Fluctuation--Response Relation in Finite-Size Noisy Coupled Phase Oscillators

Abstract

Fluctuation--response relations (FRRs) provide a fundamental relation between spontaneous fluctuations and the linear response to external perturbations, yet their validity in non-equilibrium systems remains an open problem. In a general class of noisy, finite-size, mean-field models of coupled phase oscillators, we reveal two types of FRRs in the incoherent phase below the synchronization transition. The first, corresponding to the infinite-size limit (I-FRR), involves the convolution between the external force and the correlation function in the time domain. The second, the finite-size counterpart (F-FRR), is governed by finite-size fluctuations and carries a correction factor: a spectrum function whose roots correspond to eigenvalues, or Landau poles, of the linear operator. The finite-size fluctuations are suppressed in the weak-coupling or strong-noise limits, and F-FRR reduces to I-FRR. Our analytical findings are verified by extensive numerical simulations.