Scaling of hysteresis loops at phase transitions into a quasiabsorbing state
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Scaling of hysteresis loops at phase transitions into a quasiabsorbing state
Kazumasa A. Takeuchi ∗ Department of Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan andService de Physique de l’ ´Etat Condens´e, CEA-Saclay, 91191 Gif-sur-Yvette, France (Dated: November 30, 2018)Models undergoing a phase transition to an absorbing state weakly broken by the addition ofa very low spontaneous nucleation rate are shown to exhibit hysteresis loops whose width ∆ λ depends algebraically on the ramp rate r . Analytical arguments and numerical simulations showthat ∆ λ ∼ r κ with κ = 1 / ( β ′ +1), where β ′ is the critical exponent governing the survival probabilityof a seed near threshold. These results explain similar hysteresis scaling observed before in liquidcrystal convection experiments. This phenomenon is conjectured to occur in a variety of otherexperimental systems. PACS numbers: 05.70.Jk, 05.70.Ln, 05.20.-y
Directed percolation (DP) is an archetypical model ofphase transitions into an absorbing state, i.e. a state fromwhich a system can never escape. A vast literature oftheoretical and numerical studies has enlarged the rangeof phenomena in the DP universality class [1], refiningconditions for this prominent critical behavior, knownas DP conjecture [1, 2, 3]. Experimentally, the authorand coworkers recently found that electrohydrodynamicconvection of nematic liquid crystal shows the scalingbehavior of DP at the transition between two turbulentstates (DSM1-DSM2) [4]. Applying voltages V closelyabove the threshold, spatiotemporal intermittency (STI)occurs, in which DSM2 patches move around in a DSM1background. As conjectured early by Pomeau [5], thisSTI was unambiguously mapped onto DP with DSM1playing the role of the absorbing state. This constituteda clear experimental realization of a DP-class absorbingphase transition.On the other hand, Kai et al. reported in 1989 hys-teresis phenomena around this DSM1-DSM2 transition[6]. Measuring the global light transmittance throughthe sample, increasing or decreasing the applied voltage V at a rate r , they found hysteresis loops of width ∆ V scaling roughly like ∆ V ∼ r κ with κ ≈ . . r limit, andit has been discussed whether the transition correspondsto a supercritical bifurcation or a subcritical one. This isin apparent contradiction with DSM1 being an absorb-ing state, since then one expects infinitely wide hysteresisloops. It is shown here that the scaling of hysteresis loopsis in fact in full agreement with the DP framework inwhich the DSM1 state is only quasi-absorbing, i.e. withthe existence of a small residual probability for sponta-neous nucleation of DSM2 patches either in the bulk orat the boundaries.As a first illustration, a probabilistic cellular automa-ton (PCA) version of the contact process (CP) [1, 8] is in- ∗ Electronic address: [email protected] troduced, in which an extra, small probability h to createan active site spontaneously anywhere is added. Considera two-dimensional (2D) square lattice of size L × L andassign a variable s i,j to each lattice point, encoding itslocal state, either inactive (absorbing, s i,j = 0) or active( s i,j = 1). Indices i and j denote Cartesian coordinates.The time evolution is as follows: randomly choose onesite and stochastically flip it with probabilities p i,j (0 →
1) = p s i − ,j + s i +1 ,j + s i,j − + s i,j +1 ) + h,p i,j (1 →
0) = p , (1)where p = λ/ ( λ + 1) and p = 1 / ( λ + 1). The twoterms in the first equation account for contamination byneighbors and spontaneous nucleation of active sites, re-spectively. Periodic boundary conditions s i,j = s i + L,j = s i,j + L are used throughout, and a time step (or MonteCarlo step, MCS) consists of L flipping attempts. The h = 0 case is known as the PCA version of the orig-inal (2+1)D CP, which shows a DP-class transition at λ c = 1 . L = 256 and h ′ ≡ hL = 10 − . Although, strictly speak-ing, even rare nucleation events wipe out the absorbingphase transition, in practice a significantly low nucleationrate allows us to observe the underlying critical behav-ior as we shall see in this study. The nucleation rate h theoretically corresponds to an external field [10], so aweak-field case is dealt with here.The model behaves similarly to the turbulence of liq-uid crystals in many aspects. For instance λ ≫ λ c andinitial conditions of s i,j = 0 everywhere lead to a nu-cleus growth after sufficient time has passed, which faith-fully reproduces experiments. In particular, the modelexhibits hysteresis as shown in Fig. 1(a) and Movie S1[11] when λ is increased from λ < λ c to λ > λ c at aconstant ramp rate r and then decreased at the samespeed. The hysteretic process can be decomposed intothree stages as indicated in the bottom of Fig. 2. Letus start from the uniformly inactive state and increase λ . First, active clusters do not emerge even for λ > λ c FIG. 1: Hysteresis observed in simulations and experiments. (a) Hysteresis of (2+1)D CP with h ′ = 10 − . Black regionsdenote active sites. The control parameter λ is increased and then decreased in the range of 1 ≤ λ ≤ . r = 0 .
