 + (Γ/2)2

(7)

with the decay width

Γ =g2

8πmσ

√1−

4m2q

m2σ

. (8)

Now energy-momentum conservation dictates the kine-matics of the annihilation process, defining the energy andmomentum, ∆E and ∆p, to be transferred to the classicalσ field. This is done by adding a Gaussian wave packet,taking into account the momentum in terms of a Lorentzboosted wave packet at rest. E.g., if the momentum is inx direction one uses

δφ(t,x) = A0 exp

[−γ

2(x− vxt)2 + y2 + z2

2σ2

](9)

with vx = px/E and γ = (1−v2x)−1/2. The width σ is a free

parameter of the model and characterizes an interactionvolume; A0 and v have to be determined by the change ofthe energy and momentum of the field, which is definedvia the usual energy-momentum tensor,

E[σ] =

∫V

d3x

[1

2σ2 +

1

2(∇σ)2 + U(σ)

], (10)

P [σ] =

∫V

d3xσ∇σ. (11)

Then A0 and v are determined such that

∆E = E[σ+ δσ]−E[σ], ∆P = P [σ+ δσ]−P [σ], (12)

where δσ is defined by the Gaussian wave packet (9).For the evaluation of the reverse decay process, σ →

qq, we have to translate the mean-field description intoa particle phase-space distribution function. To achievethis, we use the “coarse-graining” approach, i.e., we mapthe local energy-momentum distribution on the grid tolocal thermal-equilibrium distributions, i.e., we parame-terize the phase-space distribution function of σ particlesby the Maxwell-Juttner distribution,

fσ(x,p) = exp(−u · p

T

)= exp

[− γT

(E − v · p)], (13)

4 Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model

where

v =P [σ]

E[σ], γ =

1√1− v2

, u = γ

(1v

)(14)

denotes the collective flow-velocity field v and the four-vector flow field u of the fluid cell ∆V , respectively. Thetemperature, T , is fixed by demanding

ε =E[σ]

∆V=

∫R3

d2p

(2π)3

√m2σ + p2fσ(x,p). (15)

Hereby, the σ mass and temperature T are determinedself-consistently with the σ mass corresponding to theequilibrium value at this temperature.

Now the phase-space distribution function fσ is used ina Monte-Carlo step to determine a σ meson which decayswithin the time step according to the decay rate (σ width)(8). The corresponding energy and momentum of the de-cayed σ meson is taken out of the σ field by again subtract-ing the appropriate Gaussian wave packet (9) in the sameway as explained above in connection with the quark-anti-quark annihilation process, and the quark and anti-quarkwith their respective momenta, also determined by theMonte-Carlo step, are added to the test-particle sample.

2.2 Test calculations in equilibrium

To validate the above defined algorithm to simulate thekinetics of the linear σ model in our particle-field ap-proach, we have performed some numerical tests by simu-lating a microcanonical ensemble in a finite-size cubic boxemploying periodic boundary conditions. The number oftest particles is Ntest = 3 · 106, the numerical time steps∆t = 2 · 10−3a.u. (a.u.: arbitrary units), the spatial gridfor the field is Nσ = 1283, corresponding to a total volumeV = (6 fm)3. The total simulation time is 300 a.u. corre-sponding to 150000 time steps. The interaction volume ofthe Gaussian parameterization of the wave packets (9) is323 grid points, i.e., approximately 1.5% of the system vol-ume. As an initial condition we have chosen equilibriumconditions at a given temperature.

In Fig. 2 we show the fluctuations of the field andquark-anti-quark energy (left) and the total quark number(right) due to the σ-decay and qq annihilation processes,which show thermal fluctuations around the equilibriummean values, while the total energy is conserved with anumerical accuracy of |∆Etot/ 〈E〉 . 5 · 10−5.

As can be seen in Fig. 3, the energy distribution of thequarks and anti-quarks (left panel) follows the expectedrelativistic Maxwell-Boltzmann distribution

dN

dEd3x=

1

2π2E√E2 −m2

q exp

(−ET

)(16)

with high numerical accuracy, which demonstrates againthe good fulfillment of energy conservation as well as theprinciple of detailed balance for the elastic quark/anti-quark scattering and the “chemical” processes qq ↔ σ.In the right panel we show the spectrum of the kinetic

energy of the mean field, σ2/2. From classical field the-ory one expects a constant average energy of kBT foreach mode, which is indeed the case for small wave num-ber k, i.e., long wave lengths λ = 2π/k. At lower wavelengths the finite interaction volume encoded in the finitewidth of the Gaussian wave packets (9) used to describethe energy-momentum transfer in the interaction betweenquarks/anti-quarks and σ mesons from and to the meanfield, leads to an effective cut-off of 1/σ2

eff = σ2x+σ2

y+σ2z =

3σ2, i.e., the power spectrum is expected to follow [6]

dEσdk

=1

2kBT exp

(−k2σ2

eff/2). (17)

The deviations from this result at very high modes canbe explained by the non-linear potential in the mean-fieldequations of motion.

