Basin of attraction for turbulent thermalization and the range of validity of classical-statistical simulations
BBasin of attraction for turbulent thermalization andthe range of validity of classical-statistical simulations
J. Berges,
1, 2
K. Boguslavski, ∗ S. Schlichting, and R. Venugopalan Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany ExtreMe Matter Institute (EMMI), GSI Helmholtzzentrum f¨urSchwerionenforschung GmbH, Planckstraße 1, 64291 Darmstadt, Germany Brookhaven National Laboratory, Physics Department, Bldg. 510A, Upton, NY 11973, USA
Different thermalization scenarios for systems with large fields have been proposed in the literaturebased on classical-statistical lattice simulations approximating the underlying quantum dynamics.We investigate the range of validity of these simulations for condensate driven as well as fluctuationdominated initial conditions for the example of a single component scalar field theory. We showthat they lead to the same phenomenon of turbulent thermalization for the whole range of (weak)couplings where the classical-statistical approach is valid. In the turbulent regime we establish theexistence of a dual cascade characterized by universal scaling exponents and scaling functions. Thiscomplements previous investigations where only the direct energy cascade has been studied for thesingle component theory. A proposed alternative thermalization scenario for stronger couplings isshown to be beyond the range of validity of classical-statistical simulations.
I. INTRODUCTION
A quantum many-body system in thermal equilibriumis independent of its history in time and characterizedby a few conserved quantities only. Therefore any ther-malization process starting from a nonequilibrium initialstate requires an effective loss of details about the initialconditions at sufficiently long times. It was pointed outpreviously that an effective partial memory loss of ini-tial condition details can already be observed at earlierstages for the nonequilibrium unitary time evolution inquantum field theory [1].An extreme case occurs if at some early stage thenonequilibrium dynamics becomes self-similar . Thisamounts to an enormous reduction of the sensitivity todetails of the underlying theory and initial conditions.The time evolution in this self-similar regime is describedin terms of universal scaling exponents and scaling func-tions. While normalizations of the latter depend onmodel parameters such as couplings, masses and initialconditions, their functional form is universal. This hasthe important practical consequence that the nonequi-librium dynamics for whole ranges of different model pa-rameters and initial conditions can be grouped into uni-versality classes. The self-similar time-evolution withina given universality class of different models can thenbe mapped onto each other by simple rescalings. Theuniversal properties are described in terms of nonther-mal fixed points [2] in renormalization group theory [3].While the notion of thermal fixed points describing dif-ferent universality classes is well established for systemsin thermal equilibrium, the corresponding classificationfor nonequilibrium scaling phenomena is a rapidly pro-gressing research topic. ∗ [email protected]
Universal behavior far from equilibrium has been pre-dicted for systems ranging from early-universe inflatondynamics to table-top experiments with cold atoms. Inthese examples, attractor solutions with self-similar scal-ing behavior are associated with the phenomenon of Kol-mogorov wave turbulence [4, 5] and strong turbulencewith fluid-like behavior [2, 6–8] which leads to Bose con-densation far from equilibrium [9–11]. A turbulent ther-malization mechanism has also been predicted for non-Abelian gauge theories using classical-statistical simula-tions in a fixed box, where different groups obtain con-sistent results [12–14].Recently, the existence of a nonthermal fixed pointin longitudinally expanding non-Abelian plasmas wasdemonstrated using classical-statistical lattice simula-tions in the weak coupling limit [15, 16]. It was arguedthat heavy-ion collision experiments at sufficiently highenergies can provide the relevant initial conditions in thebasin of attraction of this nonthermal fixed point. At suf-ficiently high energies, the colliding nuclei are describedin the Color Glass Condensate framework [17]. The dy-namics of the nonequilibrium Glasma [18] created in sucha collision is that of gluon fields with typical momentumQ and a weak gauge coupling α s ( Q ). Since the charac-teristic occupancies ∼ /α s ( Q ) are large, the gauge fieldsare strongly correlated even for small gauge coupling andthe Glasma exhibits classical-statistical dynamics.The early-time behavior is then described by plasmainstabilities, which have been intensively analyzed overthe past years [19–22]. Classical-statistical simulationsshow that gluons become highly occupied ∼ /α s ( Q ) af-ter a proper time ∼ Q − log ( α − s ) for expanding sys-tems [23–25]. Subsequently, as argued in [15, 16], oneexpects a transition to the universal weak coupling at-tractor.However, in Ref. [26] it has recently been argued thatby exceeding a rather small value of the gauge coupling α = g / π with g ∼ .
