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Phase transitions in 2D linearly stable coupled map lattices
PHY$1CA EIAqEVIER
Physica D 103 (19971 369-380
Phase transitions in 2D linearly stable coupled map lattices Y. Cuche a,*, R. Livi b, 1,2,3, A. Politi a.2 4 a lstituto Nazionale di Ottica, Largo E. Fermi 5 50125, Firenze, Italy b Dipartimento di Fisica, Universitb di Bologna, via lrnerio 46, 40126 Bologna, Italy
Abstract
Interlace dynamics separating homogeneous phases is shown to be the main mechanism underlying irregular evolution in 2D linearly stable, coupled map lattices. In a fully deterministic model belonging to this class, we find evidence of at least two different regimes that we call weak and strong turbulence. The transition between the two regimes is carefully investigated revealing a direct connection with the destabilization of the interfaces separating homogeneous phases. The critical behaviour i~ analysed and compared with that of stochastic models like directed percolation.
1. I n t r o d u c t i o n
In the recent years, an increasing experimental interest for spatio-temporal phenomena has led to many theoretical efforts tackling the questions raised in this research field. The traditional approach to the study of spatially extended dynamical systems based on partial differential equations (PDEs) revealed some limitations. In particular, technical difficulties prevent a straightforward extension of methods and concepts l rom very simple to slightly more complicated models. The exploration of possible alternatives yielded lhe appearance o f new models that can be regarded as space-time discretized versions of PDEs. The most successful class of deterministic models are coupled map lattices (CMLs) 11,2].
* Corresponding author. I E-mail: [email protected] or [email protected]. 2 INFM-FORUM, 3 INFN Sezione di Bologna. 4 INFN Sezione di Firenze.
In this paper, we shall consider the following model: Xt+l(i, j ) = (1 - e ) f ( x t ( i , j ) ) +s
Z f ( x t ( i ' , J" )), (i',j' E.M(i.j)
O
I
(1)
where the pair of integers (i, j ) labels the sites of a 2D lattice, A/(i, j ) represents the set of the nearest neighbouts of the point (i, j ) , t is a discrete time variable, and xt (i, j ) is the state variable. Finally, f ( x ) is a map of the interval [0,1] into itself, and ~ is the diffusive coupling. When f ( x ) is a chaotic map, one of the most widely studied features is "chaos reduction" due to the diffusive spatial coupling [3]. In fact, upon increasing e, a transition from chaotic to periodic or quasi-periodic evolution can be observed (notice, however, that an increase of the chaoticity is again observed at larger ~-values, when two almost uncoupled sublattices exist). Intuitively, one can say that the evolution is the result of the competition between local dynamics and
0167-2789/97/$17.(10 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(96100270-9
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diffusion. Thus, one can notice a similarity with the "symmetry breaking" mechanism typical of phase transitions in statistical mechanics [4]. For e --+ 0 +, each map evolves independently. Therefore, if the invariant measure of the single map covers the whole unit interval, the asymptotic dynamics of the CMLs fills in the whole configuration space. For increasing values of e, the support of the invariant measure reduces and possibly splits into a collection of (disjoint) periodic and/or chaotic attractors of lower dimensionality. For instance, this scenario has been analysed in detail for a relatively small number N of coupled logistic maps [5]. A mechanism similar to - even if more intricate than - Feigenbaum's period-doubling cascade has been identified. However, the type and the number of attractors both depend on N in an intricate manner, so that inferring a phase transition in the thermodynamic limit is still an open question. Numerical investigations on models of diffusively coupled chaotic maps exhibit similar scenarios [6]. One might even ask if the existence of phase transitions, in the sense of statistical mechanics, is a well-defined problem for CMLs. 5 In fact, one should consider that in the realm of spatially extended dynamical systems, phase transitions are typically associated to probabilistic evolution rules (e.g., stochastic equations, probabilistic cellular automata, etc.). This is not surprising, because probability plays a role similar to temperature in equilibrium statistical mechanics, by inducing a natural measure and providing a statistical interpretation for the averages of dynamical quantities. As long as deterministic chaos is present, we know that it is equivalent to some type of stochastic process (the reader can, for instance, think of the reduction of the evolution of Axiom-A systems to Markov chains). Nevertheless, deterministic chaos cannot be fully reduced to a probabilistic process. This is the reason why spatio-temporal inter-
5 Recently it has been shown that symbolic encoding of the dynamics of chaotic CMLs with a built-in Ising symmetry can exhibit Ising-like phase transitions between two chaotic phases, see [7].
mittency, the most general scenario so far identified of order-to-chaos transition [8], is not fully equivalent to a percolation transition though the analogies and similarities are very strict [9]. Even more strikingly, quite recently, it has been shown that maps possessing suitable stable periodic orbits may exhibit a new peculiar mechanism for the onset of stochastic-like behaviour [ 10-12], when spatially coupled. In such models, for (almost) any value of e, an initial condition always evolves to a stable periodic attractor, which may or may not coincide with the homogeneous state corresponding to the attractor of the single map. In fact, the maximum Lyapunov exponent associated with the evolution in tangent space of such a class of models is strictly negative (though dependent on e), so that there must be a non-chaotic attractor. For small values of the diffusion constant, the average transient time T2 required to reach a periodic state grows at most as a power of the lattice size L. Upon increasing e, at some point, one observes a transition to a new dynamical regime characterized by a so-called "chaotic transient", where T2 ~ e ~L2 [13]. Such a phenomenon is not at all surprising whenever the finite-time maximum Lyapunov exponent is positive during the transient, as indeed observed in a lattice of logistic maps in the presence of stability windows. In such cases, we may say that the very complex structure typical of the phase diagram in generic lowdimensional attractors with interlaced chaotic and ordered regions, disappears in spatially extended systems. There, chaotic behaviour is more likely to be structurally stable. This is not, however, the case of the models discussed in [10-12], where deterministic chaos cannot be invoked. A more enlightening analogy has been suggested with cellular automata [11,14], where exponentially long transients can be observed too. More important is the "experimental" observation that the transient evolution appears to be stationary and ergodic, since ensemble and temporal averages do coincide. In the ID case, it is still unclear if one can establish a firm connection between the transition from a temporally ordered to a "chaotic transient" regime
Y Cuche et al./Physica D 103 (1997) 369-380
and standard critical phenomena like, e.g., percolation transitions. Conversely, some of the rich phenomenology exhibited by 2D models appears to be much closer to that of more realistic systems, suggesting the possibility of a successful exploration of critical behaviour. For instance, in a recent paper [13], spiral waves, nucleation and surface roughening phenomena have been shown to characterize the dynamics of a 2D CML made of maps supporting a superstable period-3 orbit (a detailed description of this model is reported in Section 2). Here, we focus our attention on a phase transition from a weak turbulence (WT) regime, dominated by spiral chaos, to a strong turbulence (ST) regime, with short-range correlations. A schematic phase diagram in the plane e, b is drawn in Fig. 1 (for a definition of the control parameter b see Section 2), where the critical line Lc separating the two phases is reported. Most of the complexity exhibited by the model can be traced back to dynamical properties of interfaces separating homogeneous period-3 regions (with different phases). For instance, Lc is the locus, where the velocity of the interface vanishes and, simultaneously, the inter-
371
face becomes unstable. This line ends up at some point Pc in the parameter plane, below which the crossover between the two phases is mediated by a nucleation phenomenon which somehow softens the transition. We understand that, inside an interface separating two regions, the third homogeneous phase may nucleate and grow, while the interface itself moves away. The critical nucleation radius plays a prominent role in the determination of overall dynamical features. Just below Pc, the radius is rather small, thus explaining why nucleation becomes so important. Conversely, the critical radius is much too large with respect to the typical sample size in the vicinity of Lc, thus leading to the observed hysteretic effects. A quantitative analysis of the whole phenomenology is presented in Section 3, while Section 4 is devoted to the study of the critical properties. The mechanisms underlying the scaling laws suggest a close relationship with other nonequilibrium phenomena occurring in dynamical systems like directed percolation.
2. The model 2.7
In this paper we investigate the behaviour of a lattice of piecewise linear maps of the type /- J
ST /
II
/
2.6 b
f(x)
WT~'Pc
I
2.5
2.4
0.'18
e
0.19
Fig. 1. Schematic phase diagram of the CML model (2). The thick solid line Lc represents the critical line separating WTfrom ST-regime. It ends in the point Pc where the transition is smeared out by nucleation. The dashed line has only the qualitative meaning of a fuzzy border between the two turbulent regimes. The thin solid lines (I) and (II) denote two paths (h = 2.5 and b(e) = 12.048e + 0.353, respectively) along which an adiabatic change of the parameters has been carefully investigated.