In Fig. 4 (left) we show the distribution of the scalar-field values around their mean as a function of time. Dueto the quark-antiquark annihilation and σ-decay processesthe mean-field value can drift slowly, while the fluctuationslead to the expected Gaussian distribution around this av-erage. The local fluctuations of the field are increased byqq annihilation and damped by σ decay. On the right-handside the corresponding volume-integrated rates are plot-ted, demonstrating the good fulfillment of the principle ofdetailed balance within our simulation, which guaranteesthe right chemical equilibrium in the long-time limit.

In Fig. 5 we demonstrate the chemical-equilibrationprocess with a contour plot of the scalar field in an x-y-plane cut. Initially we have chosen a uniform field withoutany excitations. At early times (left panel) one can see theexcitations of the field due to qq annihilation in terms ofGaussian blobs around the random interaction regions. Atlater times, when chemical equilibrium is reached (rightpanel), i.e., the particle-annihilation and creation ratesbecome equal, we observe the expected thermal fluctu-ations. Their spatial width is determined by the interac-tion volume governed by the standard deviation, σ, pa-rameterizing the interaction volume in the Gaussian wavepackets (9). The height and strength of the fluctuations inthe equilibrium limit is governed by the temperature T ofthe system in accordance with the above discussed powerspectrum (right panel of Fig. 3).

2.3 Thermal quench

As another test of the field-particle algorithm we initial-ize the kinetic equations in an off-equilibrium “thermalquench” situation in an isotropic periodic box. Initially thequarks are sampled at a temperature Tq = 140 MeV andthe σ field at its equilibrium value for Tσ = 180 MeV. Thecoupling is set to g = 3.3 corresponding to a cross-overchiral transition.The qq-annihilation cross section is de-termined with the corresponding Breit-Wigner cross sec-tion as described above. Additionally a constant elasticqq → qq-scattering cross section, σelastic = 15 mb is em-ployed. We use a number of test particles Ntest = 2 · 106

and a time-step size of ∆t = 0.002 fm/c.

Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model 5

-6

-4

-2

0

2

4

6

8

10

0 1 2 3 4 5Energ

y F

luctu

ati

on · 1

03 [

arb

. unit

s]

Time [arb. units]

Total EnergyEnergy Quarks

Energy Fields

486000

487000

488000

489000

490000

491000

492000

493000

494000

0 50 100 150 200 250

Tota

l Part

icle

Num

ber

Time [arb. units]

Total Particles

Fig. 2: (Color online) Left: Energy fluctuations of the mean field and the quarks, ∆E = E(t)−〈E〉.Due to the σ decay and quark-anti-quark annihilation processes energy is exchanged between themean field and the particles, leading to thermal fluctuations |∆E|/E ∼ 10−2 for the field and|∆E|/E ∼ 10−3 for the quarks and anti-quarks. The total energy is conserved up to numericalfluctuations of |∆Etot/ 〈E〉 . 5 · 10−5. Right: Fluctuations of the total quark number due toσ-decay and qq-annihilation processes.

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

f(E)

Energy [arb. units]

Particle energyAnalytic solution

10-610-510-410-310-210-1100

1 10

Fouri

er

am

plitu

des

Fourier modes k

Spectrum of Ekin

1/2 kb T exp(- k2 e2 / 2)

Fig. 3: (Color online) Left: The energy distribution of the quarks and anti-quarks follow theexpected relativistic Maxwell-Boltzmann distribution fq(E) ∝ exp(−E/T ), demonstrating thenumerical stability of the kinetic equations as well as the validity of the principle of detailedbalance. Right: The spectrum of the kinetic energy density of the mean field, σ2/2. For low k(long wavelengths) the distribution follows the expected equipartion theorem for a classical field,according to which each mode carries a mean energy of 1/2kT , while at higher energies the finiteinteraction volume encoded in the finite width of the Gaussian wave packets leads to an effectivecut-off σ2

eff = σ2x + σ2

y + σ2z = 3σ2.

