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Generalizing the theory to include quenched frequency-sampling fluctuations via coarse-graining

Develop a generalized mesoscopic framework that incorporates quenched frequency-sampling fluctuations by approximating a given realization of the empirical frequency distribution \(\bar{g}_{N}(\omega)=N^{-1}\sum_{j=1}^N\delta(\omega-\omega_j)\) through coarse-graining into M bins within \([−\Omega,\Omega]\), i.e., \(\bar{g}_{N}(\omega) \approx (\sum_{m=0}^{M-1} f_m)^{-1}\sum_{m=0}^{M-1} f_m\,\delta[\omega-\Omega(2m+1-M)/M]\), and derive the corresponding modifications to the reduced SDE and steady-state order-parameter distribution.

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Background

The main text and applications primarily treat dynamical (Gaussian) noise-induced finite-size fluctuations, while neglecting quenched fluctuations arising from finite-sample frequency realizations, which can be important for heavy-tailed distributions at small N.

Appendix F proposes a concrete coarse-graining scheme to approximate the sampled frequency distribution in bins as a possible remedy, and explicitly states that extending the analysis to include this generalization is left for future work.

References

A possible remedy to access the quenched fluctuations is to approximate a given realization $\bar{g}{_N}(\omega)=N{-1}\sum{j=1}N\delta(\omega-\omega_j)$ by coarse-graining it into $M$ bins within $[-\Omega,\Omega]$, and approximating it as $\bar{g}{_N}(\omega) \approx \big(\sum{m=0}{(M-1)}f_m\big){-1}\sum_{m=0}{(M-1)}f_m\delta[\omega-\Omega(2m+1-M)/M]$, where $f_m$ is the number of sampled frequencies in the $m$-th bin. This amounts to a generalization of our analysis, which we leave for future work.

— Finite-size fluctuations for stochastic coupled oscillators: A general theory (2510.02448 - Majumder et al., 2 Oct 2025) in Appendix F (Fluctuations in g(ω))