[bx
I
a
ifO
(2)
Since the right branch of the map is totally flat, it is easy to convince oneself that when b > 1 and a < 1/b, the asymptotic evolution of the single map consists of a periodic altemancy between the two branches. Again in virtue of the constancy of the right branch, the periodic orbit is superstable. In this paper we shall always consider values of a and b such that a period3 solution exists: a(A) --+ ab(B) -+ ab2(C) As soon as the spatial coupling is added, any trajectory different from the fully homogeneous one is no longer superstable, since an exactly zero multiplier can be obtained only if all lattice sites are simultaneously on the right branch. This represents the first indication
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that the space-time evolution is less trivial than that of the single map, as it was already understood from the rigorous analysis of some defect dynamics in a 1D lattice [15]. Although the maximum Lyapunov exponent is no longer - ~ , it remains negative, a feature which necessarily implies an asymptotic convergence towards a stable periodic attractor. However, a contraction of volumes does not imply any specific feature of the transient regime. In fact, it has been discovered that, in some parameter regions, the transient time increases exponentially with the system size (when starting from a random initial condition) [10,11,131. Since the discussion of this point is crucial for understanding the results presented in the next sections, let us give a rigorous definition of the transient T2. By denoting with x} L) the configuration of a lattice of linear length L at time t, the transient time T2(~) is the smallest time t such that [xl L) -x~.L)l < ~ for some s < t. While it is obvious that T2 --+ cx~ for ~ --~ 0, it is in general expected that the variation of the transient time with 3 is independent of the system size, so that it is not particularly relevant in the thermodynamic limit. For this reason, we drop from now on the dependence on 8. Moreover, let us notice that the above definition does not discriminate between a short evolution towards an orbit with an extremely long period and a long approach to a short-period trajectory. In fact, what is important is the time needed to realize that the evolution has actually fallen onto a periodic orbit, rather than the period itself, as we claim that in the thermodynamic limit it is the "transient" evolution that matters. Much more relevant is the empirical observation that for times smaller than 7"2 but larger than a pretransient T~, the dynamics appears to be stationary, i.e., any ensemble average over evolutions stemming from the same ensemble of initial conditions is almost independent of time. Since TI turns out to be almost independent of the system size, the stationary regime extends over an infinity of time scales in the thermodynamic limit, so that it is legitimate to consider it as the relevant regime.
3. P h e n o m e n o l o g y In this section we present a qualitative description of the dynamics of the CML model with periodic boundary conditions. In particular, we will focus our attention on a domain of the parameter space in which the system exhibits a complex behaviour, reminiscent of some properties observed in more realistic physical systems. To give a preliminary idea of the various dynamical regimes that can be generated by the model, we present in Fig. 2 snapshots of the evolution of a 600 × 600 toroidal CML for some values of the diffusive constant e and for the map parameters a = 0.1, b = 2.5. The pattern reported in Fig. 2(a) (obtained for e = 0.172) reveals a typical weak turbulent regime with extended homogeneous phases that are simply connected sets of lattice sites where the dynamical variable evolves according to the period-3 solution of the single map. These patches are characterized by a typical length scale and are separated by rough interfaces (sets of sites where x(i, j ) ~ {A, B, C}). In a stroboscopic view (i.e., by monitoring the evolution every third time step) the core of the homogeneous regions remains unchanged, the only evolution being associated to the interface motion. On the average, an interface moves in such a way that phase A invades B (analogously, B invades C and C invades A), but its evolution is not regular at all: the local velocity strongly fluctuates. Besides interfaces, there are special points, where the three different phases come close to each other. They appear as centres of spirals wherefrom three arms (interfaces) depart; depending on the local ordering of the phases, the spirals may rotate clockwise or anticlockwise and for such a reason will be termed vortices and antivortices, respectively. A vortex-antivortex pair may annihilate or spontaneously arise when a nucleus of the third phase is generated within the interlace (we will return to this possibility later on). A pair may also lock a finite distance apart and the corresponding "dipole" structure acts as a source of almost concentric waves (fronts of the three phases following each other). This scenario is very similar to the type of evolution observed in excitable media 116-18]. It is important to notice that the choice of a period-3 solution
Y. Cuche et al./Physica D 103 (/997) 3 6 ~ 3 8 0
(~)
373
(b) Fig. 2. Continued
(cl
(d)
Fig. 2. Typical patterns of a 600 × 600 CML for b = 2.5 and various values of e: (a) spiral waves in a WT-regime lt)r ~: = 0.172: (b~ a similar WT-regime for e = 0.180, with a smaller characteristic length: (c) disappearance of WT for F = 0.190, because of an instability in the interface which produces fragmentations; (d) ST-regime for ~ = 0.192. These pictures are obtained during the adiabatic process described in the caption of Fig. 4(a). The tour grey levels correspond to the different phases: the three states of the homogeneous solution plus the disordered phase in the interface.
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is the simplest one guaranteeing this scenario, since the two phases of a period-2 solution cannot invade each other for obvious symmetry reasons. While it is not the task of this paper to discuss in detail the relationship between this WT-regime and similar regimes found in some PDEs [17,18], we limit ourselves to notice that the spontaneous roughening of the spiral arms is, to our knowledge, a phenomenon that has not been discussed in the literature. We are currently working to study to what extent this is peculiar to our CML. The snapshot reported in Fig. 2(b) corresponds to a higher value of the coupling constant (e = 0.18). The pattern is similar to the previous one, the main difference being the characteristic length of the homogeneous regions, which is clearly smaller. Moreover, the interfaces are rougher and move faster. Upon further increasing e (see Fig. 2(c)), more and more vortices appear (the characteristic length of the homogeneous patches decreases), while larger fluctuations characterize the interfacial zone, allowing for the nucleation of homogeneous nuclei inside it. The nuclei may grow, depending on whether fluctuations allow to exceed the nucleation radius. In fact, it was observed in [ 13] that the diffusive coupling present in our model induces a surface tension which blocks the growth of sufficiently small droplets. In any case, a sort of fragmentation of the interface arises, which prevents a clear-cut identification of the separation line between different homogeneous phases. However, it is worth mentioning that, notwithstanding the fragmentation process, it is still possible to identify and measure the local thickness of the interface, which turns out to be approximately equal to that of the previous case. At still higher values of e (see Fig. 2(d)) homogeneous clusters are dramatically reduced to tiny patches with a short lifetime, while the "interface" has invaded most of the lattice, giving rise to a complex phase which, borrowing the language of fluid dynamics, we shall call strong turbulence, in contrast to the WT-regime described above. The best argument in favour of the existence of two distinct regimes is provided by the dynamics of flat interfaces separating two semi-infinite homogeneous phases. In the WT-regime, the interface thickness remains finite, while its profile roughens; in the ST-regime, the interface blows up in-
1.0
C(~) 0.5
/
b
~
..
0.0
-0.5 0
30 0
6000
~
9000
Fig. 3. Normalized time autocorrelation functions for both WT(e = 0.1885, curve (a) and ST- (e = 0.1915, curve (b) regimes. The average is computed over nine independent signals of time length 30 000, obtained by selecting nine uncorrelated sites in a single pattern. vading both phases. Moreover, the time autocorrelation function
(xt+~xt)C(z-) --~
(x) 2 (x2) _ (x)2
(3)
provides further evidence of these two distinct regimes. The symbol (.) denotes an average that has been performed both in space, over a subset of lattice points sufficiently uncorrelated with each other, and in time in order to increase the statistics. The results are reported in Fig. 3 (of course, the evolution is sampled at every third iterate, to get rid of the irrelevant cyclic evolution in the homogeneous patches). It turns out that the ST-regime is characterized by an exponential decrease of the autocorrelation function, while in the WT-regime we find stronger correlations with large amplitude oscillations around zero. Such oscillations originate from the almost regular motion of the spirals which induces a periodic alternation of the three homogeneous phases on a given lattice site. In fact, we have verified that the time interval r corresponding to the first maximum of C(r) coincides with the average time needed for the permutation of the homogeneous clusters in a given site. Similar conclusions can be drawn from the analysis of the spatial autocorrelation function, although the limited size of the lattice prevents any quantitative description.