As shown in Fig. 6a the total quark number, i.e., thenumber of quark-antiquark pairs, drops rapidly in the be-ginning due to the annihilation and creation processesqq ↔ σ, which (according to Fig. 6b) converge after about10 fm/c, fluctuating around the common mean-field value.Thermal equilibrium is roughly reached after about 20 fm/c,leading to a common global temperature of about Teq '144 MeV as demonstrated by Figs. 6c and 6e: at t =25 fm/c the quark distribution is given by a Boltzmanndistribution at this temperature, and the σ field shows aGaussian distribution around its equilibrium mean value.The temperature change of the quarks due to the transferof initial potential field energy by only about 4 MeV issmall, because the field’s initial potential energy is much

smaller as compared to the energy of the quarks. As onecan see in Figs. 6d and 6f, energy between the quarks andthe field is exchanged, while the total energy is conservedwith high accuracy. As can be seen from Fig. 6a the parti-cle number still rises slowly, and the system has not fullyreached equilibrium. Also the mean-field value is not ex-actly at the equilibrium value shown in Fig. 1. The finalapproach to exact chemical equilibrium is very slow sincethe difference in the σ potential is very small in this tem-perature range around the cross-over transition, resultingin a weak driving force. This underlines the importance ofthe “chemical” processes qq ↔ σ for a description of theglobal thermalization of the system to the expected global

6 Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model

0

200

400

600

800

1000

1200

0.0095 0.0105 0.0115 0.0125

Densit

y [

a.u

.]

Scalar Field Value [a.u.]

t=0t=100t=200

Fig. 4: (Color online) Left: The distribution of the scalar field around its equilibrium valueshows the expected Gaussian shape. The mean field can drift slowly with time due to the particle-field interactions. The quark-annihilation processes increase the local fluctuations, while particleproduction damps them by dissipating energy from the field. Right: Volume integrated rates forthe pair-production and pair-annihilation processes, σ ↔ qq, demonstrating the good fulfilment ofthe principle of detailed balance.

Fig. 5: (Color online) Contour plot of the scalar field in an x-y-plane cut. The field is initializedwith a uniform scalar field without excitations. Due to qq annihilation, field fluctuations build up,leading to single Gaussian excitations around the respective random interaction regions at earlytimes (left). The right panel shows the field fluctuations after chemical equilibrium has beenreached, leading to the expected thermal fluctuations of the field with height and strength scalingwith temperature.

equilibrium as described by the phase diagram shown inFig. 1.

In the next example we choose the same parametersas in the previous one but start with a lower Temperatureof Tq = 80 MeV for the quarks. The σ field has beeninitialized at the value 〈σ〉 ' 10 MeV in the chiral restoredphase. Initially the quark number raises slightly due to σ-decay processes as shown in Figs. 7a and 7b while the

mean field also increases moving towards the cross-overtransition to the chiral broken phase. At some point ofthe evolution the σ mass drops below the threshold forthe decay to quark-anti-quark pairs, mσ < 2mq, and thefield becomes undamped and thus keeps oscillating for therest of the time evolution (Fig. 7e). This non-equilibriumstate stays in this oscillating state after about 10 fm/c.The field shows again a Gaussian distribution around its

Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model 7

258000

260000

262000

264000

266000

268000

270000

0 5 10 15 20 25

N*N

test

Time [fm/c]

Total Quark Number

(a) Total number of quarks in the system.

0

5000

10000

15000

20000

25000

30000

0 5 10 15 20 25

Part

icle

Pro

cess R

ate

· N

test [c

/fm

]

Time [fm/c]

Pair productionPair annihilation

(b) Reaction rate for pair-production and pair-annihilation.

0

100

200

300

400

500

600

0.019 0.02 0.021 0.022 0.023 0.024 0.025

Dis

trib

uti

on o

f th

e M

ean F

ield

Sigma [GeV]

t = 25 fm/cGaussian t

(c) Spatial distribution of the sigma mean field 〈σ(x)〉 afterthermalization time t = 25 fm/c.

-0.1

-0.05

0

0.05

0.1

0 5 10 15 20 25

Energ

y F

luctu

ati

on (

E -

Et=

0)

[GeV

]

Time [fm/c]

Energy QuarksEnergy Sigma Field

(d) Deviation from the initial energy for the quarks and theenergy of the field, calculated with E(t)− 〈E(t = 0)〉.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

f(E)

Energy [a.u.]

t=0 fm/ct=25 fm/c

analytic, T=140.0 MeVanalytic, T=143.6 MeV

(e) Distribution of the quarks at the beginning of the systemand after global thermalization.