5, the dynamics changes dramati-cally compared to previous weak-coupling estimates, us- a r X i v : . [ h e p - ph ] O c t ing small initial Gaussian distributed fluctuations super-imposed to the large classical macroscopic field ∼ O (1 /g )as derived in Ref. [27]. Moreover, similar findings ofpressure isotropization for condensate driven initial con-ditions have been reported in nonequilibrium classical-statistical scalar field theory with expansion [28].Even without expansion, a different thermalization sce-nario in the presence of large initial condensates wasdiscussed using classical-statistical simulations once theself-coupling λ reaches a certain strength [29]. In con-trast, in the weak-coupling limit, the same initial con-ditions are known to lead to high occupancies ∼ /λ after a time t ∼ Q − log( λ − ) where the characteris-tic scale Q is determined by the energy density of thesystem [30, 31]. The subsequent evolution exhibits thephenomenon of turbulent thermalization [4] reminiscentof the classical-statistical non-Abelian gauge theory re-sults in the weak coupling limit [12–14]. The importantadvantage for scalars is that powerful resummation tech-niques exist for the description of the entire evolutionin quantum field theory. It has been demonstrated thatclassical-statistical descriptions have a well understoodrange of validity for which they accurately describe thedynamics of the underlying quantum theory [2, 32–34].In this work, we revisit the question of nonequilibriumdynamics from condensate driven initial conditions forthe single component theory employed in Ref. [29]. Wefirst investigate the dynamics in the weak coupling limit.In this limit, we observe turbulent thermalization, wherewe point out that1. the inverse particle cascade, which has so far onlybeen established for O ( N ) symmetric theories with N ≥ ∼ Q − log( λ − ) for the wholerange of couplings where the classical-statistical approachis valid. Thus the observed differences in thermalizationscenarios that have been proposed are not caused by dif-ferences in the initial conditions. We finally investigate4. the range of validity of classical-statistical simula-tionsand explain why the results of Ref. [29] do not lie in thisrange. II. SCALAR MODEL AND INITIALCONDITIONS
There is a restricted class of problems where thedynamics of bosonic quantum fields can be accuratelymapped onto a classical-statistical problem. The mostintuitive criteria for this can be formulated in situationswhere a kinetic description in terms of quasi-particleexcitations is applicable. The system exhibits classicaldynamics whenever the typical occupation numbers permode are much larger than unity, f ( t, p ) (cid:29) . (1)If occupation numbers fall below unity, quantum pro-cesses will dominate the dynamics. This is clearly seenin a Boltzmann transport framework where classical scat-tering processes are sub-leading to quantum ones for oc-cupation numbers smaller than unity [36, 37].It is also possible to formulate more general criteria,which do not rely on a quasi-particle picture, in theSchwinger-Keldysh formalism of nonequilibrium quan-tum field theory [33, 38–40]. This ‘classicality condition’is met whenever anti-commutator expectation values fortypical bosonic field modes are much larger than the cor-responding commutators [33, 39]. Stated differently, thisconcerns the large field or large occupancy limit, whichis relevant for important phenomena such as nonequi-librium instabilities or wave turbulence encountered inour study. The classicality condition has been verifiedin detail by comparing quantum to classical-statisticalresults in the context of scalar quantum field dynamics[2, 32, 33, 39] and coupled to fermions [41, 42].We emphasize that the condition for a system to ex-hibit classical dynamics is in general time dependent.In particular, the approach to complete thermal equi-librium is not accessible within the classical-statisticalframework. As a consequence of the Rayleigh-Jeans di-vergence, the classical thermal state is only well definedfor an ultraviolet cutoff Λ, which may be implementedby a lattice regularization. In its range of validity, ob-servables computed from classical-statistical simulationsare insensitive to the Rayleigh-Jeans divergence. Ther-mal equilibrium is a genuine quantum state which can-not be reached within classical-statistical field theory.Nevertheless, the classical-statistical regime may extendover large times such that the quantitative properties ofthe nonequilibrium quantum evolution are accurately de-scribed within the classical-statistical approach.The time scale t quant for entering the quantum regimeis not a universal quantity and depends in general onthe properties of the initial state as well as the dy-namics in the classical regime. For the consideredcases of large initial fields or large initial fluctuationsat weak coupling, this time scale is parametrically givenby t quant ∼ Q − λ − / in our case [5]. The range of valid-ity of classical-statistical techniques in time is