Role of unstable periodic orbits in phase transitions of coupled map lattices

The thermodynamic formalism for dynamical systems with many degrees of freedom is extended to deal with time averages and fluctuations of some macroscopic quantity along typical orbits, and applied to coupled map lattices exhibiting phase transitions. Thereby, it turns out that a seed of phase transition is embedded as an anomalous distribution of unstable periodic orbits, which appears as a so-called $q$-phase transition in the spatiotemporal configuration space. This intimate relation between phase transitions and $q$-phase transitions leads to one natural way of defining transitions and their order in extended chaotic systems. Furthermore, a basis is obtained on which we can treat locally introduced control parameters as macroscopic “temperature” in some cases involved with phase transitions.

Figure 1

Schematic view of an expected form of the topological entropy spectrum $H(\lambda ,A)$.

Figure 2

Local Bernoulli map at a site $i$. The time evolution of ${\Delta }_i^t$ is defined in Eq. (18).

Figure 3

(Color online) Schematic illustration of the spin reduction from an UPO $\left[\mathbf{s}\right]$ to the corresponding two-dimensional array $\left\{s_{j,k}\right\}$. (a) The spin configuration of an UPO $\left[\mathbf{s}\right]$. Spins form clusters of length $2$ in the time direction under the updating rule of Eq. (20). The red solid line and the blue broken line indicate the spin interaction that comes from the Lyapunov exponent $\lambda \left(\left[\mathbf{s}\right]\right)$ and the observable $A\left(\left[\mathbf{s}\right]\right)$, respectively, in Eq. (6). (b) The two-dimensional spin array $\left\{s_{j,k}\right\}$ obtained by the reduction, by which each cluster is reduced to a single spin located at the odd $(i-t)$. The two kinds of spin interaction turn into a helical ferromagnetic part (purple solid line) and a spatial antiferromagnetic part (purple dotted line) supposing both $(\beta +kq∕2)$ and $q$ are positive.

Figure 4

(Color online) Demonstration of Eqs. (9, 10) for the 1D Bernoulli CML with $k=1$, by means of Monte Carlo calculations. Lines correspond to (a) mean $\frac{\partial \Psi }{\partial \beta }(1,0)$ and (b) standard deviation $-\frac{{\partial }^2\Psi }{\partial {\beta }^2}(1,0)$ at $N=8,16,32,64$ evaluated via the GMF $\Psi (q,\beta )$, which is obtained by averaging results of 400 independent Monte Carlo runs with 100 000 samples after 100 steps of transients. The range of errors, estimated from standard deviation among the independent runs, is less than ${10}^{-4}$ for (a) and ${10}^{-2}$ for (b), and therefore, negligible. Symbols in both figures indicate the results of direct measurement of ${⟨A\left(\left[\mathbf{s}\right]\right)⟩}_{\mathrm{UPO}}$ and $pN\sigma {\left[A\left(\left[\mathbf{s}\right]\right)\right]}_{\mathrm{UPO}}^2$, respectively, by Monte Carlo simulations with 1 000 000 samples after 100 steps of transients. Corresponding standard deviations are denoted by error bars. The black dashed curve in (a) represents the exact value ${⟨A⟩}_{\mu }$. Note that plots for $p=32$ and 64 are nearly at the same place. Lines and symbols for fixed $p$ and $N$ are within the range of statistical errors, and thus Eqs. (9, 10) are confirmed. Furthermore, the two figures show that $\frac{\partial \Psi }{\partial \beta }(1,0)$ and $-\frac{{\partial }^2\Psi }{\partial {\beta }^2}(1,0)$ converge for sufficiently large $p$, the former of which coincides with the exact value in accordance with Eq. (9), and also for sufficiently large $N$, which indicates that the GMF $\Psi (q,\beta )$ is analytic in the limit $p,N\to \infty$.

Figure 5

(Color online) Second derivative of the GMF $-\frac{{\partial }^2\Psi }{\partial {\beta }^2}$ with $k=1$, $p=N=16$ in the 1D Bernoulli CML. $\Psi (q,\beta )$ is obtained from Eq. (24) with $\Delta q=0.02$ and $\Delta \beta =0.01$. The ensemble average in Eq. (24) is performed over 50 000 Monte Carlo steps after 100 steps discarded as transients. The derivative $\frac{{\partial }^2\Psi }{\partial {\beta }^2}$ is yielded by the three-point formula. The figure shown above is smoothed by taking its moving average over $5\times 5$ data points.

Figure 6

Second and third (inset) derivatives of the “entropy function” $H\left(A\right)$ in the weak-interaction case $k⪡1$, which indicate an anomaly in the UPO distribution of the 1D Bernoulli CML. The exact solution of the 2D Ising model [29] is used to plot these curves. Note that $H\left(A\right)$ does not depend on $k$ if we neglect $O\left(k^2\right)$.

Figure 7

(Color online) $q$-phase transition curves of the 1D Bernoulli CML with $p=N=16$ and (a) $k=0.1,0.5,1.0,2.0$, (b) $k=-0.1,-0.5,-1.0,-2.0$, which are indicated by a red solid line, green dot-and-dashed line, blue dashed line, and purple dotted line, respectively. The black cross is located on the physical situation $(q,\beta )=(1,0)$. These transition curves are obtained by detecting local maxima of $-\frac{{\partial }^2\Psi }{\partial {\beta }^2}$ in $q$ and $\beta$ direction, separately, and then eliminating false maxima that come from statistical errors in the Monte Carlo samplings and the finite-size effect. They can be distinguished from rounded-off singularity by examining their continuity and dependence on the system size $p$ and $N$. See also the caption of Fig. 5 for the way to obtain $\frac{{\partial }^2\Psi }{\partial {\beta }^2}$. The numbers of Monte Carlo samplings are 20 000, 20 000, 50 000, and 100 000 for $\mid k\mid =0.1,0.5,1.0,2.0$, respectively.

Figure 8

The local map, given by Eqs. (33, 34), for the repelling CML by Just and Schmüser. ${\Gamma }_{\pm \pm ,s_{i+1}^t}$ indicates a slope of each piecewise linear part.

Figure 9

(Color online) $q$-phase transition curves [Eq. (38)] of the 1D repelling CML defined by Eqs. (32, 33, 34, 36), with $J=0.3$, $J_c\approx 0.44,0.6,1.0$, which are drawn with a red solid line, green dot-and-dashed line, blue dashed line, and purple dotted line, respectively. The black cross indicates the physical situation $(q,\beta )=(1,0)$. The transition curves for negative $J$ are obtained by reflecting the figure over the $q$ or $\beta$ axis.