001 MCS − . The critical point for the model without nucleation is λ c = 1 . λ denote whether they are increasing or decreasing. (b) Hysteresis in the electrohydrodynamic convection, where the samecell as in Ref. [4] is used. The control parameter, applied voltage V , is ramped in the range of 22 V ≤ V ≤
75 V at the rateof r = 1 .
71 V / s with fixed frequency of 250 Hz. The critical voltage is V c ≈
35 V [4]. Darker regions correspond to DSM2, theactive state. Note that the global intensity and contrast are adjusted for the sake of clarity, and that DSM1 and DSM2 coexistin the two images at the lower left. λ c r = 1.0 x10 -3 r = 3.0 x10 -4 λ r = 2.0 x10 -4 r = 1.0 x10 -5 ∆λ * ∆λ ∆λ *t A c t i v e s i t e den s i t y , ρ (1) (2)(3) FIG. 2: (Color online) Typical hysteresis loops for four differ-ent ramp rates r in (2+1)D CP with h ′ = 10 − . Note that theratios of the four values of r are chosen to be approximatelythe same as in Fig. 1 of Ref. [6] to allow the comparison (seealso Note [12]). The hysteretic process can be decomposedinto three stages as indicated in the bottom figure. due to the very low nucleation rate (1st stage). How-ever, once a spontaneous nucleation occurs, the activenucleus grows and finally covers the whole system be-cause of λ > λ c (2nd stage). The density of active sites, ρ , saturates at the steady state value ρ steady ( λ ). On theother hand, when λ is decreased, the number of activesites decreases gradually and homogeneously contrary tothe growing process, approximately following ρ steady ( λ )(3rd stage). This strikingly resembles what is observed inthe liquid crystal experiments [Fig. 1(b), Movie S2 [11],Refs. [6, 12]]. Note that the observed hysteresis both inthe experiments and in the simulations is not a station-ary property of the system, as would imply a first ordertransition, but rather a dynamical effect owing to thesweep of the parameter.The dependence on the ramp rate r is shown in Fig.2, which is again very similar to the corresponding ex-periments [6, 12]. The widths of the hysteresis loops ∆ λ and ∆ λ ∗ , defined as in Fig. 2, clearly exhibit the powerlaw dependence ∆ λ, ∆ λ ∗ ∼ r κ [Fig. 3 (disks and trian-gles)], with κ = 0 . λ and κ = 0 . λ ∗ .Here the ranges of error correspond to 95% confidenceintervals in the sense of Student’s t. They are in goodagreement with the experimental value κ = 0 . . κ can also be derived only byassuming DP criticality with a very low probability forspontaneous nucleation. For absorbing phase transitions,the probability P ∞ with which an active site survivesforever grows algebraically as P ∞ ∼ ε β ′ for ε ≡ λ − λ c >
0, where β ′ constitutes one of the critical exponentscharacterizing these transitions. (Note that for the DPclass the so-called “rapidity” symmetry implies β ′ = β [1, 13], where β is the critical exponent corresponding tothe stationary active site density ρ steady .) Suppose ε isincreased linearly as ε ( t ) = rt and a nucleus appears andgrows at time t = T , and assume that the ramp rate r isso slow that the finite-time survival probability converges r ∆ λ -5 -4 -3 -2 -1 (x10 ) -3 κ = 0.61(1) κ = 0.56(3) κ = 0.64(5) FIG. 3: (Color online) Widths of the loops ∆ λ (disk), ∆ λ ∗ (triangle), and ∆ λ ∗ t (square) with respect to the ramp rate r , in the case of (2+1)D CP with h ′ = 10 − . The sym-bols and errorbars indicate means and standard deviations,respectively, of 50 independent runs. Dashed curves denotethe results of the fitting to the power law ∆ λ, ∆ λ ∗ , ∆ λ ∗ t ∼ r κ .The inset shows the same data in logarithmic scales. to P ∞ before the control parameter significantly changes,the following relation then approximately holds:1 ≈ Z T h ′ P ∞ ( ε ( t ))d t ∼ h ′ r β ′ T β ′ +1 , (2)and thus the width of the hysteresis is∆ λ ∗ t ≡ rT ∼ r / ( β ′ +1) . (3)It gives the exponent for the hysteresis as κ = 1 / ( β ′ +1) =0 . h ∆ λ ∗ t i based on theprobabilistic distribution. This more rigorous approachis also straightforward. With P ( t ) being the probabil-ity that a nucleus does not appear and grow until time t , the probability that such a nucleation first occurs be-tween time t and t + d t is written as − d P ( t ) = P ( t ) · h ′ P ∞ ( ε ( t ))d t = Ch ′ r β ′ t β ′ exp − Ch ′ r β ′ β ′ + 1 t β ′ +1 ! d t, (4)where C is defined by P ∞ = Cε β ′ . This gives the averageof the hysteresis width as h ∆ λ ∗ t i = r Z ∞ t (cid:18) − d P ( t )d t (cid:19) d t = Γ (cid:18) β ′ + 2 β ′ + 1 (cid:19) (cid:20) ( β ′ + 1) rCh ′ (cid:21) / ( β ′ +1) , (5)which confirms Eq. (3). Note that the standard deviationalso obeys the same power law (with a different coeffi-cient), since the stochastic process at play is essentiallyPoissonian. TABLE I: Hysteresis exponent κ for several models.Exponent κ forModel a ∆ λ ∆ λ ∗ ∆ λ ∗ t / ( β ′ + 1)(2+1)D CP (PCA) 0 . . . . b (2+1)D CP 0 . . . . b (1+1)D CP (PCA) 0 . . . . b (1+1)D site DP c . . . . b (2+1)D voter-like de . . . . a System sizes and nucleation rates are set to L = 4096 and h ′ =10 − for (1+1)D, and to L = 256 and h ′ = 10 − for (2+1)D. b Values of the DP exponent β ′ are from Ref. [15] for (1+1)D andfrom Refs. [14] for (2+1)D. c Simulations are performed in the Domany-Kinzel lattice. d Kinetic Ising model with spin-flip probability p sH is considered,where s = ± H ∈ {− , − , , , } denote a spin and its localfield, respectively. p − is swept here with the other parametersfixed at p = h = 10 − /L , p = 0 . , p = 0 . , p − = 0 .
68. Thismodel shows a transition in the voter universality class [16]. e Hysteresis is measured in terms of the density of interfaces (i.e.,the fraction of + − pairs) instead of the active site density (i.e.,magnetization), since the former characterizes the voter class better[16] and shows faster relaxation. The derived value of κ = 0 . λ and ∆ λ ∗ ), whereas, theoretically, the exact critical point λ c is used to define the lower bound. Adopting the latterdefinition for the simulations (∆ λ ∗ t in Fig. 2), κ = 0 . h ′ and r that nucleations, including those with ashort lifetime, occur several times within the range wherethe scaling P ∞ ∼ ε β ′ holds.Given that the only assumption was criticality of anabsorbing transition together with very rare spontaneousnucleations, the observed scaling of hysteresis with theexponent κ = 1 / ( β ′ + 1) is expected to be found uni-versally in systems that exhibit quasi-absorbing transi-tions. This is confirmed by performing simulations indifferent dimensions and for different models and univer-sality classes [Table I and Fig. 4 (a)(b)]. In all cases, themeasured κ values for ∆ λ ∗ t are in good agreement withthose derived by Eq. (3). The loop scaling is also robustto situations when the ramp rate r and/or the nucleationprobability h ′ vary with time or the control parameter.As long as they are nonzero and analytic at criticality,this gives only higher order corrections to Eq. (2) anddoes not affect the final result when r →
0. Even if thiscondition is not satisfied, the corrected form of Eq. (3)can be calculated, for example in the case of nonlinearramping ε ( t ) = r ′ t a ; the hysteresis exponent becomes -6 -5 -4 -2 -1 r ∆ λ -6 -5 -4 -2 -1 r ∆ p - -8 -7 -6 -5 -2 -1 r’ ∆ λ -3 -2 -1 -2 -1 r’ ∆ λ (a) (b)(c) (d) κ for ∆λ ( ): 0.69(1) ∆λ * ( ): 0.73(3) ∆λ * ( ): 0.81(4) t ∆ p ( ): 0.465(14) ∆ p * ( ): 0.460(17) ∆ p * ( ): 0.47(4) -2,t-2-2 κ for κ for ∆λ ( ): 0.46(3) ∆λ * ( ): 0.41(4) ∆λ * ( ): 0.50(7) t κ for ∆λ ( ): 0.74(1) ∆λ * ( ): 0.71(2) ∆λ * ( ): 0.78(2) t FIG. 4: (Color online) Hysteresis scaling for (1+1)D CP (a),(2+1)D voter-like(b), (2+1)D CP with quadratic ramping ε ( t ) = r ′ t (c), and (2+1)D CP with square root ramping ε ( t ) = r ′ t / (d). The estimates of the exponent κ for ∆ λ ∗ t agree with the theoretical values, namely, 0 .