Y. Cuche et al./Physica D 103 (1997) 369-380 1.0
1.0
R~
RI
0.8
0.8
0.6
0.6
04
0.4
02
0.2
375
E
O0 0.16
,
,
0.17
i
0.18
,
i
0.19
,
0.20
(~)
0.0
0.1815
0.1820
0.1825
0.1830
(b)
Fig. 4. Plot of the fraction of interface sites Rt(t(e)) versus e, along paths I (a) and II (b) in the control parameter space. The curves are obtained starting from a typical ST-regime (obtained after a 3000 step evolution from a random initial condition) at (e -- 0.196, b = 2.50) for (a) and at (e = 0.1825, b = 2.552) for (b), and slowly decreasing the parameters afterwards (Ae = 4 x 10-8 in (a) and Ae = 2 x 10-8 in (b)). In (b), the process has been inverted at the point (e = 0.1815, b = 2.540), incrementing both s and b with the same rate as before. The lattice size is 600 x 600 in (a) and 512 x 512 in (b).
Before passing on to a more detailed discussion of
is strongly favoured in the second time step. This ini-
the transition between WT- and ST-regime, let us mention that in other regions of the parameter space more
tial asymmetry would not be important, if we had the
exotic patterns have been observed, the behaviour of which will not be analyzed in this paper. However, for
possibility to simulate sufficiently large lattices. However, the inconvenience can be overtaken by means of an adiabatic modification of the control parameters,
the sake of completeness, we want to give a flavour of those scenarios which may arise in the same,
starting from the ST-regime. This process allows us to reach the typical configurations shown in Fig. 2.
seemingly trivial, model. For e ~ 0.15, the interface
The adiabatic process also represents a simple tool
becomes the source of localized "flying" objects (gliders) which move with a velocity definitely higher than
to measure possible order parameters in several re-
that of the interfaces. Such gliders give rise to several
computed the number of lattice sites occupied by the three homogeneous phases and also the number
t)pes of scattering processes (among them and with the interface), so that it is natural to interpret their evolution as a transport and processing of information. An important remark has to be made about the appropriate method for generating the patterns shown in Fig. 2. In the parameter region where we have detected the ST-regime, the dynamics rapidly converges to a typical pattern, almost independently of the (random) initial condition. Conversely, in the WT-regime, a random initial condition almost always attains the trivial homogeneous solution. This can be partly understood by considering that a random initial state does not give the same contribution to the formation of the three types of homogeneous clusters: in fact, phase A
gions of the parameter space. For instance, we have
Ni(t) of interfacial sites, i.e., of points (i, j ) where xt(i, j ) ~ {A, B, C}. Notice that such a coding procedure is meaningful, because of the superstability of the map, which forces the homogeneous regions to assume exactly the expected value. In Fig. 4(a) we have plotted the fraction R(t) = N I ( t ) / L 2 of interface sites (L is the linear size of the square lattice) versus e (for a constant b = 2.5). Starting from e = 0.196, the diffusive constant has been decreased at the rate Ae = 4 x 10 -8 per time step. The most important features of this graph are the two jumps observed around e -- 0.163 and e ~ 0.191. The left one is associated with a divergence of the
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Y Cuche et al./Physica D 103 (1997) 369-380
characteristic length of the spirals which makes impossible for our lattice size (L = 600) to sustain a WT-regime. Moreover, the jump is associated to the onset of exotic structures such as gliders, which seem to prevent a quantitative study of the transition. The right jump coincides with the "transition" between WT-regime (Figs. 2(a) and (b)) and ST-regime (Fig. 2(d)). However, the smoothness of the curve (no evidence of singularities even in the derivative) suggests that it is perhaps improper to speak of a true transition. This feeling is strengthened by the observation that an adiabatic process in the opposite direction allows reproducing exactly the same curve, while in the presence of a true phase transition, hysteretic effects are typically expected as a consequence of time and space finiteness. The results of the adiabatic process are also confirmed by the computation of the asymptotic fraction Ras = l i m t ~ R(t) made for some fixed e-values, which superpose perfectly to the curve in Fig. 4(a). A closer inspection of the local properties of the evolution reveals that the continuity of R(e) in the crossover region is a consequence of the abovementioned fragmentation mechanism which gives rise to an increase in the number of nuclei and to a reduction of the critical radius for the nucleation of a homogeneous phase. The "transition" between WT- and ST-regimes appears to be very different in other regions of the parameter space. As an example of a clearly different scenario, we report in Fig. 4(b) the results of an adiabatic process along a path in the parameter space, where both e and b vary (see dashed curve in Fig. 1). The main difference with respect to the former case is the hysteresis, which is seen when the path is followed along opposite directions. We have verified that the size of the hysteresis loop does not depend on the rate of change of the parameters. The analysis of the local dynamics reveals that the nucleation of the third phase inside the interface is absent, thus showing that another mechanism must be invoked in the transition from a WT- to a ST-regime, namely the expansion of the disordered phase (interface) which starts invading any homogeneous region. Also in this case, we have checked the robustness of the patterns obtained by the adiabatic process. We
have taken as initial conditions two typical configurations sampled on the upper and lower branch of the hysteresis, corresponding to the same values of b and e. Then, we have let them evolve, estimating Ras over a sufficiently long time interval. Again the two values fall onto the adiabatic curves. Conversely, further simulations show that the width of the hysteresis reduces when the sample size is increased. Whether a finite jump is to be expected in the thermodynamic limit cannot be concluded from our simulations. An extended inspection of the parameter space indicates the existence of a phase-transition line schematically reported in Fig. 1 (solid curve). It seems reasonable to conjecture that this line terminates in the critical point Pc, although a numerical analysis does not allow an unambiguous identification of such a point. In summary, we have found evidence of a phase transition, using the variable R as an order parameter. On the other hand, the smooth behaviour exhibited by the adiabatic curves close to the transition regions (see Figs. 4(a) and (b)) indicates that finite-size effects can render a quantitative scaling analysis extremely difficult.
4. Critical phenomena in 2D The phenomenological analysis indicates that all the relevant mechanisms underlying the phase transition are already present in the interface dynamics. Accordingly, we expect to obtain a refined quantitative analysis of the scaling properties by focusing our attention on the dynamics of a single initially fiat interface separating two semi-infinite homogeneous regions. This requires considering a cylindrical geometry of the lattice. In our numerical simulations we start from the following initial conditions:
x0(i,j) =
K r/(i,j) M
if if if
j<0, 1 < j < W, j>W,
(4)
where/ E {I . . . . . L}, K , M c { A , B , C } , K 5~ M, r/(i, j) is a random number uniformly distributed on (0, 1), and W is the initial thickness of the interface
Y. Cuche et al./Physica D 103 (1997) 3 6 ~ 3 8 0
which, without loss of generality [ 13], we have fixed equal to 3. Periodic and no-flux boundary conditions are imposed along the i and j direction, respectively. Actually, it is sufficient to follow the evolution of 1 (t) (from now on, we use 1 (t) as a shorthand notation for the collection of lattice points belonging to the interface at time t) in the smallest window which contains the interface entirely, keeping track of its displacement. A particularly useful observable in the characterization of interface dynamics is the asymptotic velocity of its centre of mass. ('1)=
lira t-~ t
~
Nl(t) (i,j)El(t)
j
,
(5)
w.here (.) represents the average over independent realizations of the interface. Notice that this quantity is ~'ell defined only in the WT-regime, since the local thickness of the interface remains finite. The analogous observable in the ST-regime is the front velocity,
(ot(K)) = ,lira t
~
(i,j)cBx(,)
where N K (t) is the number of sites in the interface boundary BK (t) defined as
BK(t) ---- {(i, j) ~ l(t) ] 3(i', j') c A/'(i, j), x(i', j ' ) = K}.