0.9995

0.9996

0.9997

0.9998

0.9999

1

1.0001

1.0002

1.0003

1.0004

1.0005

0 5 10 15 20 25

Rela

tive E

nerg

y E

/<E>

Time [fm/c]

Total Energy

(f) Relative fluctuations of the system’s total energyE(t)/〈E(t = 0)〉.

Fig. 6: (Color online) Results for a calculation with the quenched-scenario. Particles and fields caninteract and their interaction rates converge after some time, leading the same global temperatureof Teq ' 144 MeV. The temperature of the quarks is given by the Boltzmann-distribution, thetemperature of the field is given by the distribution of the field modes as expected from a powerspectrum S(k) = T/k2 as in a Langevin approach.

average (Fig. 7c), and energy is exchanged with the quarks(Fig. 7d) via the Vlasov terms in Eqs. (5) and (6). In Fig.7f one sees that the spatial distribution of the σ field showsthe corresponding oscillations of the mean value and thestochastic fluctuations which persist from the time whenthe dissipative processes σ ↔ qq cease.

3 Simulations of an expanding fireball

To investigate a situation more close to the medium pro-duced in heavy-ion collisions we initialize the system asa droplet with a Woods-Saxon like spherically symmetric

8 Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model

0

20000

40000

60000

80000

100000

120000

140000

0 5 10 15 20 25

N · N

test

Time [fm/c]

Total Quark Number

(a) Total number of quarks in the system.

0

5000

10000

15000

20000

0 5 10 15 20 25

Part

icle

Pro

cess R

ate

· N

test [c

/fm

]

Time [fm/c]

Pair productionPair annihilation

(b) Reaction rate for pair-production and pair-annihilation.

0

50

100

150

200

250

300

350

400

0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.08

Dis

trib

uti

on o

f th

e M

ean F

ield

Sigma [GeV]

t = 25 fm/cGaussian t

(c) Spatial distribution of the mean σ field 〈σ(x)〉 in the sta-tionary non-equilibrium state at t = 25 fm/c.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25

Energ

y F

luctu

ati

on (

E -

Et=

0)

[GeV

]

Time [fm/c]

Energy QuarksEnergy Sigma Field

(d) Deviation from the initial energy for the quarks and theenergy of the field, calculated with E(t)− 〈E(t = 0)〉.

0

10

20

30

40

50

60

70

80

90

100

110

0 5 10 15 20 25

Mean F

ield

[M

eV

]

Time [fm/c]

Sigma Field

(e) Volume averaged value of the σ-mean field. (f) Spatial distribution of the σ-field at t = 25 fm/c.

Fig. 7: (Color online) Results for a calculation with the quenched-scenario from the initiallyhot phase to the cold and chiral broken phase. Fields and particles can interact above Tc and theglobal shift of the σ-field is damped by particle production. At and below Tc field and particlesdecouple because the σ-particle can not decay anymore. The system becomes undamped and fallsin a coherent oscillation.

temperature distribution

T (x) = T0

[1 + exp

(|x| −R0

α

)]−1

. (18)

For this exploratory study we first work with a small sys-tem, corresponding to R0 = 0.45 fm and surface thicknessα = 0.1 fm. The total system size has been chosen to 5 fm.Also the boundary conditions have to be adapted from the

so far used periodic boundary conditions to ones that al-low for the expansion of the system. For the particles wechoose a “distance cutoff” rc = 2.75 fm from the center,where the particles are removed from the system.

The boundary conditions for the field is more compli-cated. Ideally one would choose absorbing boundary con-ditions (ABC), which perfectly absorb every wave trav-eling through the boundary and all its energy [19]. Un-

Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model 9

fortunately, such boundary conditions are very hard toimplement and are computational expensive, especially inthree dimensions and to have a good performance theirformulation needs to be non-local in time [20]. Still, theyhave been successfully applied for physical systems likethe 3D Schrodinger equation [21].

To keep the computational cost reasonable, the bound-ary conditions are therefore kept periodic, but in the outerregion of the box the σ-field is additionally damped by aneffective friction ∝ σ for any wave passing the cutoff ra-dius |x| > rc = 2.75 fm, leading to a strong dampingof the waves. The total energy in this system is not con-served any more and any energy passing the boundarywill be damped or removed from the system, preventingan interference with the rest of the system.

