Self-Organized Criticality, effective dynamics and the universality class of the Deterministic Lattice Gas
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Self-Organized Criticality, effective dynamics and the universality class of theDeterministic Lattice Gas
Andrea Giometto , ∗ and Henrik Jeldtoft Jensen † Blackett Laboratory, Department of Physics and Complexity & Networks Group,Imperial College London, London, SW7 2AZ, UK Dipartimento di Fisica G. Galilei, Universit`a di Padova, Via Marzolo 8, I-35151 Padova, Italy and Complexity & Networks Group and Department of Mathematics ,Imperial College London, London, SW7 2AZ, UK
We show that the Deterministic Lattice Gas (DLG) (Phys. Rev. Lett. , 3103 (1990)) model ofSelf-Organized Criticality (SOC) despite of its deterministic micro dynamics belongs to the Mannauniversality class of absorbing state phase transitions. This finding is consistent with our observationthat an effective stochastic term is generated in the DLG at large length scales, whereby the macrodynamics of the DLG appears closer to the other stochastic SOC models of the Manna class. It is well known that Self-Organized Criticality (SOC)was introduced by Bak, Tang and Wiesenfeld (BTW) asan attempt to explain the widespread occurrence of 1 /f temporal fluctuations and fractal spatial structure[1, 2].It was, however, soon realized that the model used byBTW did not contain a 1 /f spectrum[2, 3]. The De-terministic Lattice Gas (DLG) was introduced in an ef-fort to find another deterministic model, which couldbe used as a proof of existence for the SOC scenario.The model being deterministic is difficult to analyze an-alytically and one has to rely on a combination of sim-ulations and non-rigorously justified effective analyticinvestigations[4]. Numerical simulations found that thedensity fluctuations in the DLG did exhibit 1 /f fluctua-tions (and that dissipation did take place on a fractal).This was found to be consistent with analysis of (non-linear) diffusion equations for which one assumes the ab-sence of any bulk noise term[4].Until now it has been unclear how the DLG relatedto the universality classes of SOC[5, 6] and how the ex-istence of an absorbing state at low density influencesthe behavior of the model. In this letter we first presentsimulations which demonstrate that the density fluctua-tions at elevated densities for large systems have a powerspectrum S ( f ) ∝ /f µ with µ = 3 /
2, which is consistentwith a description in terms of diffusion equation with abulk noise term. Next we present compelling evidencethat the transition to an absorbing state is of the Mannauniversality class and use scaling relations to determinethe value of the power spectrum exponent µ for densitiesnear the absorbing state phase transition. Model –
In the DLG particles interact through anearest-neighbor repulsive central unit force and doubleoccupancy is not permitted. All particles are updatedsimultaneously by moving the particles deterministically to neighbor sites according to the vector sum of the forcesthey are subject to. If two particles want to move to the ∗ Electronic address: [email protected] † Electronic address: [email protected] same site, the particle subject to the strongest force ismoved, while in the case of equal forces no particle isdisplaced. Periodic boundary conditions are consideredand we consider the model in dimension d = 2.The number of particles N ( t ) in a sub-volume of thelattice exhibits interesting temporal fluctuations. We de-termine the power spectrum S ( f ) of N ( t ) from the squareof the absolute value of the Fourier transform. Successivetime sequences are averaged in order to achieve sufficientstatistics: S ( f ) = 12 πT *(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T X t =1 N ( t ) e − i πft (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (1)The angular brackets denote averaging over many differ-ent time series.The spectrum S ( f ) was shown[7] to satisfy S ( f ) ∼ /f µ at the particle density ρ = 0 .
