We investigate the nonequilibrium evolution of a scalar field in (2+1)
dimensions. The field is set in a double-well potential in contact (open) or
not (closed) with a heat bath. For closed systems, we observe the synchronized
emergence of coherent spatiotemporal configurations, identified with oscillons.
This initial global ordering degenerates into localized order until all
oscillons disappear. We show that the synchronization is driven by resonant
parametric oscillations of the field's zero mode and that local ordering is
only possible outside equipartition. None of these orderings occur for open
systems.Comment: 4 pages, 5 figures, LaTeX, minor corrections to eqs. 1,3,

Gleiser, Marcelo

Howell, Rafael C.

English

arXiv

We investigate the nonequilibrium evolution of a scalar field in (2+1) dimensions. The field is set in a double-well potential in contact (open) or not (closed) with a heat bath. For closed systems, we observe the synchronized emergence of coherent spatiotemporal configurations, identified with oscillons. This initial global ordering degenerates into localized order until all oscillons disappear. We show that the synchronization is driven by resonant parametric oscillations of the field’s zero mode and that local ordering is only possible outside equipartition. None of these orderings occur for open systems

Howell, Rafael C

Dartmouth Digital Commons (Dartmouth College)

Dartmouth CollegeDartmouth Digital CommonsOpen Dartmouth: Faculty Open Access Articles12-23-2003Resonant Emergence of Global And LocalSpatiotemporal Order in a Nonlinear Field ModelMarcelo GleiserDartmouth CollegeRafael C. HowellDartmouth CollegeFollow this and additional works at: https://digitalcommons.dartmouth.edu/facoaPart of the Statistical, Nonlinear, and Soft Matter Physics CommonsThis Article is brought to you for free and open access by Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: FacultyOpen Access Articles by an authorized administrator of Dartmouth Digital Commons. For more information, please contactdartmouthdigitalcommons@groups.dartmouth.edu.Recommended CitationGleiser, Marcelo and Howell, Rafael C., "Resonant Emergence of Global And Local Spatiotemporal Order in a Nonlinear Field Model"(2003). Open Dartmouth: Faculty Open Access Articles. 1996.https://digitalcommons.dartmouth.edu/facoa/1996arXiv:hep-ph/0209176v3  8 Sep 2003Resonant emergence of global and local spatiotemporal order in a nonlinear fieldmodelMarcelo Gleiser∗ and Rafael C. Howell†Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA(Dated: March 19, 2018)We investigate the nonequilibrium evolution of a scalar field in (2+1) dimensions. The field is setin a double-well potential in contact (open) or not (closed) with a heat bath. For closed systems,we observe the synchronized emergence of coherent spatiotemporal configurations, identified withoscillons. This initial global ordering degenerates into localized order until all oscillons disappear.We show that the synchronization is driven by resonant parametric oscillations of the field’s zeromode and that local ordering is only possible outside equipartition. None of these orderings occurfor open systems.PACS numbers: 11.10.Lm, 05.45.Xt, 98.80.CqThe emergence of spatiotemporal ordered structuresin nonlinear systems is an ideal laboratory for investigat-ing the trend toward complexification observed in natureat the physical, chemical, and biological level [1]. Forthese ordered structures to survive, they must interactwith an external environment, which maintains the lo-cal nonequilibrium conditions. Examples can be foundin hydrodynamics, in networks of chemical reactions [2],and in living organisms [3].In field theory and cosmology, most of the interest inordered configurations has focused on topological or non-topological static solutions of the equations of motion [4].An exception to this trend are oscillons, long-lived time-dependent localized field configurations that have beenfound in field theory [5, 6], soft condensed-matter sys-tems [7], and stellar interiors [8]. Here, we show howoscillons spontaneously emerge as a nonlinear scalar fieldapproaches thermal equilibrium. We observe not only thelocal emergence of spatiotemporal order (oscillons), butalso that this emergence is initially synchronized (global).Our results are applicable to any system modeled by ascalar order parameter φ with amplitude-dependent non-linearities, if its potential V (φ) satisfies ∂2V/∂φ2 < 0 fora range of φ.Consider a (2+1) dimensional real scalar field φ(x, t),which evolves under the equation of