thus nat-urally confined to weak couplings. In general the use ofclassical-statistical methods at large couplings requiresgreat care since genuine quantum effects may dominatethe dynamics already at rather early times. This will beinvestigated in detail below after we present the physicalweak coupling results.We study the dynamics of a massless real scalar fieldtheory with quartic interaction. The classical action isgiven by S [ ϕ ] = (cid:90) d x (cid:18) ∂ µ ϕ∂ µ ϕ − λ ϕ (cid:19) , (2)summing over µ = 0 , , , δS/δϕ = 0 for the single-component field reads (cid:3) ϕ ( x ) + λ ϕ ( x ) = 0 . (3)Following Ref. [29], we consider Gaussian initial con-ditions. They are fully determined by the initial one-and two-point correlation functions of the fields ϕ ( x ) and π ( x ) = ∂ϕ ( x ) /∂x at time t = x = 0. For a spatiallyhomogeneous system the most general initial Gaussianconditions can be characterized by φ ( t = 0) = (cid:104) ϕ (0 , x ) (cid:105) , ˙ φ (0) = (cid:104) π (0 , x ) (cid:105) (4) F ( t = t (cid:48) = 0 , x − y ) = (cid:104) ϕ (0 , x ) ϕ (0 , y ) (cid:105) − φ (0) (5) K ( t = t (cid:48) = 0 , x − y ) = (cid:104) π (0 , x ) π (0 , y ) (cid:105) − ˙ φ (0) (6)as well as (cid:104) ϕ (0 , x ) π (0 , y ) + π (0 , x ) ϕ (0 , y ) (cid:105) − φ (0) ˙ φ (0) . (7)The values for these initial correlation functions are takento coincide with those from the corresponding quantuminitial value problem. In the classical-statistical theorythe correlation functions are obtained by performing aphase-space average over the initial configurations for agiven observable (cid:104) O cl ( ϕ, π ) (cid:105) = (cid:90) D ϕ (0) D π (0) W ( ϕ (0) , π (0)) O cl ( ϕ, π )(8)with respect to the distribution function of initial fields W ( ϕ (0) , π (0)). The latter is chosen to give the prescribedcorrelation functions (4–7). Accordingly, we sample theinitial field configurations until convergence to the pre-scribed initial correlations is achieved. Each initial con-figuration is separately evolved in time and the time evo-lution of correlation functions is obtained from the en-semble average.Using the corresponding definitions also for t >
0, wemay extract the time evolution of occupation numbers f ( t, p ) in spatial Fourier space with p = | p | from [38] f ( t, p ) + 12 = (cid:113) ˜ F ( t, p ) ˜ K ( t, p ) , (9) The antisymmetric combination ∼ (cid:104) ϕ ( x ) π ( y ) − π ( x ) ϕ ( y ) (cid:105) is fixedby the corresponding field commutation relation in the quantumtheory (or Poisson bracket in the classical theory) [38]. as well as the dispersion ω ( t, p ) = (cid:115) ˜ K ( t, p )˜ F ( t, p ) , (10)where ˜ F ( t, p ) = (cid:82) d x e − i px F ( t, x ) denotes the Fouriertransform. With these definitions, the above initial con-ditions specify the initial field amplitude φ ( t = 0) anddistribution function f ( t = 0 , p ). The initial frequencyreads ω ( t = 0 , p ) = (cid:112) p + m , where the effective massterm is given by m = λ (cid:32) φ ( t = 0) + (cid:90) Λ p F ( t = 0 , p ) (cid:33) + δm . (11)In this setup the initial correlator (7) vanishes and wewill also employ ˙ φ ( t = 0) = 0 in the following.The counter-term δm in Eq. (11) can be chosen tocancel the leading quadratic Λ-dependence of the three-dimensional momentum integral over the initial F ( t =0 , p ). The same counter term is then used to cancel theassociated divergence in the equation of motion (3). Ifnot stated otherwise, we will perform such renormalizedsimulations. In addition, we will also present results for δm = 0 as employed in Ref. [29].In the classical equation of motion (3) all model de-pendence on the coupling constant λ could be scaled outby the reparametrization ϕ → ϕ/ √ λ . The correspondinginformation is then entirely encoded in the initial condi-tions. The coupling drops out everywhere except for the‘quantum half’ in the initial conditions, which would be-come ‘ λ/ (cid:15) is used to calcu-late the characteristic scale Q = √ λ (cid:15) . (12)For the initial conditions, which we consider in the fol-lowing, this scale is independent of the coupling constant.The first set of initial conditions is characterized by alarge macroscopic field φ ( t = 0) = σ √ λ , f ( t = 0 , p ) = 0 , (13)(Condensate IC)with σ of order one. For this initial condition the modeoccupancies are zero such that all modes are initializedwith the vacuum ‘quantum half’ up to the lattice ultra-violet cutoff as in Ref. [29].The second class of initial conditions is taken to be fluc-tuation dominated for comparison, with an overoccupieddistribution up to some initial momentum Q , φ ( t = 0) = 0 , f ( p ) = n λ Θ( Q − p ) , (14)(Fluctuation IC)and the initial amplitude of the particle distribution n with Q ∼ √ n Q . For both types of initial conditionsone finds that the energy density scales with the couplingconstant as (cid:15) ∼ /λ . III. TURBULENT THERMALIZATION