3, while previous worksshowed that the same result is obtained in a wide range ofdensities with a drive at the boundary[8]. In those papersthe maximum linear system size considered is L = 128,we will see that different behaviors are observed for largersystems. In Fig. 1 we show how the effective exponent µ changes from µ = 1 to µ = 3 / L increases. It isinteresting to relate this change in the temporal fluctua-tions to the appropriate macroscopic equation of motionfor the particle density ρ ( r , t ).Simulations show that although particles move deter-ministically in the DLG model they do behave like ran-dom walkers in the sense that their square displacementis linear in time. This suggests that ρ ( r , t ) evolves ac-cording to a diffusion equation and integrating the diffu-sion equation allows us to extract the temporal evolutionof N ( t ) = R V d r ρ ( r , t ), where V denotes the measuringsub-volume. Assume that ρ ( r , t ) satisfies the diffusion ∂ρ∂t = D ∇ ρ + ξ. (2)It has been shown[2, 4] that the power spectrum of N ( t )depends on the details of the diffusion equation. If theequation is driven by white noise at the boundary of V and the noise term ξ is absent the power spectrum S ( f ) f Slope -1Slope -1.5L=32L=64L=250L=1000 FIG. 1: Scaling behavior of the spectrum S ( f ) ∝ f − µ of thetotal number of particles N ( t ) in the DLG for increasing linearsizes of the lattice L . A crossover from µ = 1 for small L to µ = 1 . L is observed. Particle density ρ = 0 . exponent derived from Eq. (2) is µ = 1. If in contrasta conservative bulk noise term is included in Eq. (2)one obtains instead µ = 3 /
2. Including non-linearitiesin Eq. (2) will not influence the behavior of µ , see [4].It is this observation that allows us to conclude that aconservative noise term is generated in the macroscopicLangevin description when the sub-volume V becomessufficiently large. S ( f ) scaling for large lattices – We now consider theDLG for much larger lattices than previously studied andour aim is to understand the fluctuations spectrum asfunction of density. We plot S ( f ) in Figure 2 and findthat the value of µ depends on the density: S ( f ) ∼ ( f − . ρ ≥ . f − . ρ ≃ ρ c = 0 .
245 (3)where ρ c is the critical density of the absorbing phasetransition (APT in the following).The observed exponent at high densities is the sameof a gas of random walkers [9], while at low densities thepower spectrum scaling exponent is determined by thecritical properties of the DLG at the APT, as we shalldiscuss in the following. Absorbing Phase Transition –
At very low densitiesthe DLG enters configurations in which all particles arefar away from each other and, due to the short rangeinteractions, the particles become unable to move, i.e.the dynamics is frozen. Near the transition the densityof active particles ρ a behaves according to ρ a ( δρ ) ∼ δρ β for ρ > ρ c and ρ a ( δρ ) = 0 for ρ < ρ c with δρ = ρ − ρ c .We determine the critical density ρ c from a log-log plotof ρ a versus δρ and obtain ρ c = 0 . S ( f ) - A r b i t r a r y un i t s fSlope -1.8Slope -1.5 ρ =0.24506 ρ =0.27 ρ =0.3 ρ =0.5 FIG. 2: Scaling behavior of the spectrum S ( f ) ∝ f − µ of thetotal number of particles N ( t ) in the DLG for different par-ticle densities ρ . A crossover from µ ≃ . ρ ≃ ρ c = 0 . µ = 1 . ρ ≫ ρ c is observed. S ( f ) has been multipliedby different constants for different densities ρ to visualize thescaling exponents properly. Lattice size L = 1000. β = 0 . ρ a = L (cid:2) h ρ a i − h ρ a i (cid:3) ∼ δρ − γ ′ with γ ′ = 0 . hL particles on the latticeand move each of them to one of its empty neighbors.The external field prevents the system from falling intoan absorbing state. We assume that the order parameterand its fluctuations satisfy ρ a ( δρ, h ) = λ − β ˜ R ( δρλ, hλ σ )and ∆ ρ a ( δρ, h ) = λ γ ′ ˜ D ( δρλ, hλ σ ), where λ >
0. Fromthis scaling ansatz we conclude that ρ a ( δρ = 0 , h ) ∼ h β/σ and ∆ ρ a ( δρ = 0 , h ) ∼ h − γ ′ /σ . Our simulations lead to σ = 2 . γ ′ = 0 . λ = h − /σ :see Fig. 3 for ∆ ρ and [12] for ρ a and more details. Finite Size Scaling –
