Finite-Size Effects in Synchronization Transitions

• Finite-size effects in synchronization transitions are deviations observed in finite oscillator networks compared to the ideal thermodynamic limit, characterized by sample-dependent fluctuations.
• They are quantified by robust scaling laws, such as δKc ~ N^-2/5 and Δ0 ~ N^-1/5, which underscore the critical influence of system size on the synchronization threshold.
• These phenomena highlight the need for precise finite-size scaling frameworks to accurately interpret experimental and numerical studies of synchronization in realistic, finite systems.

Finite-size effects in synchronization transitions refer to the deviations and fluctuation phenomena observed in coupled oscillator networks when the system size $N$ is finite, rather than in the thermodynamic limit. In nonlinear dynamical systems such as the Kuramoto model, these effects manifest as sample-dependent fluctuations, threshold shifts, and modifications to scaling behavior near the transition to global entrainment. Notably, such finite-size phenomena play a determining role in the collective dynamics observed in real systems, where $N$ is necessarily large but finite; they affect both the sharpness of the synchronization transition and the universality class of the critical behavior.

1. Finite-Size Scaling in the Globally Coupled Kuramoto Model

In the canonical Kuramoto model with all-to-all (global) coupling, the transition to macroscopic synchronization is characterized by a complex order parameter: $$\Delta e^{i\theta} \equiv \frac{1}{N}\sum_{j=1}^N e^{i\phi_j}$$ where $\phi_j$ is the phase of oscillator $j$. Close to the synchronization threshold at $K = K_c$, finite $N$ induces strong sample-to-sample fluctuations in the density of oscillators near the critical frequency interval $|\omega| < K\Delta$. This leads to a sample-dependent shift in the effective threshold, denoted $\delta K_c$, which scales anomalously as: $\delta K_c \sim N^{-2/5}$ Since $\Delta_0 \sim (\delta K_c)^{1/2}$ at criticality, the critical order parameter scales as $\Delta_0 \sim N^{-1/5}$. The corresponding finite-size scaling form is: $$\Delta(K, N) = N^{-1/5} f\left( (K - K_c) N^{2/5} \right)$$ where the "correlation size exponent" is $\nu' = 5/2$. This scaling indicates that, at the nominal threshold, only a sub-extensive fraction $N^{4/5}$ of the total oscillators are entrained.

2. Critical Properties and Dimensional Dependence

High Dimensions ($d > 4$)

For oscillators placed on a $d$-dimensional lattice with $d > 4$ and local coupling, the transition is a traditional symmetry breaking. Fluctuations in the phase field remain bounded, and the critical behavior parallels that of the mean-field (globally coupled) Kuramoto model, with critical exponents: $$\beta = \frac{1}{2}, \quad \nu = \frac{5}{2d}, \quad \nu' = d \nu = \frac{5}{2}$$ Here, the order parameter scaling is conventionally written as: $$\langle \Delta^2 \rangle = L^{-2\beta/\nu} \Phi\left( (K - K_c) L^{1/\nu} \right)$$ for system side length $L$ and total size $N \sim L^d$.

Lower Dimensions ($d = 3,4$)

For $d = 3$ and $d = 4$, phase fluctuations in the thermodynamic limit become unbounded, invalidating the classical order parameter as a measure of global synchronization. Synchronization is then more appropriately measured by the Edwards-Anderson order parameter: $$\Delta_{\text{EA}} = \lim_{t - t_0 \rightarrow \infty} \frac{1}{N} \left| \sum_j e^{i[\phi_j(t) - \phi_j(t_0)]} \right|$$ The finite-size scaling for $\langle \Delta_{\text{EA}}^2 \rangle$ remains as in higher dimensions, but the exponents are distinct: $$\beta = 0, \quad \nu = \frac{2}{d-2} \quad (d \leq 4)$$ The vanishing $\beta$ signals a discontinuous jump in the macroscopic entrained fraction in the infinite system, with finite-size systems exhibiting critical-like signatures due to the aggregation and percolation of locally entrained domains.

3. Role of Fluctuations and Scaling Theory

Finite-size fluctuations in oscillator populations at the synchronizing frequency fundamentally alter the observed transition. The density of oscillators within the entrainment window fluctuates as $N^{-1/2}$, but the critical sample-to-sample shift in $K_c$ arises from rare events, scaling as $N^{-2/5}$. This leads to unusual scaling exponents not seen in conventional (equilibrium) phase transitions. The finite-size scaling ansatz captures both the dependence of the order parameter on system size and on the reduced coupling: $\Delta(K,N) = N^{-1/5} f( (K-K_c) N^{2/5} )$ This generalized notion of "correlation volume" and its associated exponent $\nu'$ is essential for accurate interpretation of simulations and experiments.

4. Synchronization on Complex Networks and Finite-Dimensional Lattices

The critical exponents and scaling relations found for globally coupled systems also manifest in randomly connected networks (e.g., Erdős–Rényi or $z$-regular graphs), where they are controlled by the interplay of link disorder and frequency disorder (Hong et al., 2013). For locally coupled oscillators on regular lattices, as dimension decreases, the effect of diverging phase disorder becomes pivotal, eroding the viability of traditional global order parameters and necessitating alternative statistical measures such as the Edwards-Anderson parameter.

A summary of dimensional regimes is shown below:

Regime	Order Parameter	Key Exponents	Scaling Relation
$d > 4$ (mean-field)	$\Delta$	$\beta=1/2$, $\nu'=5/2$	$\langle \Delta^2\rangle = L^{-2\beta/\nu}$
$d=3,4$	$\Delta_{\rm EA}$	$\beta=0$, $\nu=2/(d-2)$	$\langle \Delta^2\rangle = L^{-2\beta/\nu}$

5. Implications, Limitations, and Real-World Relevance

The smoothing of the entrainment transition in finite systems implies that the observed onset of synchronization in realistic (large but finite) oscillator assemblies is always a crossover rather than a true singularity. The number of oscillators in the synchronized cluster at threshold is always sub-extensive, and local fluctuations in either the network topology or the oscillator frequency density can dominate the macroscopic behavior.

For finite-dimensional systems ($d \leq 4$), critical scaling persists in global observables only via locally entrained clusters, not global phase coherence. In all regimes, the correct extraction and interpretation of scaling exponents ($\nu', \beta$) and careful distinction between different types of order parameters are crucial for the analysis of experiments and numerical results.

6. Summary

Finite-size effects in synchronization transitions produce critical phenomena that depart from thermodynamic-limit predictions, with scaling exponents (notably $\nu' = 5/2$) governed by oscillator density fluctuations near the synchronizing frequency. While global order emerges via symmetry breaking in sufficiently high dimensions, in lower dimensions it is mediated by the percolation of synchronized domains with the usual order parameter vanishing due to unbounded phase disorder. A unified scaling framework, adapted to both finite-size and network topology, is essential for describing and predicting the onset and character of collective synchronization in real-world oscillator populations (Tang, 2010).