We have performed simulations for this expanding-fireball situation using both the model without and withthe chemical processes σ ↔ qq. The initial temperaturewas chosen Tinitial = 175 MeV, the number of test particlesNtest = 105, and a time step of ∆t = 2 · 10−3 fm/c. Ne-glecting the chemical processes, the σ-field shows a rapidoscillation in the beginning of the expansion and losesenergy mainly by radiating long-wavelength oscillationswhich propagate out of the system. Without any interac-tion processes the system stays nearly isotropic, only smallspatial fluctuations of the quark-density lead to small de-viations in the symmetry.

Additionally, for larger couplings the system can fallinto a meta-stable state in which cold quarks are trappedin a potential well of the chiral field which has a longlifetime.

Here we concentrate on the results of simulations tak-ing into account chemical processes σ ↔ qq. These pro-cesses should lead to additional particle production when-ever the σ-field shows strong fluctuations and particle an-nihilation in areas of high quark density, inducing addi-tional fluctuations on the field. There are a couple of quali-tative differences between the calculations with and with-out chemical processes. For all three different couplings(g = 3.3, 3.63, and 5.5 corresponding to a cross-over,2nd- and 1st-order chiral phase transition, respectively) thetypes of fluctuations are very different. The systems with-out chemical processes show strong global fluctuations,leading to shell-like structures with a fast oscillation ofthe chiral field at the origin of the matter droplet. Thefluctuations are much less present in the calculations withchemical processes. The decay process σ → qq damps suchstrong fluctuations by removing kinetic energy from thefield. In contrast, the process qq → σ creates strong lo-cal fluctuations on the field, increasing with a strongercoupling constant g. Additionally, all calculations withoutchemical processes stay nearly isotropic through their timeevolution while the calculations with chemical processesshow a breaking of this symmetry after a very short time.This symmetry is broken by strong and space-dependentfluctuations of the quark density and the field distribu-tion and due to a collective drift of both particles andthe field disturbance. Such a drift is surprising but is ex-plained by the conservation of energy and momenta in the

interactions between fields and particles. The σ-field canemit particles, generating a momentum-kick in the oppo-site direction of the particles, leading to a random-walkphenomena of the droplet. These outcomes are remark-able because no direct random processes are involved inthe numerical simulation, which would create these fluc-tuations. All stochastic and fluctuating processes are theresult of microscopic interaction kernels and cross-sectionswhich are sampled via Monte-Carlo methods.

By comparing the calculations with chemical processeswith different couplings g, qualitative differences are vis-ible, as well. The fluctuations in the field and particledensity increase for stronger couplings. For a coupling ofg = 5.5 the quark density forms small regions with higherdensities, which merge with time. These observations areconsistent with calculations of the linear σ-model witha hydrodynamic background [16,17,18], in which the au-thors find the strongest fluctuations for a medium witha first-order phase transition. Classical theories of phasetransitions in thermal equilibrium predict the strongestfluctuations at and near the phase transition for second-order transitions. Calculations presented here show thestrongest fluctuations for the highest coupling associatedwith the first-order phase transition. At first, this scenariocan not be directly compared to a phase transition. Aphase transition is a phenomenon described by equilib-rium physics for very large systems which evolve in theadiabatic limit on large time scales. In off-equilibrium sit-uations fluctuations need time to build up via interactionswithin the bulk medium. Most important, the correla-tion length is often largely enhanced at the phase tran-sition, which is no problem for systems which are muchlarger than this correlation length. The scenario of thehot-matter droplet is quite the opposite. The system sizeis in the order of the interaction length and therefore itscorrelation length. The quark matter expands rapidly andits dynamics creates distributions which strongly deviatefrom equilibrium, and the total lifetime of the system isat most in the order of the equilibration timescale. Addi-tionally, a rapid expansion leads to a highly anisotropicsystem with gradients and parts of the system separate toregions with very different densities and temperatures. Allthese circumstances do not allow a consistent descriptionof the system in terms of an equilibrium phase transition,especially not if the quarks are described by particles withnon-equilibrium distributions and not by a fast equilibrat-ing medium. Any deviation from equilibrium descriptionshave a strong impact on the thermal properties of themedium.