We assume the fol-lowing finite size scaling form ρ a ( δρ, h, L ) = λ − β ˜ R pbc ( δρλ, hλ σ , Lλ − ν ⊥ ) and ∆ ρ a ( δρ, h, L ) = λ γ ′ ˜ D pbc ( δρλ, hλ σ , Lλ − ν ⊥ ), where the exponent ν ⊥ describes the divergence of the spatial correlation length,i.e., ξ ⊥ ∝ δρ − ν ⊥ . The universal scaling functions dependon the particular choice of boundary conditions, althoughin the thermodynamic limit ˜ R pbc ( x, y, ∞ ) = ˜ R ( x, y ) and˜ D pbc ( x, y, ∞ ) = ˜ D ( x, y ).Following [5] we consider the fourth order cumulant Q , which is defined as: Q = 1 − h ρ a i h ρ a i . For non-vanishing order-parameter the cumulant tends to Q =2 / Q ( δρ, h, L ) =˜ Q pbc ( δρλ, hλ σ , Lλ − ν ⊥ ). Choosing λ = L /ν ⊥ at δρ = 0 h - γ ‘ / σ ∆ ρ a ( δ ρ , h ) δρ h -1/ σ h=2 10 -5 h=2.8 10 -5 h=4 10 -5 h=4.8 10 -5 ∆ ρ a ( δ ρ , h ) δρ FIG. 3: Data collapse of ∆ ρ a . In the inset data before thecollapse are shown. Lattice linear size L = 1000. we obtain the following equation: Q (0 , h, L ) = ˜ Q pbc (0 , hL σ/ν ⊥ ,
1) (4)which enables us to determine ν ⊥ through a data collapseby plotting Q (0 , h, l ) against hL σ/ν ⊥ as in Fig. 4. Bestresults are obtained for ν ⊥ = 0 . Q ( , h , L ) h L σ / ν Infinite system Q=2/3L=1000L=500L=250 0.64 0.65 0.66 0.67 1e-06 0.0001 0.01 Q ( , h , L ) hInfinite system Q=2/3 FIG. 4: (lin-log) Data collapse of Q (0 , h, L ) for the determi-nation of ν ⊥ . In the inset data before the collapse are shown. Dynamical scaling –
Starting simulations of the DLGfrom a random distribution of particles above the crit-ical point ρ c the density of active sites ρ a decreases intime and tends to its steady state value. At the criticalpoint ρ = ρ c the order parameter decays algebraically as ρ a ( δρ = 0 , h = 0 , t ) ∼ t − α . Simulating a lattice of linearsize L = 4000 we obtain α = 0 . ρ a ( L, t ) = L − αz ˜ R ′ pbc ( tL − z , z = ν k /ν ⊥ .The best data collapse is obtained for z = 1 . DLG universality class –
We are now able to addressthe question concerning which universality class the DLG belongs to. In table I we compare the measured valuesof the DLG critical exponents with those of the Mannauniversality class[13], showing that they are compatiblewith each other - for further details see [12]. The shape β ν ⊥ ν k σ Manna 0.639(9) 0.799(14) 1.225(29) 2.229(32)DLG 0.634(2) 0.83(5) 1.2(1) 2.19(1) γ ′ γ α z Manna 0.367(19) 1.590(33) 0.419(15) 1.533(24)DLG 0.37(1) 1.54(1) 0.41(1) 1.5(1)TABLE I: The measured critical exponents for the DLG andthe corresponding critical exponents for the Manna universal-ity class in d = 2 [13]. of the universal scaling functions is remarkably similar tothose that can be found in the literature for the Mannauniversality class. This is seen e.g. by comparing ourFig. 3 with Fig. 4 in [10], see also [13]. From the scalingexponents and the scaling functions we find compellingevidence that the DLG belongs to the Manna universalityclass. Power spectrum at criticality –
Finally we relate theexponent of the power spectrum at low density to thescaling properties of the DLG at the critical point ρ c of the APT[14]. The scaling behavior of the correlationfunction C a ( r , t ) = h ρ a ( r , t ) ρ a (0 , i − h ρ a (0 , i at thecritical density ρ = ρ c determines the power law exponentin the scaling of the spectrum S a ( f ) of the total numberof active particles N a ( t ) in the lattice. Assume C a ( r , t ) = λ − η ˜ C a ( λρ, λ − ν k t, λ − ν ⊥ r ) (5)In stationary directed percolation processes above criti-cality it has been observed[13] that the correlation func-tion C a ( r , t ) at t = 0 first decays in space algebraically as r − β/ν ⊥ , until it saturates at a constant value at r > ξ ⊥ .In the saturated regime the two sites become uncorre-lated so that this value is just equal to the squared sta-tionary density of active sites ρ a . One can use this factto extract the scaling exponent