motion∂2φ∂t2−∇2φ = −∂Vdw∂φ. (1)The double-well potential Vdw(φ) =14(φ2 − 1)2 has min-ima at φ = ±1. It has been shown that this system cangenerate oscillons, characterized by a persistent oscilla-tory behavior at their core [5]. To see this, prepare thefield with a Gaussian profile, φ(r) = φa exp(−r2/R2)−1,and let it evolve via Eq. (1). All that is needed is thatV′′dw(φa) < 0 and that the initial radius R be larger thana bifurcation value Rosc [5, 9]. The properties of these so-called deterministic oscillons have been extensively stud-ied in two [9, 10] and three dimensions [5, 11].We investigate open (canonical) and closed (micro-canonical) systems. In both cases, the field is initiallythermalized in a single-well potential Vsw(φ) = (φ + 1)2,symmetric about φ = −1, chosen so that V′′sw(−1) =V′′dw(−1). The thermalization is achieved by coupling thefield to an external heat bath via a Langevin equation∂2φ∂t2+ γ∂φ∂t−∇2φ = −V′sw(φ) + ξ, (2)where the viscosity coefficient γ is related to thestochastic force of zero mean ξ(x, t) by the fluctuation-dissipation relation (kB = 1 and T is the temperature ofthe heat bath), 〈ξ(x, t)ξ(x′, t′)〉 = 2γT δ2(x−x′)δ(t− t′).The numerical evolution is implemented on a square lat-tice with periodic boundary conditions, using a stag-gered leapfrog method. Integrations are performed withδx = 0.1 and δt = 0.01 with 1024 lattice sites per side.The coupling to the heat bath continues until equiparti-tion is satisfied at temperature T . Results are ensembleaverages from 50 independent realizations, which, giventhe large number of degrees of freedom, have energy Ewithin ∆E/E < 10−3.The potential is then switched from Vsw to Vdw, andthe system is tossed again out of equilibrium. This switchcan be interpreted as an instantaneous quench on thesystem, implemented by varying some control param-eter α, as is customarily done in many applications.To see this, shift φ → φ + 1 so that Vsw = φ2 andVdw = 1/4[φ2(αφ − 2)2|α=1. Note that when α = 0,Vdw = Vsw. The quench then occurs instantaneously bysending α = 0 to α = 1 (or equivalently, Vsw → Vdw).For the closed system the coupling to the bath is re-moved at the time of the switch (by setting γ = 0), whilefor the open system it is kept on throughout the simula-tion. After the switch occurs, energy exchange betweenthe nonlinearly coupled modes will again drive the systemto equipartition. (We note that the initial and final equi-librium temperatures never differ more than ∼ 0.15%.Thus, each experiment is referred to by its initial ther-malization temperature.) For open systems, equilibriumis attained within a time scale of O(γ−1), without any2FIG. 1: Bottom: lattice-averaged field for various tempera-tures. Top: fraction of the field above the inflection point.emergence of ordered configurations. This is in markedcontrast with closed systems.Figure 1 (lower half) shows the area-averaged fieldφav(t) for closed systems at various initialization temper-atures. The time is set so that the switch to the double-well potential occurs at t = 0. Note that φav(t) displaysdamped oscillations, indicating that the zero mode of thefield transfers its energy to higher modes until it reachesits temperature-dependent final equilibrium value 〈φ〉T .At low temperatures, such as T = 0.1, the oscillationshave small amplitude and remain nearly periodic. Athigher temperatures, such as T = 0.2, φav(t) oscillatesbeyond the left inflection point and fluctuations in thefield probe the unstable region of the double-well poten-tial. This can also be seen in the upper half of Fig. 1,where the time evolution of the fraction of the field abovethe inflection point finf(t) is shown. For temperaturesabove T ≃ 0.185, over half the field probes the unsta-ble region. For even larger temperatures, T & 0.25, thewhole field goes above the inflection point, signaling theapproach to criticality. Above this temperature the fieldseparates into large, slowly evolving thin-walled domains.(The critical temperature of this system, where φav → 0,is Tc ≃ 0.270± 0.005.)Figure 2 shows a sequence of snapshots of the field atT = 0.2, spanning in time about half an oscillation pe-riod of an oscillon. In order to generate this figure andrelate the emergent configurations to oscillons, the fieldis smoothed with an optimal (Wiener) filter [12]. This fil-tering technique is especially useful in our situation, sinceits implementation requires knowledge of the unwantedthermal noise in the system. With our choice of thermalinitial conditions, Boltzmann statistics provides us withtFIG. 