The condensate driven initial conditions (13) lead tothe well known phenomenon of parametric resonance. Inthe context of inflationary cosmology, the correspondingpreheating dynamics [30] and the process of turbulentthermalization has been studied in great detail both us-ing classical-statistical simulations [2, 4, 5, 8, 9, 43, 44]as well as directly in quantum field theory using resum-mation techniques based on the two-particle irreducible(2PI) effective action [2, 31, 32]. In the cosmologicalcontext, the preheating dynamics has also been inves-tigated in the presence of fermion matter [45, 46] andtheir consequences deduced for fermion production andthe phenomenon of turbulent thermalization [41, 42].Initially, the energy density is stored in the largecoherent field φ ( t ) for small coupling λ . Since φ ( t ) israpidly oscillating, we will also consider the behavior ofthe envelope of the maximum field amplitude φ ( t ) as afunction of time. For the subsequent evolution one canidentify three characteristic ranges in time, which areparametrically given as follows.1. Instability for 0 (cid:28) t (cid:28) Q − log( λ − ): In thisparametric resonance regime, the field φ ( t ) is a periodicoscillating function with frequency characterized by theinitial rescaled field amplitude σ [47, 48]. This initialamplitude also determines the initial resonance band inmomenta for which an exponential growth of f ( t, p ) canbe observed [48]. The rapid growth of fluctuations leadsto strong nonlinearities which broaden the resonanceband and produce enhanced growth rates [31]. As aconsequence, a wide range of growing modes lead to aprethermalization of the equation of state at the end ofthis early stage [1, 49, 50] while the distribution functionitself is still far from equilibrium.2. Turbulence for Q − log( λ − ) (cid:28) t (cid:28) Q − λ − / : Inthis regime, the field amplitude approaches a power-lawbehavior and the distribution function becomes self-similar [5]. Elastic scattering processes dominate anda dual cascade forms in distinct momentum ranges: Adirect energy cascade towards the ultraviolet modes [5]and an inverse particle cascade towards the infrareddevelop [2]. These cascades are separated in momentumspace by the characteristic momentum of the dominantresonance peak in the distribution function. Thedirect energy cascade drives the thermalization processby energy transport towards higher momenta. Theinverse particle cascade leads to the phenomenon ofBose condensation in this far from equilibrium regime [9]. λ / φ / Q Q t t -1/3 Λ / Q = 10.2 Λ / Q = 15.3 FIG. 1. The rescaled field amplitude √ λ φ ( t ) for λ = 10 − .We employ two different cutoffs in order to demonstrate theinsensitivity of the results on the lattice regularization. Alsoshown is the predicted power law ∼ t − / (gray dashed line),where one observes very good agreement with data at latertimes. In the inset, the same data is shown to better resolveearly times. Thermalization for t (cid:29) Q − λ − / : In this regimeelastic and inelastic processes lead to a Bose-Einsteindistribution. This regime is beyond the range of validityof classical-statistical simulations. At sufficiently latetimes the classical evolution will always end up showinga classical thermal distribution with a temperatureparameter T Λ depending on the Rayleigh-Jeans cutoffΛ. The inability of the classical approach to describethis late-time thermalization stage has been studied indetail in the literature [33, 38].The characteristic time scales are given as weak cou-pling parametric estimates for our purposes. In addition,some of the prefactors are also available that can be sig-nificantly different from one for the case of the single com-ponent scalar field theory. For instance, the time scalefor the end of the instability regime is more accuratelygiven by t (cid:39) (2 γ ) − log( λ − ), where γ (cid:39) . Q [48] isthe largest growth rate of the primary instabilities.The following numerical results are obtained usingstandard lattice discretization techniques [38]. In Fig. 1we show the time-dependent rescaled field amplitude √ λ φ ( t ) for a weak coupling λ = 10 − and σ = 2 . Q .Two different data sets are shown corresponding to twodifferent lattice momentum cutoffs Λ /Q = 10 . /Q = 15 . The comparison confirms that the weakcoupling results are insensitive to the Rayleigh-Jeans cut-off in classical-statistical field theory, which is a crucial These results are obtained on 768 lattices. The volumes aredifferent for each cutoff but the lattice is large enough such thatthe evolution of the system is seen to be independent of thevolume. ingredient for its ability to describe the correspondingquantum field dynamics [33].The oscillating behavior with constant maximum am-plitude at early times is visible from the inset of Fig. 1.Subsequently, for t ∼ Q − log( λ − ) the corrections fromclassical-statistical fluctuations change this behavior dra-matically and trigger a transient rapid field decay. Thishappens when the size of fluctuations has grown suchthat their contribution to the energy density becomescomparable to that from the macroscopic field. At thisstage the evolution becomes strongly non-linear whicheven leads to a temporary