(7)
Notice that, for symmetry reasons, Ivf(A)l = Iof(B)l = Ivr(C)l. In fact, at the borderline between the disordered (interface) and the homogeneous phases, the latter ones cyclically permute, so that the interface invades them with the same average velocity. Hence, the front velocity is exactly one half of the growth rate of the interface thickness and we can drop everywhere the dependence on the homogeneous phase. Still for symmetry reasons, we expect that the centre of mass of the interface does not move so that ivl) = 0 in the ST-regime. Conversely, WT is characterized by a front velocity (vr) = 0 , because the average thickness of the interface remains constant. We have computed (vl) and (vf) along the path corresponding to the second adiabatic process. Fig. 5
377
0.02
0.01
O Q
O O
O
O
E 0.00
o.1815
I
o.1;2o
i
e
o1~25
o18ao
Fig. 5. Average interlace (v|) (circles) and front 2(vf) velocity (diamonds) versus s, for some parameter values along path II. The duration of the simulations ranges from t = 1000 to about t = 10000, while the average has been taken over 50 realizations. The initial condition of each realization is a 512 × 3 random stripe separating homogeneous phases of types A and C.
shows that (VI> and (vf) go to zero (from the left and right, respectively) exactly at the same value e~. = 0 . 1 8 2 3 . . . Both quantities, used as order parameters, point to a continuous phase transition. Moreover, it is worth observing that ec is approximately placed in the middle of the hysteresis loop found when the same path is adiabatically followed in parameter space (see Fig. 4(b)). Notice, instead, that the fragmentation mechanism observed in the first adiabatic process (Fig. 4(a)) makes the quantities (vl) and (v¢) ill-defined in the crossover region. For this reason and in order to elucidate a possible analogy with directed percolation phenomena, we have monitored a further observable, the number Nl(t) of sites in the interfacial phase. This variable is not dissimilar to the average position of the front profile BE which contributes to define yr. However, at variance with BE, N1 is not affected by the onset of homogeneous nuclei and is a meaningful quantity in any regime. The evolution of Nl(t) is reported in Fig. 6(a) for some parameter values along the path of the first adiabatic process. The upper curves, that correspond to large e-values, display a linear growth almost from the very beginning. The lower curves, instead, exhibit a crossover from an initially constant to an
378
Y. Cuche et al./Physica D 103 (1997) 369-380 10 5
10 5 Ni
NI
10 4 10 4
2+1 10310'
......
i02
......
i03
t
(a)
101
102
103
104
t
lOS
(b)
Fig. 6. Log-log plots of the average of the absolute number of interface sites (Nl(t)) versus time t. The curves in (a) and (b) refer to b -- 2.5 and 2.55, respectively. From bottom to top, the curves in (a) correspond to e = 0.185, 0.188, 0.190, 0.192 and 0.195, while those in (b) correspond to e = 0.18220, 0.18230, 0.18231, 0.18232, 0.18234, 0.18240 and 0.1830. Each curve is the result of an average over 100 realizations. For comparision we have drawn the power-law behaviour expected for directed percolation in 1 + 1 and in 2 + 1 dimensions.
asymptotic unbounded growth. The sudden increase of such curves is due to the nucleation process. The crossover time increases for decreasing e, since the probability of nucleation becomes very small (partly because the nucleation radius becomes very large). Accordingly, these results confirm the lack of a clearcut distinction between WT- and ST-regimes in this region of the parameter space. A different scenario appears for b = 2.55, when e is varied in the interval (0.1822, 0.1825). This path intersects the critical line approximately in the same place where the second adiabatic process does intersect. Accordingly, we can also check whether the phase-transition features depend on the way the critical point is approached. Both the front and the interface velocities display a behaviour similar to that one depicted:in Fig. 5, going to 0 at e = 0.18233(2). At variance with the previous case, the lack of nucleation processes allows now distinguishing the STfrom the WT-regime. In the former regime, Nt grows linearly with time, while in the latter, it eventually saturates. This is consistent with the observation that Ras should exhibit a discontinuity in the thermodynamic limit at ec. Moreover, a power-law behaviour,
Ni(t) cxt ~,
(8)
is observed at the critical point. It is tempting to interpret the above findings in the spirit of directed percolation (DP). In fact, we can view the WT-ST transition as the passage from a regime where the three absorbing states A, B and C eventually win, to a regime where the disordered interfacial behaviour "percolates" through the entire lattice. Such an analogy is reinforced by the proof that the velocity of the front separating the disordered from the ordered phase is zero at DP threshold [20]. Our numerical estimate of the critical exponent, r/ 0.29, differs significantly from the known results for DP in 2 + 1 dimensions, ~ = 0.214(8). However, one should notice the peculiarity of our initial condition: the linear interface corresponds to a line of "defects", which differs from the typical initial condition used in numerical experiments on percolation, where the defects are concentrated within a bounded region. The structure of the initial condition makes the geometry of our problem to resemble more DP in 1 ÷ 1 dimensions (the critical behaviour is in fact observed only transversally to the interface). Our exponent is indeed
Y. Cuehe et al./Physica D 103 (1997) 369-380
closer to the r/ value in 1 + 1 dimensions, 0.317(2), and we cannot exclude that the difference is due to systematic errors. Moreover, we should recall that the W T - S T transition does not separate a strictly ordered regime from a chaotic one: infinitely many patterns are generated in the WT-regime and could be assimilated to different absorbing states. Models with many absorbing states (e.g., transfer threshold process [21 ], and the pair contact process [22]) have been recently shown to belong to DP universality class [23]. Nonetheless, it has also been stressed that the critical behaviour may strongly depend on the choice of the initial condition. A last remark on the observed scaling law calls into play a I D probabilistic CA proposed by Grassberger [24] for modelling a kinetic growth process, where the number of particles is conserved modulo 2. In analogy with our case, where an interface separating two different homogeneous phases cannot disappear, there, a single defect can never die. This simple fact implies that the resulting critical phenomenon falls into a universality class other than DP. Actually, in Grassberger model it turns out to be r/G = 0.272(12), a value close to that obtained for our model.
379
terface as a height field of a suitable growth process, one could view the W T - S T transition as a roughening transition. The complete description of the phase transition is a further open problem. The study of interfacial dynamics clearly points to a continuous transition: this, in turn, suggests that the hysteretic behaviour displayed by the fraction of interface sites should converge (in the thermodynamic limit) to a continuous non-analytic function. However, on the basis of our numerical simulations, we cannot rule out the hypothesis that R exhibits a discontinuous jump, i.e., undergoes a firstorder transition. A final remarkable property of the studied model is the disappearance of the phase transition below Pc. The dashed line drawn in Fig. 1 has a purely qualitative meaning. It indicates that in the corresponding parameter region the WT- and ST-regimes are separated by a complex phase dominated by the nucleation mechanism described in the previous sections. This phase can be viewed as a peculiar feature of this model, strongly resembling the mechanism of defect mediated transition to fully developed turbulence. Comparison with PDE models is expected to provide a deeper insight into the physical relevance of this phase.
5. Conclusion and perspectives Acknowledgements We have shown that superstable coupled maps on a 2D lattice exhibit a phase transition from a weak to a strong turbulence regime. It is important to stress that neither an external stochastic force is invoked nor an intrinsic linear instability is present in our model. Thus, we can conclude that nonlinear mechanisms different from the usual deterministic chaos may be responsible for the irregular behaviour of extended systems. How all such ingredients cooperate to the macroscopic evolution is still an open question. The nature of the phase transition as well as the values of the critical exponents suggest a parallel with contact processes. Within the analogies with stochastic models, it is worth investigating the possible similarity with ID non-equilibrium processes like the one recently studied in [25]. In fact, by interpreting the distance between upper and lower profiles of an in-
This work has been completed with the financial support of FORUM. One of us (YC) acknowledges support from the Swiss Physics Foundation.
References [1] K. Kaneko, Progr. Theoret. Phys. 72 (1984) 480. [2] I. Waller and R. Kapral, Phys. Rev. A 30 (1984) 2047. 13] A. Politi and A. Torcini, Chaos 2 (1992) 293. [4] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, London, 1971). [5] C. Giberti and C. Vernia, Chaos 4 (1994) 651; L. Bunimovich, V. Franceschini, C. Giberti and C. Vernia, On the stability of structures and patterns in extended systems, preprint (1995), submitted to Physica D. [6] K. Kaneko, Coupled map lattice, in: Chaos, Order and Patterns, ed. R. Artuso et al. (Plenum Press, New York, 1991), and references therein.
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[7] J. Miller and D.A. Huse, Phys. Rev. E 48 (1993) 2528. [8] H. Chat6 and P. Manneville, Physica D 32 (1988) 409; Physica D 37 (1989) 33; J.M. Houlrik, I. Webman and M.H. Jensen, Phys. Rev. A 41 (1990) 4210. [9] P. Grassberger and T. Schreiber, Physica D 50 ( 1991 ) 177. [10] J.P. Crutchfield and K. Kaneko, Phys. Rev. Lett. 60 (1988) 2715. [11] A. Politi, R. Livi, G.-L. Oppo and R. Kapral, Europhys. Lett. 22 (1993) 571. [121 E Bignone, J. Theoret. Biol. 161 (1993) 231. [13] R. Kapral. R. Livi, G.L. Oppo and A. Politi, Phys. Rev. E 49 (1994) 2009. [14] R. Bonaccini and A. Politi, Physica 103 (1997) 362. [15] L.A. Bunimovich, R. Livi, G. Martinez-Mekler and S. Ruffo, Chaos 2 (1992) 283. [161 E. Meron, Phys. Rep. 218 (1992) I.
117] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1980). [18] R. FitzHugh, Biophys. J. I (1961) 445; J. Nagumo, S. Arimoto and S. Yoshizawa, Proc. IRE 53 (1962) 2061. [191 C.H. Bennett, G. Grinstein, Y. He, C. Jayaprakash and D. Mukamel, Phys. Rev. A 41 (1990) 1932. [201 R. Durrett, Ann. Probab. 12 (1984) 999. [21] LEE Mendes, R. Dickman, M. Henkel and M.C. Marques, J. Phys. A 27 (1994) 3019. 122] I. Jensen and R. Dickman, Phys. Rev. E 48 (1993) 1710. [23] M.A. Mufioz, R. Dickman, G. Grinstein and R. Livi, Phys. Rev. Lett. 76 (1996) 451. [24] P. Grassberger, J. Phys. A 22 (1989) L1103. [25] U. Alon, M.R. Evans, H. Hinrichsen and D. Mukamel, Phys. Rev. Lett. 76 (1996) 2746.
Physica D 103 (19971 369-380
Phase transitions in 2D linearly stable coupled map lattices Y. Cuche a,*, R. Livi b, 1,2,3, A. Politi a.2 4 a lstituto Nazionale di Ottica, Largo E. Fermi 5 50125, Firenze, Italy b Dipartimento di Fisica, Universitb di Bologna, via lrnerio 46, 40126 Bologna, Italy
Abstract
Interlace dynamics separating homogeneous phases is shown to be the main mechanism underlying irregular evolution in 2D linearly stable, coupled map lattices. In a fully deterministic model belonging to this class, we find evidence of at least two different regimes that we call weak and strong turbulence. The transition between the two regimes is carefully investigated revealing a direct connection with the destabilization of the interfaces separating homogeneous phases. The critical behaviour i~ analysed and compared with that of stochastic models like directed percolation.
1. I n t r o d u c t i o n
In the recent years, an increasing experimental interest for spatio-temporal phenomena has led to many theoretical efforts tackling the questions raised in this research field. The traditional approach to the study of spatially extended dynamical systems based on partial differential equations (PDEs) revealed some limitations. In particular, technical difficulties prevent a straightforward extension of methods and concepts l rom very simple to slightly more complicated models. The exploration of possible alternatives yielded lhe appearance o f new models that can be regarded as space-time discretized versions of PDEs. The most successful class of deterministic models are coupled map lattices (CMLs) 11,2].
* Corresponding author. I E-mail: [email protected] or [email protected]. 2 INFM-FORUM, 3 INFN Sezione di Bologna. 4 INFN Sezione di Firenze.
In this paper, we shall consider the following model: Xt+l(i, j ) = (1 - e ) f ( x t ( i , j ) ) +s
Z f ( x t ( i ' , J" )), (i',j' E.M(i.j)
O
I
(1)
where the pair of integers (i, j ) labels the sites of a 2D lattice, A/(i, j ) represents the set of the nearest neighbouts of the point (i, j ) , t is a discrete time variable, and xt (i, j ) is the state variable. Finally, f ( x ) is a map of the interval [0,1] into itself, and ~ is the diffusive coupling. When f ( x ) is a chaotic map, one of the most widely studied features is "chaos reduction" due to the diffusive spatial coupling [3]. In fact, upon increasing e, a transition from chaotic to periodic or quasi-periodic evolution can be observed (notice, however, that an increase of the chaoticity is again observed at larger ~-values, when two almost uncoupled sublattices exist). Intuitively, one can say that the evolution is the result of the competition between local dynamics and
0167-2789/97/$17.(10 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(96100270-9
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diffusion. Thus, one can notice a similarity with the "symmetry breaking" mechanism typical of phase transitions in statistical mechanics [4]. For e --+ 0 +, each map evolves independently. Therefore, if the invariant measure of the single map covers the whole unit interval, the asymptotic dynamics of the CMLs fills in the whole configuration space. For increasing values of e, the support of the invariant measure reduces and possibly splits into a collection of (disjoint) periodic and/or chaotic attractors of lower dimensionality. For instance, this scenario has been analysed in detail for a relatively small number N of coupled logistic maps [5]. A mechanism similar to - even if more intricate than - Feigenbaum's period-doubling cascade has been identified. However, the type and the number of attractors both depend on N in an intricate manner, so that inferring a phase transition in the thermodynamic limit is still an open question. Numerical investigations on models of diffusively coupled chaotic maps exhibit similar scenarios [6]. One might even ask if the existence of phase transitions, in the sense of statistical mechanics, is a well-defined problem for CMLs. 5 In fact, one should consider that in the realm of spatially extended dynamical systems, phase transitions are typically associated to probabilistic evolution rules (e.g., stochastic equations, probabilistic cellular automata, etc.). This is not surprising, because probability plays a role similar to temperature in equilibrium statistical mechanics, by inducing a natural measure and providing a statistical interpretation for the averages of dynamical quantities. As long as deterministic chaos is present, we know that it is equivalent to some type of stochastic process (the reader can, for instance, think of the reduction of the evolution of Axiom-A systems to Markov chains). Nevertheless, deterministic chaos cannot be fully reduced to a probabilistic process. This is the reason why spatio-temporal inter-
5 Recently it has been shown that symbolic encoding of the dynamics of chaotic CMLs with a built-in Ising symmetry can exhibit Ising-like phase transitions between two chaotic phases, see [7].
mittency, the most general scenario so far identified of order-to-chaos transition [8], is not fully equivalent to a percolation transition though the analogies and similarities are very strict [9]. Even more strikingly, quite recently, it has been shown that maps possessing suitable stable periodic orbits may exhibit a new peculiar mechanism for the onset of stochastic-like behaviour [ 10-12], when spatially coupled. In such models, for (almost) any value of e, an initial condition always evolves to a stable periodic attractor, which may or may not coincide with the homogeneous state corresponding to the attractor of the single map. In fact, the maximum Lyapunov exponent associated with the evolution in tangent space of such a class of models is strictly negative (though dependent on e), so that there must be a non-chaotic attractor. For small values of the diffusion constant, the average transient time T2 required to reach a periodic state grows at most as a power of the lattice size L. Upon increasing e, at some point, one observes a transition to a new dynamical regime characterized by a so-called "chaotic transient", where T2 ~ e ~L2 [13]. Such a phenomenon is not at all surprising whenever the finite-time maximum Lyapunov exponent is positive during the transient, as indeed observed in a lattice of logistic maps in the presence of stability windows. In such cases, we may say that the very complex structure typical of the phase diagram in generic lowdimensional attractors with interlaced chaotic and ordered regions, disappears in spatially extended systems. There, chaotic behaviour is more likely to be structurally stable. This is not, however, the case of the models discussed in [10-12], where deterministic chaos cannot be invoked. A more enlightening analogy has been suggested with cellular automata [11,14], where exponentially long transients can be observed too. More important is the "experimental" observation that the transient evolution appears to be stationary and ergodic, since ensemble and temporal averages do coincide. In the ID case, it is still unclear if one can establish a firm connection between the transition from a temporally ordered to a "chaotic transient" regime
Y Cuche et al./Physica D 103 (1997) 369-380
and standard critical phenomena like, e.g., percolation transitions. Conversely, some of the rich phenomenology exhibited by 2D models appears to be much closer to that of more realistic systems, suggesting the possibility of a successful exploration of critical behaviour. For instance, in a recent paper [13], spiral waves, nucleation and surface roughening phenomena have been shown to characterize the dynamics of a 2D CML made of maps supporting a superstable period-3 orbit (a detailed description of this model is reported in Section 2). Here, we focus our attention on a phase transition from a weak turbulence (WT) regime, dominated by spiral chaos, to a strong turbulence (ST) regime, with short-range correlations. A schematic phase diagram in the plane e, b is drawn in Fig. 1 (for a definition of the control parameter b see Section 2), where the critical line Lc separating the two phases is reported. Most of the complexity exhibited by the model can be traced back to dynamical properties of interfaces separating homogeneous period-3 regions (with different phases). For instance, Lc is the locus, where the velocity of the interface vanishes and, simultaneously, the inter-
371
face becomes unstable. This line ends up at some point Pc in the parameter plane, below which the crossover between the two phases is mediated by a nucleation phenomenon which somehow softens the transition. We understand that, inside an interface separating two regions, the third homogeneous phase may nucleate and grow, while the interface itself moves away. The critical nucleation radius plays a prominent role in the determination of overall dynamical features. Just below Pc, the radius is rather small, thus explaining why nucleation becomes so important. Conversely, the critical radius is much too large with respect to the typical sample size in the vicinity of Lc, thus leading to the observed hysteretic effects. A quantitative analysis of the whole phenomenology is presented in Section 3, while Section 4 is devoted to the study of the critical properties. The mechanisms underlying the scaling laws suggest a close relationship with other nonequilibrium phenomena occurring in dynamical systems like directed percolation.