2: Snapshots displaying synchronous emergence of os-cillons in the 2-dimensional field, starting at t = 15.5 andspanning half the period of an oscillon. Simulations can beviewed in Ref. [14].the power spectrum at the beginning of the experiment,〈|φ(k, t = 0)|2〉 = T/(k2 + 2), which is to be removed bythe filter. Throughout the experiment, only modes with0 < |k| . 0.8 amplify above the noise, sometimes by asmuch as two orders of magnitude. Their spectrum hasa shape and width that coincide with those of determin-istic oscillons. The transformation back into real spaceclearly reveals the emergence of localized field configura-tions within a smooth background.With the filtered field, we can catalog and track thelocation of all local extrema at each instant in time, thevast majority of which are seen to correspond to the cen-ters of localized configurations. We then measure thevalue of the field at each extrema and the correspond-ing configuration radius (at half maximum). With am-ple sorting, we compile a library containing each large-amplitude fluctuation during the entire evolution of thesystem (0 < t < 1500), from which we can obtain theirnucleation times, sizes, periods of oscillation, and life-times. A deterministic oscillon is characterized bothby its large-amplitude oscillations (the field at its coreprobes the positive half of the potential, φa > 1) and itsextreme longevity. We thus establish two criteria to se-lect the subset of all configurations which correspond tooscillons: they must have a maximum amplitude abovethe background satisfying φa > 1 and they must survivefor at least ten oscillations (t > 60).The inset in Fig. 3 shows the probability distributionfunctions for the radii R and periods of oscillation P forall the oscillons present throughout the simulations, withbinwidth δR = δP = 0.1. The fitted curves are Gaussian3FIG. 3: The number of oscillons nucleated between t and t+δtat T = 0.2, with δt = 1. The global emergence is evident earlyin the simulations. Inset: the probability distribution of radiiand periods of oscillation.functions with centers R0 = 2.86 and P0 = 6.12 andwidths σR = 0.33 and σP = 1.19, respectively.In Fig. 3 we also show the distribution of nucleationtimes for these oscillons at T = 0.2. This function givesthe number of oscillons nucleated between t and t + δt,with δt = 1. The sharp peaks at early times, t < 60, cor-respond to the synchronous emergence of oscillons, whilefor t > 60 this global ordering gives way to only localordering, in which oscillons emerge at arbitrary timeswith similar probability. Even this local emergence dis-appears after approximately t & 500 (discussed below).Notice the correlation between the nucleation activity ofoscillons and the energy loss from the zero mode (Fig.1) at early times. This is observed for all temperatureswithin 0.185 . T . 0.25.To understand the origin of this synchronous emer-gence, we decompose the field as φ(x, t) = φav(t) +δφ(x, t) and first investigate the behavior of φav(t), priorto relinquishing its energy to higher modes, by approx-imating δφ(x, t) as remaining statistically equivalent toits initial symmetric state. Upon substituting this formof φ(x, t) into Eq. (1) and performing a statistical av-erage over the fluctuations, we arrive at the mean-fieldequation of motion for φav(t): φ¨av = [1−3〈δφ2〉]φav−φ3av,where 〈δφ2〉 is the two-point correlation function at t = 0and depends linearly on the temperature T . Thus, φav(t)starts at φ = −1 and oscillates anharmonically in theleft-hand well of an effective double-well potential, withminima at φ± = ±√1− 3〈δφ2〉. This mean-field ap-proximation works very well at describing the evolutionof φav(t) at temperatures sufficiently far from Tc (c.f.FIG. 4: Lines of constant amplification rate for small-amplitude modes at various temperatures, beginning withη−1 ≃ 27 for the bottom-most contour and increasing in in-crements of δη = 0.05.T = 0.1 in Fig. 1). It also gives 〈φ〉T ≈ φ− to within 4%even at the highest temperatures we considered.We now examine the behavior of small fluctuationsabout φav(t). Linearizing Eq. (1) with respect to δφ(x, t)and taking the Fourier transform we obtain (for k > 0)δ¨φ(k, t) +[k2 + V′′dw (φav(t))]δφ(k, t) = 0 . (3)Equations of this type, generalized Mathieu equations,are known to exhibit parametric resonance, which canlead to exponential amplification (∼ exp ηt) in the os-cillations of δφ(k, t) at certain wavelengths, in responseto the time-dependent harmonic term. These equationshave been of great interest in reheating studies of infla-tionary cosmologies [13]. To verify that this is the mech-anism behind the synchronous amplification of oscillonmodes early in the simulations, we use in Eq. (3) the an-harmonic solution for φav(t) obtained in the mean-fieldapproximation above.Figure 4 shows lines of constant amplification rate ηof the fluctuations δφ(k, t) for various k and T . Atlow temperatures, T < 0.165, no modes are ever am-plified. As the temperature is increased, so is the am-plitude of oscillation in φav(t), eventually causing theband 0.6 . k . 