Considering the system dynamics in terms of its trans-port properties would be a more adequate approach tothe characteristics of its behavior. The strong fluctua-tions only occur if the chemical processes σ ↔ qq arepresent, their microscopic interaction kernels are derivedfrom a Breit-Wigner cross-section. Larger couplings leadto a higher interaction rate between fields and particles(in our model σσ↔qq ∝ g2). This already implies that thelargest fluctuations build up for the calculations with thecoupling g = 5.5 and quite similar fluctuations for g = 3.3

10 Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model

Fig. 8: Volumetric 3D rendering of the quark density for the hot matter droplet scenario with acoupling of g = 5.5, a first-order transition and chemical processes. The time progresses for 1 fm/cfor every image. Regions with high particle densities are colored red, the scale is constant over allimages. Visualization was created using the yt-toolkit [22].

and g = 3.63, even though these different couplings wouldshow very different kind of phase transitions in an equili-brated system.

Furthermore, two other aspects play an important rolein the comparison of the different calculations. The cou-pling has a direct impact on the mass of the quarks. Ahigher coupling leads to a larger quark mass in the chiralbroken phase. This implies both a larger particle num-

ber at the same temperature for a higher coupling and alarger potential energy for the chiral fields. The second as-pect has its origin in the phase diagram. Higher couplingslead to a lower Tc in the linear σ-model, which stronglychanges the dynamics of the chemical interactions as theyare only possible above the mass threshold, meaning aboveTc. For lower Tc, in comparison to other couplings, thequarks and the fields have more time to stay in the chi-

Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model 11

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Part

icle

Count

Energy [GeV]

t=10 fm/ct=15 fm/ct=20 fm/ct=30 fm/ct=40 fm/ct=55 fm/c

Fig. 9: (Color online) Left: Total quark number in the matter-droplet scenario. Solid lines show thesimulations with chemical processes, the dashed lines the simulations without the σ ↔ qq process.In the scenario without chemical processes the droplet radiates most of the quarks in shell-likestructures, as reflected in the quark-number plateaus which drop suddenly. Due to condensationprocesses the system can form meta-stable states, in which cold quarks are trapped in a chiral-potential well, which can be observed in the stable, non-zero quark number for g = 5.5 overlong times. The behavior is completely different for the calculations with chemical processes, inwhich the systems lose quarks in a steady and continuous process, the formation of quark-numberplateaus is washed out. Right: Energy distribution functions of the quarks in the simulation forthe matter-droplet scenario with g = 5.5 and chemical processes. After 10 fm/c most of the initialparticles have left the system, and the remaining particles have formed a drop of cold quarks,which are trapped in a chiral-potential well. One can see a non-thermal distribution function inthe beginning, which slowly thermalizes due to elastic interactions. However, mainly high-energyquarks can leave the potential well, leading to both a slow evaporation of the drop and to aneffective cooling of the remaining particles.

ral restored phase, more time to interact with each otherand therefore more time to build up fluctuations via theseinteractions. This already implies stronger fluctuations,regardless of the type of phase transition, which would begiven by the corresponding coupling. A fair comparisonbetween the scenarios is not given by comparing the sys-tem at different temperatures. Better approaches could bea comparison with same energies or same particle number.

In the calculations neglecting chemical processes theformation of meta-stable drops of quarks could be ob-served, especially at the high coupling, g = 5.5. Thisbehavior can be seen in the simulations including chem-ical reactions, too. For g = 5.5 with chemical reactionsthe system even shows something like bubble formation.Figure 8 shows a volumetric 3D rendering of the quarkdensity which projects the full three dimensional densityand visualizes how small areas of high quark-density startto merge into larger areas, having some similarity to thecondensation of water drops in steam. The mechanism ofthis condensation is that it is the energetically favorableconfiguration for the σ-field. Interestingly, the condensa-tion progresses and the σ-field looses energy by radiatingwave excitations. The process seems to stop after 10 fm/cby forming a quasi-stable drop. The left panel in Figure9 shows the total quark number of the system, showinga steady decrease of quarks in the system, indicating akind of particle evaporation from the drop. This evapora-tion seems to be quite slow and calculations have shown

a lifetime of up to 50 fm/c before the drop bursts. Theright panel Figure 9 displays the evolution of the quark-distribution function with time after a condensation drophas been formed. The figure shows a decrease of the quarknumber with time, and mainly particles with high ener-gies leave the system. This is reasonable because particleswith high energy can escape the potential well, leading toa collective cooling of the remaining medium.