field growth [31]. The non-linear dynamics finally leads to a power-law decay of thefield amplitude φ ( t ) ∼ Q ( Qt ) − δ (15)with the predicted exponent δ = 1 / Q − log( λ − ) (cid:28) t (cid:28) Q − λ − / the distributionfunction approaches a self-similar behavior, f ( t, p ) = ( Qt ) α f S (( Qt ) β p ) , (16)with the universal scaling exponents α , β and the sta-tionary scaling function f S . The dynamical scaling expo-nents describe the evolution of typical occupation num-bers and momenta. For hard modes, which dominatethe energy density, β = − / α = − / t quant ∼ Q − λ − / at which the typical occupanciesbecome order one and the classicality condition (1) is nolonger fulfilled.Wave turbulence [51] describes the transport of con-served quantities such as energy density. Accordingly,in momentum space power-law cascades form. This canbe either a ‘direct cascade’, for transport towards highermomenta, or an ‘inverse cascade’ into the infrared. Inthe inertial range of momenta, where the distributionfunction is described as a momentum power-law, one canwrite f ( p ) ∼ (cid:18) Qp (cid:19) κ (17)with a universal scaling exponent κ . The proportionalityfactor depends in general on time for isolated systemsas in our case, i.e. without applied sources or sinks thatcould lead to stationary cascades.If there is more than one conserved quantity, the sys-tem may accommodate this by forming distinct cascadesin different momentum regimes. It has been shownthat for characteristic momenta p (cid:46) Q the dynamics isdominated by elastic scatterings [2]. This leads to anadditional conserved quantity that emerges during thenonequilibrium time evolution even though total parti-cle number is not conserved in the relativistic scalar fieldtheory. As a consequence, apart from a direct energy -2 p / Q λ f p -4 p -4/3 p -3/2 Bose condensationfar from equilibriuminverse particlecascade direct energycascade
Q t = 160Q t = 240Q t = 1350Q t = 3400Q t = 7100
FIG. 2. The rescaled distribution function λf at differenttimes for λ = 10 − . Power-laws with exponents 3 /
2, 4 / cascade towards the ultraviolet, an inverse particle cas-cade towards the infrared can be observed. For the singlecomponent theory, this analysis was previously only donefor the direct cascade [4, 5]; the presence of the inversecascade was not established.In Fig. 2 we show our results for the evolution of therescaled distribution function λf ( t, p ) for different fixedtimes. At times Qt = 160 and 240 one observes the res-onance behavior of the instability stage. The resonancepeak occurs first around p ∗ /Q ≈ . p ∗ = 0 . σ [48]. Since the macroscopic field am-plitude decreases with time, the peak is shifted to softermomenta [5].At later times, the peak serves as a source for energyand particles, leading to momentum-scale invariant en-ergy and particle fluxes towards the ultraviolet and theinfrared, respectively. Accordingly, different momentumpower-laws emerge to the left and to the right of the stillvisible dominant resonance peak in Fig. 2. On the dou-ble logarithmic plot the power-laws are well describedby straight lines with different slopes, corresponding todifferent values for the exponent of (17) in distinct mo-mentum ranges. The spectrum at Qt = 7100 is comparedto three power laws with the exponents κ S = 4 , κ M = 43 , κ H = 32 . (18)The hard scale exponent κ H = 3 / ↔ (1 + soft) processes involving the conden-sate [4, 5]. The inverse particle cascade towards lowermomentum modes shows two distinct momentum regimesdepending on the size of the occupation numbers f ( p )in each regime. For λf ( p ) (cid:46) O (1) the weak wave tur-bulence exponent κ M = 4 / λf ( p ) (cid:38) O (1) the strong turbulence expo-nent κ S governs the dynamics [2].We emphasize that the entire inverse particle cascade,which is described by κ M and κ S , can be understoodin terms of elastic processes only. The enhanced scal-ing exponent κ S > κ M is a consequence of an emergenteffective scattering matrix element for 2 ↔ λ eff ( p ) ∼ p [2, 8, 9]. This analyt-ical prediction is based on a systematic large- N expan-sion to next to leading order of the 2PI effective actionfor the N -component scalar quantum field theory, whichgives κ S = d + 1 in d space dimensions [52]. Since wehave N = 1, and in view of some reported deviations for N = 2 [35], it is remarkable that we find the predictedexponent with the observed accuracy.In order to display the whole inverse particle cascadewith both the weak and the strong turbulence regimes,we used rather large lattices up to 768 . Moreover, tocover the entire range of momenta shown in Fig. 2, weactually combined the data from two simulations withdifferent lattice cutoffs Λ /Q = 10 . /Q = 15 .
3. Inthe overlapping momentum regions the two simulationsagree to very good accuracy. We emphasize that thisis only done for presentational purposes of the physicalweak coupling results. In particular, all results presentedin the upcoming sections come from simulations includ-ing the entire range of displayed momenta.