0.9 to resonate. We note that thecharacteristic wavenumber of oscillon configurations iskosc = 2/R0 [5]. Using the result in Fig. 3 for the aver-age configuration radius R0 = 2.86, we obtain kosc ≃ 0.7.From Fig. 4, modes with kosc ≃ 0.7 are excited forT & 0.18, the temperatures above which we see thesynchronized emergence of oscillons. For T > 0.22, a4FIG. 5: The change of Π(t) from the initial state for closedsystems at T = 0.2. The exponential approach to equilibriumis clear at late times. The inset illustrates the role of oscillonsas a bottleneck to equipartition.second band of longer wavelength modes becomes ex-cited as well, signaling the onset of criticality. Thesegeneral results corroborate our earlier findings: for tem-peratures 0.185 . T . 0.25, oscillations of the zeromode drive, via parametric resonance, amplification ofthe modes comprising oscillon configurations, ultimatelyleading to the usual breakdown of the linear approxima-tion. The growth of these modes corresponds to large-amplitude fluctuations that probe the unstable regions ofthe potential (∂2V/∂φ2 < 0), which coordinate to formthe observed coherent structures.Finally, we introduce a measure of the partitioning ofthe kinetic energy Π(t), which we use to describe thenonequilibrium evolution of the system:Π(t) = −∫d2k p(k, t) ln p(k, t), (4)where p(k, t) = K(k, t)/∫d2kK(k, t) and K(k, t) is thekinetic energy of the kth mode. Π(t) attains its maxi-mum [Πmax = ln(N) on a lattice with N degrees of free-dom] when equipartition is satisfied. This occurs bothat the initial thermalization (t = 0) and final equilib-rium states, since in this case all modes carry the samefractional kinetic energy. In Fig. 5 we show the changeof Π(t) from the initial state, Π(t = 0) − Π(t), for theclosed system at T = 0.2. At late times (t & 150), wehave found that the system equilibrates exponentially ina time scale τ ≃ 500. At early times, the localizationof energy at lower k modes, corresponding to the globalemergence of oscillons, prolongs this approach to equipar-tition. The inset of Fig. 5 shows the large variations inΠ(t) (dotted line) that arise due to the synchronous os-cillations in the kinetic energy of these configurations.Also shown (solid line) is the average between successivepeaks of Π(t), with a plateau at 20 . t . 70 that coin-cides with the maximum oscillon presence in the system.Thus, oscillon configurations serve as early bottlenecksto equipartition, temporarily suppressing the diffusion ofenergy from low (0 < |k| . 0.8) to higher modes.We have investigated the nonequilibrium evolution ofa scalar field with a double-well potential. For a range oftemperatures, the approach to equilibrium is character-ized by three stages: first, the synchronized emergence ofoscillons; second, the loss of the initial synchronicity, butthe persistence of oscillons; and third, their disappear-ance as the system approaches equipartition. It wouldbe interesting to investigate the possibility of controllingthe duration of the synchronization stage and search forthis emergent behavior in laboratory systems, rangingfrom vibrating grains to ferromagnetic materials.MG was supported in part by NSF grants PHY-0070554 and PHY-0099543.∗ Electronic address: mgleiser@dartmouth.edu† Electronic address: rhowell@dartmouth.edu[1] D. Walgraef, Spatio-Temporal Pattern Formation(Springer, New York, 1997).[2] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65,851 (1993).[3] N. M. Shnerb, Y. Louzoun, E. Bettelheim, and S.Solomon, Proc. Natl. Acad. Sci. U.S.A. 97, 10 322 (2000).[4] R. Rajamaran, Solitons and Instantons (North-Holland,Amsterdam, 1987); T. D. Lee and Y. Pang, Phys. Rep.221, 251 (1992).[5] M. Gleiser, Phys. Rev. D 49, 2978 (1994); E. J. Copeland,M. Gleiser, and H.-R. Muller, ibid. 52, 1920 (1995).[6] D. Bettison and G. Rowlands, Phys. Rev. E 55, 5427(1997); C. Crawford and H. Riecke, ibid. 65, 066307(2002).[7] P. 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Resonant emergence of global and local spatiotemporal order in a nonlinear field model

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