783 (a), 0 . . . then κ = 1 / ( aβ ′ + 1), which is numerically confirmed[Fig. 4 (c)(d)].Some experimental systems expected to belong to theDP class seem to lack strictly absorbing states due toresidual nucleations [1, 17]. This suggests that the samehysteresis may be observed in such systems, for example with different alignments or at other transitions in theelectrohydrodynamic convection [18, 19]. A much moreintriguing candidate can be found in the field of quan-tum turbulence [20]. Recently, a number of experimen-tal studies on transitions to turbulence in superfluid Hehave reported hysteresis [21, 22, 23, 24] and temporal in-termittency in local state of turbulence [22, 23, 24, 25].The existence of a (quasi-)absorbing state is also ex-pected due to the quantum topological constraint. All ofthese facts suggest that an absorbing transition to STImay take place in this superfluid system. Although itseems technically difficult to examine conventional crit-ical phenomena of absorbing transitions directly there,scaling of hysteresis loops may be more easily accessi-ble and would allow to decide about the correspondinguniversality class.In conclusion, the hysteresis loop scaling experimen-tally observed before at the DSM1-DSM2 transition ofliquid cristal convection was explained by assuming DPdynamics with very rare spontaneous nucleations. Thisimplies that DSM1 is probably only quasi-absorbing inthe liquid crystal system. Moreover, scaling of hysteresisloops ∆ λ ∼ r κ with κ = 1 / ( β ′ + 1) was demonstrated tobe able to decide the universality class of transitions intoa quasi-absorbing state. These results may also be usedto analyze critical phenomena in systems where measur-able quantities are so limited that usual approaches toabsorbing phase transitions cannot be adopted, such asin superfluid turbulence.I am grateful to M. Sano, H. Chat´e, I. Dornic, S.Kai, M. Kobayashi, M. Kuroda, N. Oikawa, H. Park,M. Tsubota, and H. Yano for fruitful discussions. Thiswork is partly supported by JSPS Research Fellowshipsfor Young Scientists. [1] H. Hinrichsen, Adv. Phys. , 815 (2000).[2] H. K. Janssen, Z. Phys. B , 151 (1981).[3] P. Grassberger, Z. Phys. B , 365 (1982).[4] K. A. Takeuchi, M. Kuroda, H. Chat´e, and M. Sano,Phys. Rev. Lett. , 234503 (2007).[5] Y. Pomeau, Physica D , 3 (1986).[6] S. Kai, W. Zimmermann, M. Andoh, and N. Chizumi,Phys. Rev. Lett. , 1111 (1990).[7] S. Kai (private communication).[8] T. E. Harris, Ann. Prob. , 969 (1974).[9] R. Dickman, Phys. Rev. E , R2441 (1999).[10] S. L¨ubeck, Int. J. Mod. Phys. B ρ byturning the figure upside down.[13] P. Grassberger and A. de la Torre, Ann. Phys. , 373(1979). [14] P. Grassberger and Y. C. Zhang, Physica A , 169(1996); C. A. Voigt and R. M. Ziff, Phys. Rev. E ,R6241 (1997).[15] I. Jensen, J. Phys. A: Math. Gen. , 5233 (1999).[16] I. Dornic, H. Chat´e, J. Chave, and H. Hinrichsen, Phys.Rev. Lett. , 045701 (2001).[17] P. Rupp, R. Richter, and I. Rehberg, Phys. Rev. E ,036209 (2003).[18] D. E. Lucchetta, N. Scaramuzza, G. Strangi, and C. Ver-sace, Phys. Rev. E , 610 (1999).[19] N. Oikawa, Y. Hidaka, and S. Kai, Prog. Theor. Phys.Suppl. , 320 (2006).[20] W. F. Vinen and R. J. Donnelly, Phys. Today (4), 43(2007); W. F. Vinen and J. J. Niemela, J. Low Temp.Phys. , 167 (2002).[21] J. J¨ager, B. Schuderer, and W. Schoepe, Phys. Rev. Lett. , 566 (1995).[22] D. I. Bradley et al. , J. Low Temp. Phys. , 493 (2005).[23] N. Hashimoto et al. , J. Low Temp. Phys. , 299 (2007).[24] For a brief review, see R. H¨anninen, M. Tsubota, and W.F. Vinen, Phys. Rev. B , 064502 (2007). [25] M. Niemetz and W. Schoepe, J. Low Temp. Phys.135