2. The model 2.7
In this paper we investigate the behaviour of a lattice of piecewise linear maps of the type /- J
ST /
II
/
2.6 b
f(x)
WT~'Pc
I
2.5
2.4
0.'18
e
0.19
Fig. 1. Schematic phase diagram of the CML model (2). The thick solid line Lc represents the critical line separating WTfrom ST-regime. It ends in the point Pc where the transition is smeared out by nucleation. The dashed line has only the qualitative meaning of a fuzzy border between the two turbulent regimes. The thin solid lines (I) and (II) denote two paths (h = 2.5 and b(e) = 12.048e + 0.353, respectively) along which an adiabatic change of the parameters has been carefully investigated.
[bx
I
a
ifO
(2)
Since the right branch of the map is totally flat, it is easy to convince oneself that when b > 1 and a < 1/b, the asymptotic evolution of the single map consists of a periodic altemancy between the two branches. Again in virtue of the constancy of the right branch, the periodic orbit is superstable. In this paper we shall always consider values of a and b such that a period3 solution exists: a(A) --+ ab(B) -+ ab2(C) As soon as the spatial coupling is added, any trajectory different from the fully homogeneous one is no longer superstable, since an exactly zero multiplier can be obtained only if all lattice sites are simultaneously on the right branch. This represents the first indication
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that the space-time evolution is less trivial than that of the single map, as it was already understood from the rigorous analysis of some defect dynamics in a 1D lattice [15]. Although the maximum Lyapunov exponent is no longer - ~ , it remains negative, a feature which necessarily implies an asymptotic convergence towards a stable periodic attractor. However, a contraction of volumes does not imply any specific feature of the transient regime. In fact, it has been discovered that, in some parameter regions, the transient time increases exponentially with the system size (when starting from a random initial condition) [10,11,131. Since the discussion of this point is crucial for understanding the results presented in the next sections, let us give a rigorous definition of the transient T2. By denoting with x} L) the configuration of a lattice of linear length L at time t, the transient time T2(~) is the smallest time t such that [xl L) -x~.L)l < ~ for some s < t. While it is obvious that T2 --+ cx~ for ~ --~ 0, it is in general expected that the variation of the transient time with 3 is independent of the system size, so that it is not particularly relevant in the thermodynamic limit. For this reason, we drop from now on the dependence on 8. Moreover, let us notice that the above definition does not discriminate between a short evolution towards an orbit with an extremely long period and a long approach to a short-period trajectory. In fact, what is important is the time needed to realize that the evolution has actually fallen onto a periodic orbit, rather than the period itself, as we claim that in the thermodynamic limit it is the "transient" evolution that matters. Much more relevant is the empirical observation that for times smaller than 7"2 but larger than a pretransient T~, the dynamics appears to be stationary, i.e., any ensemble average over evolutions stemming from the same ensemble of initial conditions is almost independent of time. Since TI turns out to be almost independent of the system size, the stationary regime extends over an infinity of time scales in the thermodynamic limit, so that it is legitimate to consider it as the relevant regime.
3. P h e n o m e n o l o g y In this section we present a qualitative description of the dynamics of the CML model with periodic boundary conditions. In particular, we will focus our attention on a domain of the parameter space in which the system exhibits a complex behaviour, reminiscent of some properties observed in more realistic physical systems. To give a preliminary idea of the various dynamical regimes that can be generated by the model, we present in Fig. 2 snapshots of the evolution of a 600 × 600 toroidal CML for some values of the diffusive constant e and for the map parameters a = 0.1, b = 2.5. The pattern reported in Fig. 2(a) (obtained for e = 0.172) reveals a typical weak turbulent regime with extended homogeneous phases that are simply connected sets of lattice sites where the dynamical variable evolves according to the period-3 solution of the single map. These patches are characterized by a typical length scale and are separated by rough interfaces (sets of sites where x(i, j ) ~ {A, B, C}). In a stroboscopic view (i.e., by monitoring the evolution every third time step) the core of the homogeneous regions remains unchanged, the only evolution being associated to the interface motion. On the average, an interface moves in such a way that phase A invades B (analogously, B invades C and C invades A), but its evolution is not regular at all: the local velocity strongly fluctuates. Besides interfaces, there are special points, where the three different phases come close to each other. They appear as centres of spirals wherefrom three arms (interfaces) depart; depending on the local ordering of the phases, the spirals may rotate clockwise or anticlockwise and for such a reason will be termed vortices and antivortices, respectively. A vortex-antivortex pair may annihilate or spontaneously arise when a nucleus of the third phase is generated within the interlace (we will return to this possibility later on). A pair may also lock a finite distance apart and the corresponding "dipole" structure acts as a source of almost concentric waves (fronts of the three phases following each other). This scenario is very similar to the type of evolution observed in excitable media 116-18]. It is important to notice that the choice of a period-3 solution
Y. Cuche et al./Physica D 103 (/997) 3 6 ~ 3 8 0
(~)
373
(b) Fig. 2. Continued
(cl
(d)
Fig. 2. Typical patterns of a 600 × 600 CML for b = 2.5 and various values of e: (a) spiral waves in a WT-regime lt)r ~: = 0.172: (b~ a similar WT-regime for e = 0.180, with a smaller characteristic length: (c) disappearance of WT for F = 0.190, because of an instability in the interface which produces fragmentations; (d) ST-regime for ~ = 0.192. These pictures are obtained during the adiabatic process described in the caption of Fig. 4(a). The tour grey levels correspond to the different phases: the three states of the homogeneous solution plus the disordered phase in the interface.
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is the simplest one guaranteeing this scenario, since the two phases of a period-2 solution cannot invade each other for obvious symmetry reasons. While it is not the task of this paper to discuss in detail the relationship between this WT-regime and similar regimes found in some PDEs [17,18], we limit ourselves to notice that the spontaneous roughening of the spiral arms is, to our knowledge, a phenomenon that has not been discussed in the literature. We are currently working to study to what extent this is peculiar to our CML. The snapshot reported in Fig. 2(b) corresponds to a higher value of the coupling constant (e = 0.18). The pattern is similar to the previous one, the main difference being the characteristic length of the homogeneous regions, which is clearly smaller. Moreover, the interfaces are rougher and move faster. Upon further increasing e (see Fig. 2(c)), more and more vortices appear (the characteristic length of the homogeneous patches decreases), while larger fluctuations characterize the interfacial zone, allowing for the nucleation of homogeneous nuclei inside it. The nuclei may grow, depending on whether fluctuations allow to exceed the nucleation radius. In fact, it was observed in [ 13] that the diffusive coupling present in our model induces a surface tension which blocks the growth of sufficiently small droplets. In any case, a sort of fragmentation of the interface arises, which prevents a clear-cut identification of the separation line between different homogeneous phases. However, it is worth mentioning that, notwithstanding the fragmentation process, it is still possible to identify and measure the local thickness of the interface, which turns out to be approximately equal to that of the previous case. At still higher values of e (see Fig. 2(d)) homogeneous clusters are dramatically reduced to tiny patches with a short lifetime, while the "interface" has invaded most of the lattice, giving rise to a complex phase which, borrowing the language of fluid dynamics, we shall call strong turbulence, in contrast to the WT-regime described above. The best argument in favour of the existence of two distinct regimes is provided by the dynamics of flat interfaces separating two semi-infinite homogeneous phases. In the WT-regime, the interface thickness remains finite, while its profile roughens; in the ST-regime, the interface blows up in-
1.0
C(~) 0.5
/
b
~
..
