Synchronization transition in space-time chaos in the presence of quenched disorder

Technical Summary: Synchronization Transition in Space-Time Chaos with Quenched Disorder

Problem Statement
The synchronization of coupled map lattices (CML) is a prototypical model for studying spatiotemporal chaos and nonequilibrium phase transitions. In the absence of disorder, the synchronization transition between two identical replicas of CMLs with continuous maps (e.g., tent and logistic maps) is known to belong to the Multiplicative Noise (MN) universality class. Conversely, discontinuous maps exhibit transitions in the Directed Percolation (DP) class. While the Harris criterion suggests that quenched disorder is a relevant perturbation for systems where the correlation length exponent $\nu$ satisfies $d\nu < 2$, the specific effect of quenched disorder on the MN universality class in the context of replica synchronization has not been previously explored. This work addresses the gap in understanding how static (quenched) disorder in the coupling strength alters the critical behavior and universality class of synchronization transitions in coupled map lattices.

Methodology
The authors investigate the synchronization of two replicas of coupled map lattices ($u_1$ and $u_2$) governed by the evolution equation:
$$\begin{pmatrix} u_1(x, t+1) \\ u_2(x, t+1) \end{pmatrix} = \begin{pmatrix} 1-\gamma & \gamma \\ \gamma & 1-\gamma \end{pmatrix} \begin{pmatrix} (1-2\epsilon(i))f(u_1) + \epsilon(i)[f(u_1(x+1)) + f(u_1(x-1))] \\ (1-2\epsilon(i))f(u_2) + \epsilon(i)[f(u_2(x+1)) + f(u_2(x-1))] \end{pmatrix}$$
where $\gamma$ is the inter-replica coupling strength and $\epsilon(i)$ represents the intra-lattice coupling. The study introduces quenched disorder by assigning $\epsilon(i)$ as a random variable drawn from a uniform distribution (specifically between 0 and 0.5) that remains identical for both replicas at each site but varies across the lattice.

The authors simulate one-dimensional (1D), two-dimensional (2D), and globally coupled (mean-field) lattices using both tent maps ($f(x) = ax$ for $x<1/2$, $a(1-x)$ otherwise) and logistic maps ($f(x) = \mu x(1-x)$).

• Parameters: Simulations were performed for system sizes up to $L=1200$ (2D) and $L=5 \times 10^5$ (1D and global), running for $10^6$ time steps.
• Averaging: Results were averaged over approximately 500 distinct disorder configurations and initial conditions.
• Order Parameter: The synchronization error is defined as the spatial average of the absolute difference between replicas: $\langle |w(x, t)| \rangle = \frac{1}{N} \sum |u_1(x, t) - u_2(x, t)|$.
• Analysis: Critical exponents were determined by analyzing the power-law decay of the order parameter at the critical point ($\langle |w| \rangle \sim t^{-\delta}$), the scaling of the asymptotic order parameter near criticality ($\langle |w| \rangle \sim |\gamma - \gamma_c|^\beta$), and off-critical scaling relations to extract $\nu_\parallel$.

Key Results
The presence of identical quenched disorder in the coupling parameter fundamentally alters the universality class of the synchronization transition for continuous maps:

1. New Universality Class: The transition no longer belongs to the MN universality class (where $\delta \approx 1.1$). Instead, a new universality class emerges with distinct critical exponents that depend on the specific map and its parameters.
2. Critical Exponents:
   • Tent Map ($a=2$): The decay exponent is $\delta = 2$ and the order parameter exponent is $\beta = 2$.
   • Logistic Map ($\mu=4$): The decay exponent is $\delta = 3$ and the order parameter exponent is $\beta = 3$.
   • Correlation Length: The dynamic exponent is found to be $z = 0$ (implying no size dependence in the critical region), and the correlation length exponent is $\nu_\parallel = 1$.
3. Dimensional Independence: These critical exponents ($\delta, \beta, \nu_\parallel$) are observed to be identical across 1D, 2D, and globally coupled (mean-field) systems. The authors describe these exponents as "superuniversal," indicating independence from spatial dimensionality for the cases studied.
4. Parameter Dependence: The critical exponents are sensitive to the internal parameters of the map. For example, when the tent map parameter is changed to $a=1.5$, the critical point shifts ($\gamma_c \approx 0.167$), but $\delta$ remains 2. However, for the logistic map with $\mu=3.8$, the transition becomes discontinuous in the globally coupled case, and in 1D/2D, the exponent changes to $\delta = 1.2$.
5. Comparison with DP: The paper notes that while quenched disorder in DP systems often leads to Griffiths phases with continuously varying exponents, the disorder in the MN class (for these specific continuous maps) leads to a distinct, fixed-exponent universality class rather than a Griffiths phase.

Significance and Claims
The paper claims that quenched disorder in the coupling strength is a relevant perturbation for the replica synchronization of coupled map lattices. The primary significance of this work is the identification of a new universality class for synchronization transitions in the presence of disorder, distinct from both the Multiplicative Noise (MN) and Directed Percolation (DP) classes.

The authors emphasize that:

• The critical exponents are different from the standard MN class ($\delta=1.1$).
• The transition remains continuous (second-order) for the studied cases, but the critical behavior is governed by the disorder.
• The independence of exponents from dimensionality ($z=0$) suggests a mean-field-like behavior or a specific mechanism driven by the disorder that overrides spatial dimensionality effects in these systems.
• This finding is theoretically significant because disorder is inevitable in experimental systems, and understanding its effect on synchronization is crucial for real-world applications ranging from neuronal populations to power grids.

The study concludes that the synchronization transition in the presence of quenched disorder constitutes a distinct universality class characterized by exponents that are specific to the map dynamics but universal across spatial dimensions.