All calculations discussed so far are done for systemswith a fireball which has an initial diameter of roughlyd ≈ 1 fm. In comparison to a heavy-ion collision such asystem is very small. Both lead and gold nuclei have a di-ameter of around dAu ≈ 14 fm, resulting in a volume twoorders of magnitude larger than in this scenario. There-fore quantified predictions for a heavy-ion collision cannot be derived from these calculations. Both the numberof involved quarks and the energy density in the systemcould lead to different phenomena. Additionally, a muchlarger spatial and time scale may change the behavior atthe phase transition because the system has potentiallymore time to evolve and form structures in the particledensities.

Now we simulate the model with a larger system vol-ume of V = (36 fm)3 with an initial size of the matterdroplet of d ' 14 fm, roughly resembling a central Au-Au collision. The initial temperature was again chosento be T = 175 MeV, as well as the couplings g = 3.3,g = 3.63, and g = 5.5 to investigate the cases of cross-

12 Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model

over, 2nd, and 1st order phase transitions. The grid size isincreased to Ngrid = 1923, and the time step is reducedto ∆t = 10−3 fm/c. Also the interaction volume has beenslightly increased to Vinteraction = (0.4 fm)3. The totalnumber of particles was ∼ 5 · 106 with a test-particle mul-tiplicator of 200.

Figure 10 and 11 show the result for a calculation witha large expanding matter droplet. Both figures comparethe same calculations with and without chemical interac-tions and different couplings, g = 3.3 with a cross-overtransition in Figure 10 and g = 5.5 in Figure 11.

The calculations without chemical interactions do notchange very much in comparison to the smaller systems.The overall symmetry of the system stays effectively in-tact, and known structures like the shell-structures [23,24]and cold quark droplets are still present.

The qualitative picture changes somewhat for calcula-tions with chemical interactions, especially for larger cou-plings like g = 5.5. After a short expansion phase thematter droplet starts to form strong and non-isotropicstructures in the quark density due to the annihilationand creation process. In Figure 10 first structures are al-ready present at t ≈ 1 fm/c. These structures blur whilethe matter expands.

In Figure 11 these structures are much stronger andmuch finer. The reason is the strong coupling betweenthe field and particles, leading to fast annihilation andstrong damping of the field by σ-decay. While expanding,the fluctuations on the σ-field become stronger. Aroundt = 5 fm/c the annihilation-rate of the quarks is negligi-ble because of the low particle density due to expansion.The σ-decay becomes the dominating effect, leading to astrong damping of the field and to a quasi-freeze out of thefield leading to some kind of local bubble-formation. Theresulting field distribution stays stable for several 10 fm/clocal bubbles converge slowly to a big drop. This finaldrop contains the already known cold quarks which werenot able to escape the kinetic potential.

In the calculations for larger systems a qualitative dif-ference between the couplings can be seen in the timeevolution. While calculations for g = 3.3 and g = 3.63 be-have similar, calculations for g = 5.5 show a very differentbehavior: the quark matter forms bubbles and freezes outin a relative long-living structure.

The calculations in this section have shown the veryinteresting complexity of such a simple initial conditionlike an expanding matter droplet. The chemical reactionsbetween particles have a very strong impact on the systembehavior and dramatically change both fluctuations andmedium propagation within the expansion. However, char-acteristic signatures which would allow an event-by-eventdiscrimination between the different coupling strengths(implying different phase transitions in equilibrium) inthis scenario has not been found, at least not for calcula-tions with the cross-over coupling g = 3.3 and the second-order transition coupling g = 3.63. As a first attempt toquantify the quark-number fluctuations in Fig. 12 we plotthe angular momentum distribution of quarks leaving thefireball volume for the case of g = 3.3. The qualitative fea-

tures of the distribution do not significantly vary with thecoupling strength. Only for the first-order value g = 5.5show significantly larger magnitude of the fluctuations, asquantified by the power spectra of the angular correlationfunction.

To quantify the angular distributions further, we eval-uate the angular power spectrum, via

Clm =

∫Ω

dΩf(ϑ, ϕ)Ylm(ϑ, ϕ) (19)

with the usual normalized spherical harmonics Ylm(ϑ, ϕ)and

f(ϑ, ϕ) =

∫ ∞0

dpd

d2pNquark(p, ϑ, ϕ). (20)

The normalized power spectrum is then defined by

Cl =1

2l + 1

l∑m=−l

|Clm|2. (21)

4 Conclusions and outlook

In this paper we have used a novel numerical algorithmto simulate the off-equilibrium dynamics of a quark-mesonlinear-σ model (DSLAM) inspired by “wave-particle dual-ity” of “old quantum mechanics”, guaranteeing accurateenergy-momentum conservation and the principle of de-tailed balance by treating the quarks and anti-quarks, cou-pled to a σ mean-field with the test-particle Monte-Carloapproach to evaluate the two-body elastic-collision termsand a combined method using wave-particle duality andcoarse graining to evaluate chemical processes of quark-antiquark pair annihilation and σ-meson decay, qq ↔ σ.On top also the mean-field equation for the σ field hasbeen solved consistently.

