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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2603.05668 |
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| _version_ | 1866910060489211904 |
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| author | Sanz, Veronica |
| author_facet | Sanz, Veronica |
| contents | Collective synchronization is often summarized by a complex order parameter $R e^{iΨ}$, implicitly treating the global phase $Ψ$ as a meaningful macroscopic coordinate. Here we ask when $Ψ$ becomes \emph{operationally well-defined} in oscillator networks whose coupling varies in time. We study damped (and optionally inertial) phase-oscillator models on graphs with time-dependent coupling $K(t)$, covering standard Kuramoto dynamics as a limit and including network and spatial topologies relevant to engineered settings.
We propose an operational emergence criterion: a macroscopic phase is emergent only when it is robustly estimable, which we quantify via gauge-fixed phase-lag fluctuations under weak noise and finite sampling. This yields a quantitative threshold controlled by $NR^2$ and makes explicit why $Ψ$ is ill-posed in incoherent states even when formally definable. Nonautonomous coupling introduces a ramp timescale that competes with relaxation. Using a Laplacian-mode reduction near coherence, we derive a graph-spectral rate criterion: ordering tracks the protocol when $K(t)λ_2$ dominates the ramp rate, while faster ramps induce freeze-out. Numerically, we extract an operational freeze-out time from an energy-based tracking diagnostic and show that, for non-spatial networks, the residual incoherence at freeze-out collapses when plotted against the spectral protocol parameter $λ_2τ$ across Erdős--Rényi and small-world graph families. Finally, on periodic lattices we show that topological sectors and defect-mediated ordering obstruct complete alignment, leading to protocol-dependent, long-lived partially synchronized states and systematic deviations from spectral collapse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05668 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Operational Emergence of a Global Phase under Time-Dependent Coupling in Oscillator Networks Sanz, Veronica Statistical Mechanics High Energy Physics - Theory Chaotic Dynamics Physics and Society Collective synchronization is often summarized by a complex order parameter $R e^{iΨ}$, implicitly treating the global phase $Ψ$ as a meaningful macroscopic coordinate. Here we ask when $Ψ$ becomes \emph{operationally well-defined} in oscillator networks whose coupling varies in time. We study damped (and optionally inertial) phase-oscillator models on graphs with time-dependent coupling $K(t)$, covering standard Kuramoto dynamics as a limit and including network and spatial topologies relevant to engineered settings. We propose an operational emergence criterion: a macroscopic phase is emergent only when it is robustly estimable, which we quantify via gauge-fixed phase-lag fluctuations under weak noise and finite sampling. This yields a quantitative threshold controlled by $NR^2$ and makes explicit why $Ψ$ is ill-posed in incoherent states even when formally definable. Nonautonomous coupling introduces a ramp timescale that competes with relaxation. Using a Laplacian-mode reduction near coherence, we derive a graph-spectral rate criterion: ordering tracks the protocol when $K(t)λ_2$ dominates the ramp rate, while faster ramps induce freeze-out. Numerically, we extract an operational freeze-out time from an energy-based tracking diagnostic and show that, for non-spatial networks, the residual incoherence at freeze-out collapses when plotted against the spectral protocol parameter $λ_2τ$ across Erdős--Rényi and small-world graph families. Finally, on periodic lattices we show that topological sectors and defect-mediated ordering obstruct complete alignment, leading to protocol-dependent, long-lived partially synchronized states and systematic deviations from spectral collapse. |
| title | Operational Emergence of a Global Phase under Time-Dependent Coupling in Oscillator Networks |
| topic | Statistical Mechanics High Energy Physics - Theory Chaotic Dynamics Physics and Society |
| url | https://arxiv.org/abs/2603.05668 |