0.0
-0.5 0
30 0
6000
~
9000
Fig. 3. Normalized time autocorrelation functions for both WT(e = 0.1885, curve (a) and ST- (e = 0.1915, curve (b) regimes. The average is computed over nine independent signals of time length 30 000, obtained by selecting nine uncorrelated sites in a single pattern. vading both phases. Moreover, the time autocorrelation function
(xt+~xt)C(z-) --~
(x) 2 (x2) _ (x)2
(3)
provides further evidence of these two distinct regimes. The symbol (.) denotes an average that has been performed both in space, over a subset of lattice points sufficiently uncorrelated with each other, and in time in order to increase the statistics. The results are reported in Fig. 3 (of course, the evolution is sampled at every third iterate, to get rid of the irrelevant cyclic evolution in the homogeneous patches). It turns out that the ST-regime is characterized by an exponential decrease of the autocorrelation function, while in the WT-regime we find stronger correlations with large amplitude oscillations around zero. Such oscillations originate from the almost regular motion of the spirals which induces a periodic alternation of the three homogeneous phases on a given lattice site. In fact, we have verified that the time interval r corresponding to the first maximum of C(r) coincides with the average time needed for the permutation of the homogeneous clusters in a given site. Similar conclusions can be drawn from the analysis of the spatial autocorrelation function, although the limited size of the lattice prevents any quantitative description.
Y. Cuche et al./Physica D 103 (1997) 369-380 1.0
1.0
R~
RI
0.8
0.8
0.6
0.6
04
0.4
02
0.2
375
E
O0 0.16
,
,
0.17
i
0.18
,
i
0.19
,
0.20
(~)
0.0
0.1815
0.1820
0.1825
0.1830
(b)
Fig. 4. Plot of the fraction of interface sites Rt(t(e)) versus e, along paths I (a) and II (b) in the control parameter space. The curves are obtained starting from a typical ST-regime (obtained after a 3000 step evolution from a random initial condition) at (e -- 0.196, b = 2.50) for (a) and at (e = 0.1825, b = 2.552) for (b), and slowly decreasing the parameters afterwards (Ae = 4 x 10-8 in (a) and Ae = 2 x 10-8 in (b)). In (b), the process has been inverted at the point (e = 0.1815, b = 2.540), incrementing both s and b with the same rate as before. The lattice size is 600 x 600 in (a) and 512 x 512 in (b).
Before passing on to a more detailed discussion of
is strongly favoured in the second time step. This ini-
the transition between WT- and ST-regime, let us mention that in other regions of the parameter space more
tial asymmetry would not be important, if we had the
exotic patterns have been observed, the behaviour of which will not be analyzed in this paper. However, for
possibility to simulate sufficiently large lattices. However, the inconvenience can be overtaken by means of an adiabatic modification of the control parameters,
the sake of completeness, we want to give a flavour of those scenarios which may arise in the same,
starting from the ST-regime. This process allows us to reach the typical configurations shown in Fig. 2.
seemingly trivial, model. For e ~ 0.15, the interface
The adiabatic process also represents a simple tool
becomes the source of localized "flying" objects (gliders) which move with a velocity definitely higher than
to measure possible order parameters in several re-
that of the interfaces. Such gliders give rise to several
computed the number of lattice sites occupied by the three homogeneous phases and also the number
t)pes of scattering processes (among them and with the interface), so that it is natural to interpret their evolution as a transport and processing of information. An important remark has to be made about the appropriate method for generating the patterns shown in Fig. 2. In the parameter region where we have detected the ST-regime, the dynamics rapidly converges to a typical pattern, almost independently of the (random) initial condition. Conversely, in the WT-regime, a random initial condition almost always attains the trivial homogeneous solution. This can be partly understood by considering that a random initial state does not give the same contribution to the formation of the three types of homogeneous clusters: in fact, phase A
gions of the parameter space. For instance, we have
Ni(t) of interfacial sites, i.e., of points (i, j ) where xt(i, j ) ~ {A, B, C}. Notice that such a coding procedure is meaningful, because of the superstability of the map, which forces the homogeneous regions to assume exactly the expected value. In Fig. 4(a) we have plotted the fraction R(t) = N I ( t ) / L 2 of interface sites (L is the linear size of the square lattice) versus e (for a constant b = 2.5). Starting from e = 0.196, the diffusive constant has been decreased at the rate Ae = 4 x 10 -8 per time step. The most important features of this graph are the two jumps observed around e -- 0.163 and e ~ 0.191. The left one is associated with a divergence of the
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characteristic length of the spirals which makes impossible for our lattice size (L = 600) to sustain a WT-regime. Moreover, the jump is associated to the onset of exotic structures such as gliders, which seem to prevent a quantitative study of the transition. The right jump coincides with the "transition" between WT-regime (Figs. 2(a) and (b)) and ST-regime (Fig. 2(d)). However, the smoothness of the curve (no evidence of singularities even in the derivative) suggests that it is perhaps improper to speak of a true transition. This feeling is strengthened by the observation that an adiabatic process in the opposite direction allows reproducing exactly the same curve, while in the presence of a true phase transition, hysteretic effects are typically expected as a consequence of time and space finiteness. The results of the adiabatic process are also confirmed by the computation of the asymptotic fraction Ras = l i m t ~ R(t) made for some fixed e-values, which superpose perfectly to the curve in Fig. 4(a). A closer inspection of the local properties of the evolution reveals that the continuity of R(e) in the crossover region is a consequence of the abovementioned fragmentation mechanism which gives rise to an increase in the number of nuclei and to a reduction of the critical radius for the nucleation of a homogeneous phase. The "transition" between WT- and ST-regimes appears to be very different in other regions of the parameter space. As an example of a clearly different scenario, we report in Fig. 4(b) the results of an adiabatic process along a path in the parameter space, where both e and b vary (see dashed curve in Fig. 1). The main difference with respect to the former case is the hysteresis, which is seen when the path is followed along opposite directions. We have verified that the size of the hysteresis loop does not depend on the rate of change of the parameters. The analysis of the local dynamics reveals that the nucleation of the third phase inside the interface is absent, thus showing that another mechanism must be invoked in the transition from a WT- to a ST-regime, namely the expansion of the disordered phase (interface) which starts invading any homogeneous region. Also in this case, we have checked the robustness of the patterns obtained by the adiabatic process. We
have taken as initial conditions two typical configurations sampled on the upper and lower branch of the hysteresis, corresponding to the same values of b and e. Then, we have let them evolve, estimating Ras over a sufficiently long time interval. Again the two values fall onto the adiabatic curves. Conversely, further simulations show that the width of the hysteresis reduces when the sample size is increased. Whether a finite jump is to be expected in the thermodynamic limit cannot be concluded from our simulations. An extended inspection of the parameter space indicates the existence of a phase-transition line schematically reported in Fig. 1 (solid curve). It seems reasonable to conjecture that this line terminates in the critical point Pc, although a numerical analysis does not allow an unambiguous identification of such a point. In summary, we have found evidence of a phase transition, using the variable R as an order parameter. On the other hand, the smooth behaviour exhibited by the adiabatic curves close to the transition regions (see Figs. 4(a) and (b)) indicates that finite-size effects can render a quantitative scaling analysis extremely difficult.
4. Critical phenomena in 2D The phenomenological analysis indicates that all the relevant mechanisms underlying the phase transition are already present in the interface dynamics. Accordingly, we expect to obtain a refined quantitative analysis of the scaling properties by focusing our attention on the dynamics of a single initially fiat interface separating two semi-infinite homogeneous regions. This requires considering a cylindrical geometry of the lattice. In our numerical simulations we start from the following initial conditions:
x0(i,j) =
K r/(i,j) M
if if if
j<0, 1 < j < W, j>W,
(4)
where/ E {I . . . . . L}, K , M c { A , B , C } , K 5~ M, r/(i, j) is a random number uniformly distributed on (0, 1), and W is the initial thickness of the interface
Y. Cuche et al./Physica D 103 (1997) 3 6 ~ 3 8 0
which, without loss of generality [ 13], we have fixed equal to 3. Periodic and no-flux boundary conditions are imposed along the i and j direction, respectively. Actually, it is sufficient to follow the evolution of 1 (t) (from now on, we use 1 (t) as a shorthand notation for the collection of lattice points belonging to the interface at time t) in the smallest window which contains the interface entirely, keeping track of its displacement. A particularly useful observable in the characterization of interface dynamics is the asymptotic velocity of its centre of mass. ('1)=
lira t-~ t
~
Nl(t) (i,j)El(t)
j
,
(5)
w.here (.) represents the average over independent realizations of the interface. Notice that this quantity is ~'ell defined only in the WT-regime, since the local thickness of the interface remains finite. The analogous observable in the ST-regime is the front velocity,
(ot(K)) = ,lira t
~
(i,j)cBx(,)
where N K (t) is the number of sites in the interface boundary BK (t) defined as
BK(t) ---- {(i, j) ~ l(t) ] 3(i', j') c A/'(i, j), x(i', j ' ) = K}.