IV. BASIN OF ATTRACTION FOR THENONTHERMAL FIXED POINT
A crucial property of the turbulent regime is its strictindependence of model parameters such as the value ofthe coupling constant λ or the initial conditions such asthe initial field value σ . This is a very powerful con-sequence of universality, which finds its manifestation inthe self-similar behavior (16). The latter represents anenormous reduction of the possible dependence of the dy-namics on variations in time and momenta, since it statesthat ( Qt ) − α f ( t, p ) only depends on the product ( Qt ) β p instead of separately depending on time and momenta.In general, renormalization group theory tells us that thefixed point distribution f S appearing in (16) will dependon all ’relevant’ parameters of the system. For instance,if the initial field value σ would represent such a rele-vant parameter then f S would in addition depend on theproduct ( Qt ) ζ σ with some new exponent ζ . Plotting( Qt ) − α f ( t, p ) only as a function of ( Qt ) β p/Q would thenfail to describe the data. Therefore (16) represents a verystrong statement about the loss of information about theparameters of the underlying system already at this tran-sient stage of the nonequilibrium time evolution.Since self-similarity has been extensively discussed forour theory already in the literature [5], we only addresshere those aspects of universality that are relevant forclarifying the conflicting statements mentioned in the in- λ / φ / Q Q tCondensate IC Fluctuation IC , n = 39 Fluctuation IC , n = 7.5 -2 λ f p / Q Q t = 0Q t = 6100
FIG. 3. The rescaled field amplitude as a function of time forthree different initial conditions at fixed coupling λ = 10 − .The inset shows the corresponding occupation number distri-butions λf ( t, p ) at Qt = 6100. For the two fluctuation domi-nated initial conditions the inset shows in addition the initialspectra (gray). One observes that the evolution becomes in-dependent of the initial conditions for the condensate drivenas well as for the fluctuation dominated initializations. troduction. In particular, we want to confirm that boththe class of condensate driven initial conditions given by(13) as well as fluctuation dominated initial conditions(14) belong to the basin of attraction for this nonther-mal fixed point.So far we have used the initial conditions characterizedby a large field and small (vacuum) fluctuations. Thesecond class of initial conditions we now consider are de-scribed in terms of highly occupied modes with initialamplitude n /λ up to the momentum Q . We consider n = 39 as well as n = 7 . The characteristic scale Q given by (12) is taken to be always the same, thus fixing Q for given n . We employ a weak coupling λ = 10 − .Using both types of initial conditions, the evolutionof the rescaled field amplitude is shown in Fig. 3. Oneobserves that for each of the fluctuation dominated ini-tial conditions a condensate builds up , which is a con-sequence of the inverse particle cascade as describedabove [9]. Vice versa, the condensate driven initial con-ditions lead to fluctuations by parametric resonance. Atlate times the curves for all of the different initial condi-tions fall on top of each other to very good accuracy. Thisis further illustrated by showing the corresponding dis- The simulations for the fluctuation dominated initial conditionswere performed on 256 lattices, while for the condensate driveninitial condition we employ a 768 lattice but we checked thatsmaller lattices lead to the same observations. We use the latticecutoff Λ /Q = 15 .
3, where we have verified that the results areinsensitive to the cutoff. Any homogeneous field domain can only form as rapidly as al-lowed by causality. tribution functions in the inset of the figure. While thesingle particle spectra are very different at initial time( Qt = 0), at late times the curves for all three distribu-tions fall on top of each other. The fact that the evolutionbecomes independent of the details of the initial condi-tions already at this transient stage is a striking exampleof the phenomenon of universality far from equilibrium. V. LIMITATIONS OFCLASSICAL-STATISTICAL SIMULATIONS
We now come back to the condensate driven initial con-ditions, as employed in Ref. [29], and explore the rangeof validity of classical-statistical simulations that providea reliable description of the underlying quantum dynam-ics. Classical-statistical descriptions are restricted to theweak coupling regime as explained above. Therefore, it isimportant to verify up to which size of the coupling thesemethods can be applied. This is a time-dependent ques-tion since for sufficiently late times all classical-statisticaldescriptions break down once typical occupancies becomeorder one – as is the case for thermal equilibrium – andgenuine quantum processes dominate. For weak couplings, the time at which the instabil-ity regime ends depends logarithmically on the inversecoupling constant as explained above. The subsequentevolution becomes universal and thus independent ofthe coupling. The system exits the turbulent regimeand enters the quantum one around the parametric time t quant ∼ Q − λ − / . Increasing the coupling thereforemeans that the range of validity in time shrinks. Equiva-lently, for some fixed time one should observe deviationsfrom universal behavior as the coupling is increased. Theonset of any coupling dependence also signals the break-down of the classical-statistical method.This is illustrated in Fig. 4 where we show the rescaleddistribution function λf for different values of the cou-pling constant λ at fixed time Qt = 1200. One ob-serves that for weak couplings λ (cid:46) . In contrast,for λ = 0 . λ ≤ . λ = 0 . /Q = 15 . An important exception concerns dynamic critical phenomena.Since the universal real-time properties at the critical point arecontrolled by the infrared dynamics, they can be accurately com-puted within the classical-statistical framework [53]. Here we do not consider the frequently employed possibility toinitialize the ’quantum 1/2’ only in the instability band [32]. For a vanishing coupling constant ( λ = 0) instabilities do notoccur and the macroscopic field follows a classical evolution. -2 p / Q λ f Q t = 1200 λ = 10 -4 λ = 10 -2 λ = 0.1 λ = 0.5 FIG. 4. The rescaled distribution function λf ( t, p ) at time Qt = 1200 for different values of the coupling λ . For λ (cid:46) . λ = 0 . ε quan t / ( ε φ + ε pa r t ) Q t λ = 0.3 λ = 0.2 λ = 0.1 λ = 0.05 λ = 0.02 λ = 0.01 λ = 10 -4 FIG. 5. The quantum part of the energy density (cid:15) quant asdefined in (19) divided by the sum of the condensate andfluctuation part given in the text as a function of time fordifferent values of the coupling λ for the cutoff Λ /Q = 15 . (1) implies that modes with low occupancies of f ( p ) (cid:46) (cid:15) quant ( t ) = (cid:90) d p (2 π ) ω ( t, p ) f ( t, p ) Θ (1 − f ( t, p )) . (19)This ‘quantum’ part should be compared to the totalenergy density contained in quasi-particle fluctuations (cid:15) part ( t ) = (cid:82) p ω ( t, p ) f ( t, p ) and in the macroscopic field,which we approximate by (cid:15) φ ( t ) = λφ ( t ) /