The algorithm has been validated in “box calculations”by checking the energy-momentum conservation, the sta-bility of the expected equilibrium solutions for the meanfield and quark-antiquark phase-space distribution func-tions, and the approach to the correct thermal and chem-ical equilibrium in the long-time limit starting with sim-ple off-equilibrium initial conditions (“thermal quench”),demonstrating excellent fulfillment of energy-momentumconservation and of the principle of detailed balance. Theimplementation of the chemical processes qq ↔ σ arefound to be crucial for the correct description of the phasetransition and the approach of the equilibrium state as ex-pected from the corresponding equilibrium quantum-fieldtheoretical evaluation of the model on the mean-field level.

Afterward the model has been applied to the simula-tion of expanding fireballs mimicking the behavior of thestrongly interacting medium as created in heavy-ion colli-sions. We have found that with this model we can simulatethe expected local quark-antiquark, i.e., baryon numberfluctuations (under strict obedience of the global baryon-number conservation law) due to the implementation ofthe quark-antiquark annihilation and σ-meson decay pro-cesses. Particularly for strong couplings of the σ-meson to

Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model 13

Fig. 10: Evolution of the quark density for a large hot matter-droplet scenario with g = 3.3.The initial size of the droplet has an diameter of around d ≈ 14 fm which corresponds to thesize of a gold nuclei, the total size of the system is V = (36 fm)3. Left: Simulation withoutchemical interactions Right: simulation with chemical interactions. Chemical interactions lead tothe formation of anisotropic structures, but the overall expansion of the system is not stronglyaltered.

14 Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model

Fig. 11: Quark density for a large hot matter-droplet scenario with g = 5.5. The initial size ofthe droplet has a diameter of about d ≈ 14 fm which corresponds to the size of a gold nucleus, thetotal size of the system is V = (36 fm)3. Left: Simulation without chemical interactions Right:simulation with chemical interactions. Chemical interactions lead to a generation of strong anddetailed structures. After 5 fm/c the expansion of the system is stopped by the strong damping ofthe field, leading to the formation of local bubbles.

Christian Wesp et al.: Kinetics of the chiral phase transition in a linear σ model 15

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

0 10 20 30 40 50 60 70 80 90 100

Cl

l

g=3.3g=3.63g=5.5

Fig. 12: (Color online) Angular distribution (left panel) of quarks leaving the fireball for thecoupling g = 3.3 and the power spectrum of the angular correlation functions (right panel).

the quarks and antiquarks, corresponding to a 1st-orderphase transition in equilibrium, quasi-stable bubbles of“cold” quarks and anti-quarks, trapped in the chiral po-tential due to the mean field has been observed. As a firstattempt to find a possible observable in heavy-ion colli-sions for these fluctuations we have evaluated the angulardistributions of the quarks and antiquarks as a putativemeasure for net-baryon-number fluctuations and the asso-ciate power spectrum of the angular-density correlations.Unfortunately we have not found qualitative differencesin the shape of these power spectra for different couplingsassociated with cross-over, 1st, and 2nd order chiral phasetransitions but only in the absolute strength of the fluctu-ations as expected from the associated different couplingstrengths between σ mesons and quarks/antiquarks.

Acknowledgment: This work was partially supportedby the Bundesministerium fur Bildung und Forschung(BMBF Forderkennzeichen 05P12RFFTS) and by theHelmholtz International Center for FAIR (HIC for FAIR)within the framework of the LOEWE program (Landes-

offensive zur Entwicklung Wissenschaftlich-OkonomischerExzellenz) launched by the State of Hesse. C. W. andA. M. acknowledge support by the Helmholtz GraduateSchool for Hadron and Ion Research (HGS-HIRe), andthe Helmholtz Research School for Quark Matter Studiesin Heavy Ion Collisions (HQM). Numerical computationshave been performed at the Center for Scientific Comput-ing (CSC). H. v. H. has been supported by the DeutscheForschungsgemeinschaft (DFG) under grant number GR1536/8-1. C. W. has been supported by BMBF undergrant number 05P12RFFTS.

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