(7)
Notice that, for symmetry reasons, Ivf(A)l = Iof(B)l = Ivr(C)l. In fact, at the borderline between the disordered (interface) and the homogeneous phases, the latter ones cyclically permute, so that the interface invades them with the same average velocity. Hence, the front velocity is exactly one half of the growth rate of the interface thickness and we can drop everywhere the dependence on the homogeneous phase. Still for symmetry reasons, we expect that the centre of mass of the interface does not move so that ivl) = 0 in the ST-regime. Conversely, WT is characterized by a front velocity (vr) = 0 , because the average thickness of the interface remains constant. We have computed (vl) and (vf) along the path corresponding to the second adiabatic process. Fig. 5
377
0.02
0.01
O Q
O O
O
O
E 0.00
o.1815
I
o.1;2o
i
e
o1~25
o18ao
Fig. 5. Average interlace (v|) (circles) and front 2(vf) velocity (diamonds) versus s, for some parameter values along path II. The duration of the simulations ranges from t = 1000 to about t = 10000, while the average has been taken over 50 realizations. The initial condition of each realization is a 512 × 3 random stripe separating homogeneous phases of types A and C.
shows that (VI> and (vf) go to zero (from the left and right, respectively) exactly at the same value e~. = 0 . 1 8 2 3 . . . Both quantities, used as order parameters, point to a continuous phase transition. Moreover, it is worth observing that ec is approximately placed in the middle of the hysteresis loop found when the same path is adiabatically followed in parameter space (see Fig. 4(b)). Notice, instead, that the fragmentation mechanism observed in the first adiabatic process (Fig. 4(a)) makes the quantities (vl) and (v¢) ill-defined in the crossover region. For this reason and in order to elucidate a possible analogy with directed percolation phenomena, we have monitored a further observable, the number Nl(t) of sites in the interfacial phase. This variable is not dissimilar to the average position of the front profile BE which contributes to define yr. However, at variance with BE, N1 is not affected by the onset of homogeneous nuclei and is a meaningful quantity in any regime. The evolution of Nl(t) is reported in Fig. 6(a) for some parameter values along the path of the first adiabatic process. The upper curves, that correspond to large e-values, display a linear growth almost from the very beginning. The lower curves, instead, exhibit a crossover from an initially constant to an
378
Y. Cuche et al./Physica D 103 (1997) 369-380 10 5
10 5 Ni
NI
10 4 10 4
2+1 10310'
......
i02
......
i03
t
(a)
101
102
103
104
t
lOS
(b)
Fig. 6. Log-log plots of the average of the absolute number of interface sites (Nl(t)) versus time t. The curves in (a) and (b) refer to b -- 2.5 and 2.55, respectively. From bottom to top, the curves in (a) correspond to e = 0.185, 0.188, 0.190, 0.192 and 0.195, while those in (b) correspond to e = 0.18220, 0.18230, 0.18231, 0.18232, 0.18234, 0.18240 and 0.1830. Each curve is the result of an average over 100 realizations. For comparision we have drawn the power-law behaviour expected for directed percolation in 1 + 1 and in 2 + 1 dimensions.
asymptotic unbounded growth. The sudden increase of such curves is due to the nucleation process. The crossover time increases for decreasing e, since the probability of nucleation becomes very small (partly because the nucleation radius becomes very large). Accordingly, these results confirm the lack of a clearcut distinction between WT- and ST-regimes in this region of the parameter space. A different scenario appears for b = 2.55, when e is varied in the interval (0.1822, 0.1825). This path intersects the critical line approximately in the same place where the second adiabatic process does intersect. Accordingly, we can also check whether the phase-transition features depend on the way the critical point is approached. Both the front and the interface velocities display a behaviour similar to that one depicted:in Fig. 5, going to 0 at e = 0.18233(2). At variance with the previous case, the lack of nucleation processes allows now distinguishing the STfrom the WT-regime. In the former regime, Nt grows linearly with time, while in the latter, it eventually saturates. This is consistent with the observation that Ras should exhibit a discontinuity in the thermodynamic limit at ec. Moreover, a power-law behaviour,
Ni(t) cxt ~,
(8)
is observed at the critical point. It is tempting to interpret the above findings in the spirit of directed percolation (DP). In fact, we can view the WT-ST transition as the passage from a regime where the three absorbing states A, B and C eventually win, to a regime where the disordered interfacial behaviour "percolates" through the entire lattice. Such an analogy is reinforced by the proof that the velocity of the front separating the disordered from the ordered phase is zero at DP threshold [20]. Our numerical estimate of the critical exponent, r/ 0.29, differs significantly from the known results for DP in 2 + 1 dimensions, ~ = 0.214(8). However, one should notice the peculiarity of our initial condition: the linear interface corresponds to a line of "defects", which differs from the typical initial condition used in numerical experiments on percolation, where the defects are concentrated within a bounded region. The structure of the initial condition makes the geometry of our problem to resemble more DP in 1 ÷ 1 dimensions (the critical behaviour is in fact observed only transversally to the interface). Our exponent is indeed
Y. Cuehe et al./Physica D 103 (1997) 369-380
closer to the r/ value in 1 + 1 dimensions, 0.317(2), and we cannot exclude that the difference is due to systematic errors. Moreover, we should recall that the W T - S T transition does not separate a strictly ordered regime from a chaotic one: infinitely many patterns are generated in the WT-regime and could be assimilated to different absorbing states. Models with many absorbing states (e.g., transfer threshold process [21 ], and the pair contact process [22]) have been recently shown to belong to DP universality class [23]. Nonetheless, it has also been stressed that the critical behaviour may strongly depend on the choice of the initial condition. A last remark on the observed scaling law calls into play a I D probabilistic CA proposed by Grassberger [24] for modelling a kinetic growth process, where the number of particles is conserved modulo 2. In analogy with our case, where an interface separating two different homogeneous phases cannot disappear, there, a single defect can never die. This simple fact implies that the resulting critical phenomenon falls into a universality class other than DP. Actually, in Grassberger model it turns out to be r/G = 0.272(12), a value close to that obtained for our model.
379
terface as a height field of a suitable growth process, one could view the W T - S T transition as a roughening transition. The complete description of the phase transition is a further open problem. The study of interfacial dynamics clearly points to a continuous transition: this, in turn, suggests that the hysteretic behaviour displayed by the fraction of interface sites should converge (in the thermodynamic limit) to a continuous non-analytic function. However, on the basis of our numerical simulations, we cannot rule out the hypothesis that R exhibits a discontinuous jump, i.e., undergoes a firstorder transition. A final remarkable property of the studied model is the disappearance of the phase transition below Pc. The dashed line drawn in Fig. 1 has a purely qualitative meaning. It indicates that in the corresponding parameter region the WT- and ST-regimes are separated by a complex phase dominated by the nucleation mechanism described in the previous sections. This phase can be viewed as a peculiar feature of this model, strongly resembling the mechanism of defect mediated transition to fully developed turbulence. Comparison with PDE models is expected to provide a deeper insight into the physical relevance of this phase.
5. Conclusion and perspectives Acknowledgements We have shown that superstable coupled maps on a 2D lattice exhibit a phase transition from a weak to a strong turbulence regime. It is important to stress that neither an external stochastic force is invoked nor an intrinsic linear instability is present in our model. Thus, we can conclude that nonlinear mechanisms different from the usual deterministic chaos may be responsible for the irregular behaviour of extended systems. How all such ingredients cooperate to the macroscopic evolution is still an open question. The nature of the phase transition as well as the values of the critical exponents suggest a parallel with contact processes. Within the analogies with stochastic models, it is worth investigating the possible similarity with ID non-equilibrium processes like the one recently studied in [25]. In fact, by interpreting the distance between upper and lower profiles of an in-
This work has been completed with the financial support of FORUM. One of us (YC) acknowledges support from the Swiss Physics Foundation.
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