24. In the rangeof applicability of classical-statistical descriptions one ex-pects that the ratio (cid:15) quant / ( (cid:15) φ + (cid:15) part ) (cid:28)
1. Of course,such a simple separation into fluctuation and field parts -2
0 0.5 1 1.5 2 2.5 3 3.5 f p / Q T / ( ω - µ ) - / T = . Q m = . Q µ = . Q λ = 1 Λ / Q = 5.1 Q t = 130Q t = 700Q t = 1400Q t = 6800renorm. mass FIG. 6. The occupation number distribution f ( t, p ) for λ = 1at different times for the same parameters as in Ref. [29]. Ourresults (lines) agree to very good accuracy with the previousstudy (points). The black dashed curve gives the correspond-ing result with mass renormalization to show that no signif-icant differences can be observed at Qt = 6800. The orangedashed curve is a fit to the spectrum as employed in Ref. [29]to extract temperature T , mass m and chemical potential µ parameters. is only applicable if there is a good quasi-particle de-scription for weak couplings and such an analysis shouldalways be taken with great care. For instance, we willsee below that for λ = 1 very strong cutoff dependenceoccurs and the occupation number distribution becomesnegative such that this analysis is inapplicable. However,we find that for the range of couplings λ (cid:46) . f ( p ) ≥ λ . While for λ = 10 − this ratio is practically zero for alldisplayed times, one observes that for λ = 0 . Qt = 1200. From this timeon, for λ = 0 . (cid:15) φ + (cid:15) part and a description in terms of classical-statisticalsimulations becomes questionable. These findings are invery good agreement with the above observation aboutemerging coupling dependencies of the results beyond therange of validity of the classical-statistical approach. VI. RAYLEIGH-JEANS CUTOFFDEPENDENCE AT COUPLING λ = 1 We have seen above that quantum corrections becomemore important at later times. Therefore, one mighthope that for short enough times one could extend therange of validity of classical-statistical simulations tostronger couplings. However, since the coupling controls f + / p / Q λ = 1 Λ / Q = 5.1 Q t = 130Q t = 6800Q t = 27000Q t = 47000Q t = 81000T(t) / ( ω (t) - µ (t)) FIG. 7. f ( t, p )+1 / λ = 1 at different times with the samelattice cutoff as used in Fig. 6. Blue curves denote growingoccupancies at early times, black curves show the spectra atlater times when occupation numbers are decreasing in time.The dashed orange lines are classical thermal functions, fittedto the black curves. The dashed green line denotes f = 0. the relative size of the occupation number per mode ascompared to the ‘quantum half’ and the latter are cutoff by the Rayleigh-Jeans regulator Λ, there is the dan-ger that the sizeable cutoff dependence leads to a break-down of classical-statistical simulations already at rela-tively early times in this case.In Ref. [29] it is argued that at short enough timesand strong enough couplings one does not enter the weakcoupling attractor for turbulent thermalization and an al-ternative thermalization scenario can be observed withinthe framework of classical-statistical simulations. Theauthors find that for λ = 1 the classical-statistical ther-malization dynamics is very different than what has beenpreviously derived for weaker couplings. In particular, itis claimed that no turbulent cascades form and Bose con-densation occurs as a consequence of the formation of atransient chemical potential.In the following we reconsider the calculations ofRef. [29]. First we also use their employed 20 latticesand cutoff Λ /Q = 5 . δm = 0). In Fig. 6 our results are plotted by linesalong with the data from the referenced study given bypoints of the same colors. Both results agree to verygood accuracy. We also give the corresponding curvewith mass renormalization (black dashed curve) to showthat no significant differences can be observed. The or-ange dashed curve is a fit to the spectrum as employed inRef. [29]. One finds that f ( p ) + 1 / T / ( ω ( p ) − µ ) withtemperature parameter T , frequency ω ( p ) = (cid:112) p + m ,(squared) mass parameter m = λ (cid:10) ϕ (cid:11) + δm and chem-ical potential µ .We then extend these studies to larger lattices up -0.04-0.02 0 0.02 0.04 3 3.5 4 4.5 5 f p / Q λ = 1 Λ / Q = 5.1 Q t = 70Q t = 190Q t = 350Q t = 700Q t = 2000 FIG. 8. The tail of the spectrum f at different times for latticecutoff Λ /Q = 5 . λ = 1. The gray line indicates f = 0. to the size 512 in order to also resolve the infraredphysics. In particular, this will enable us to vary theRayleigh-Jeans cutoff while being insensitive to finite vol-ume effects. Our results with the same lattice cutoff butlarger volume and mass renormalization are summarizedin Fig. 7. The spectra start with the occupied quantum1 /
2. The blue curves denote earlier times when the dis-tribution function grows with time. For Qt = 130, onefinds the resonance structure while at Qt = 6800 hardmomenta can already be fitted by a classical thermal dis-tribution, as indicated in Fig. 6.The entire spectrum can be described by a thermalfunction starting from the time Qt = 27000. Going toeven later times, we find that the classical thermal fit-ting function changes its parameters with time since theeffective mass decreases. Therefore, the temperature andthe chemical potential of the fitting function become timedependent, where T grows while µ decreases. The differ-ence between µ and m grows and the chemical poten-tial decreases faster than the effective mass. The valuesof the thermal fits denoted by dashed orange lines are T /Q = 1 . m/Q = 0 .