for S a ( f ), making use ofthe Wiener-Khinchin theorem: C a ( t ) = h N a ( t ) N a (0) i − h N a (0) i = (cid:28)Z V d r Z V d r ′ ρ a ( r , t ) ρ a ( r ′ , (cid:29) + . . . = Z V d r C a ( r , t ) + · · · ∼ t /z (2 − β/ν ⊥ ) so that: S a ( f ) = 12 π Z dt C a ( t ) e − i πft ∼ f − − z (cid:16) − βν ⊥ (cid:17) ∼ f − µ with µ = 1 + z (cid:16) − βν ⊥ (cid:17) . With our estimates of thecritical exponents for the DLG we find µ = 1 . S ( f ) , S a ( f ) f Slope -1.8Total number of particles spectrum S(f)Total number of active particles spectrum S a (f) FIG. 5: DLG: Scaling behavior of the spectrum S ( f ) ∝ f − µ of the total number of particles in the box N ( t ) and of thespectrum S a ( f ) of the total number of active particles nearthe critical density ρ = 0 . ≃ ρ c . The spectra have beenmultiplied by arbitrary factors to visualize the scaling expo-nent properly. Lattice size L = 1000. while with the best estimates of the Manna universal-ity class exponents [13] one finds µ = 1 . N ( t ) in the lattice are triggered by activeparticles, therefore we expect the two spectra S ( f ) = π R dt N ( t ) e − i πft and S a ( f ) = π R dt N a ( t ) e − i πft to show the same scaling behavior at ρ ≃ ρ c , as it isconfirmed by simulations (see Fig. 5). Computing thepower spectrum of the total number of particles in the box at ρ = 0 . ≃ ρ c we observe a power spectrumcompatible with the predictions, confirming that fluctua-tions in the total number of particles in the sub-volume V are determined by the scaling behavior of the correlationfunction at criticality. Summary –
We have investigated the scaling prop-erties of the power spectrum S ( f ) of the total numberof particles in a sub-volume of the lattice for the DLG:our simulations show that the DLG is not characterizedby 1 /f fluctuations as had been previously observed, butpresent a much more variegated picture. At high densi-ties the power spectrum is the same as observed in a gasof random walkers, while at low densities the spectrumscales as S ( f ) ∼ /f . and we have shown that this ex-ponent is determined by the decay of the density-densitycorrelation function near criticality.We have shown that the deterministic lattice gas is inthe same universality class as the Manna model[15]. Toour knowledge this is the first example of a completelydeterministic and non-chaotic system in this universal-ity class. Perhaps this is not too surprising given thatwe found that a stochastic bulk noise is generated inthe effective Langevin description as the system size isincreased. From this point of view the DLG becomessimilar to the other members of the Manna class, whichall involved a stochastic element in their microscopicdynamics. Examples of models in the Manna class in-cludes the Oslo model[16] and the Manna model[15]. Inthe Oslo model the local threshold for relaxation is up-dated stochastically at every relaxation and in the Mannamodel particles move to stochastically chosen neighborsites. [1] P. Bak, C. Tang, K. Wiesenfeld, in Physical Review Let-ters 59, 381:384 (1987)[2] H.J. Jensen, “Self-Organized Criticality. Emergent Com-plex Behavior in Physical and Biological Systems”, Cam-bridge Lecture Notes in Physics, ISBN 0-521-48371-9(1998)[3] H.J. Jensen, K. Christensen, H.C. Fogedby, in PhysicalReview B 40, 7425:7427 (1989)[4] G. Grinstein, T Hwa, H.J. Jensen, in Physical Review A45, R559:R562 (1992)[5] S. L¨ubeck, P.C. Heger, in Physical Review E 68, 056102(2003)[6] Gunner Pruessner, “Self-Organized Criticality. Theory,Models and Charaterisation”, Cambridge UniversityPress, ISBN 9780521853354 (2011)[7] T. Fiig, H.J. Jensen, in Journal of Statistical Physics 71,653:682 (1993)[8] H.J. Jensen, in Physical Review Letters 64, 3103:3106(1990)[9] J.V. Andersen, H.J. Jensen, O.G. Mouritsen, in Physical Review B 44, 439:442 (1991)[10] S. L¨ubeck, in Physical Review E 65, 046150 (2002)[11] The error on β/σ is computed as the dispersion on itsvalue when measured at ρ c, = 0 . ρ c, =0 . ρ c = 0 ..