44 and µ/Q = 0 .
43 at Qt = 27000and T /Q = 1 . m/Q = 0 .
41 and µ/Q = 0 .
29 at thetime Qt = 81000.With classical equilibration in the total occupationnumber, the ultraviolet tail of the distribution function f ( p ) becomes negative. Stated differently, the vacuum‘quantum half’ decays [34]. This failure of the classical-statistical approximation to describe the otherwise stablevacuum of the quantum theory is illustrated in Fig. 8,where we zoom into the hard momentum region. Whilefor λ = 1 this unphysical decay of the ‘quantum half’ canalready be observed at rather early times Qt ∼ λ / φ / Q Λ / Q = . Λ / Q = . Λ / Q = . Λ / Q = . Λ / Q = . λ = 1 FIG. 9. The rescaled field amplitude as a function of time for λ = 1 and different lattice cutoffs. On the right hand side ofthe figure, a logarithmic time scale is used. The inset revealsa strong cutoff dependence of the results already at ratherearly times. classical-statistical setup is controlled by the Rayleigh-Jeans cutoff, one expects the results to become stronglycutoff dependent once the unphysical decay of quantummodes sets in. We will now compare the dynamics forvarious values of the cutoff Λ, with the mass renormal-ization as outlined above. Fig. 9 shows the field ampli-tude as a function of time. One finds that the condensatebecomes zero at a finite time while the decay time itselfcan be varied to practically any value in a vast range bychanging Λ. Since the decay happens faster the larger thecutoff, the prediction for Bose condensation of Ref. [29]has to be considered as an artifact of the employed reg-ularization.In Fig. 10 we show the spectra f ( t, p ) + 1 / Qt = 10to Qt = 5500 exhibit cutoff dependencies. The table withthe extracted fit parameters at Qt = 5500 reveals thatno sensible prediction independent of the cutoff can bemade. VII. CONCLUSION
Classical-statistical simulations provide an importanttool for ab initio descriptions of nonequilibrium quantumdynamics. However, they have a well defined range of va-lidity which is restricted to weak enough couplings. Sincethe coupling controls the relative size of the occupationnumber per mode as compared to the ‘quantum half’ andthe latter are cut off by the Rayleigh-Jeans regulator, thesizeable cutoff dependence indicates a breakdown of theclassical-statistical approach for larger couplings. For thesetup considered in the single component scalar field the-ory, we find that quantitative results can be obtained for0
Q t = 10 Λ / Q = 5.1 Λ / Q = 10.2 Λ / Q = 15.3 Λ / Q = 20.5 λ = 1 Q t = 100 Q t = 1000p / Q f + /
1 10
Q t = 5500
T / ( ω - µ ) T / Q m / Q µ / Q 1.8 0.57 0.51 3.8 0.53 -0.3210.6 0.49 -10.227.4 0.47 -40.0 FIG. 10. The occupation number distribution as a functionof momentum for λ = 1 and different lattice cutoffs. Resultsare given for four different times. The table in the inset showsthe dependence of the classical thermal fit parameters on theRayleigh-Jeans cutoff. λ (cid:46) . t (cid:28) Q − λ − / . In partic-ular, our analysis shows that the results of Refs. [29, 50]for λ = 1 are based on the application of the classical-statistical approximation beyond its range of validity.We have demonstrated that both the condensatedriven as well as fluctuation dominated initial conditionsbelong to the basin of attraction of the same nonthermalfixed point within the range of validity of the classical-statistical description. This leads to the phenomenonof turbulent thermalization, where the evolution is char- acterized by universality. Our results show the exis-tence of a dual cascade for the single component scalarfield theory. While this has been intensively studied for O ( N ) symmetric N -component field theories, with a de-tailed analytic understanding using large- N techniques,our analysis points out that these provide also a remark-ably accurate description of the universal properties for N = 1.It would be extremely valuable to have a similar anal-ysis for the range of validity of the classical-statisticalapproach describing the nonequilibrium quantum dy-namics of longitudinally expanding systems. The latterare crucial for our understanding of ultrarelativisticcollision experiments of heavy nuclei in the laboratory.Simulations in the theoretically clean weak couplinglimit demonstrate the existence of a nonthermal fixedpoint in the space-time evolution of non-Abelian plas-mas [15, 16]. Alternative thermalization scenarios withthe gauge coupling exceeding a certain strength [26] canbe analyzed along the lines of the present work. ACKNOWLEDGMENTS
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