Background continuum phase transition

PDF p. 2: Section 1, complete-graph discussion

The article attributes to Kuramoto the following background claim: for a complete graph of $n$ oscillators with random intrinsic frequencies drawn according to a unimodal density $g$ symmetric about its mean, the model undergoes a transition as $n\to \mathrm{\infty }$ from a completely disordered state to a partially synchronized state when the width of $g$ falls below a threshold. The passage does not specify a width convention or hypotheses for the continuum limit.

A precise, more limited version is available for stationary mean-field self-consistency. Assume countable choice for the cited mathematical interfaces. Let $K>0$, and let $g_0$ be an even probability density on $\mathbb{R}$ with respect to Lebesgue measure $\lambda$, nonincreasing on $[0,\mathrm{\infty })$, smooth near zero, with $g_0(0)>0$ and $g_0^{\mathrm{\prime }\mathrm{\prime }}(0)<0$. For the scale family $g_{\sigma }(\omega )={\sigma }^{-1}{g}_0(\omega \mathrm{/}\sigma )$ with $\sigma >0$, take $\sigma$ as the width parameter. In the stationary all-to-all calculation with sinusoidal coupling $K\mathrm{/}n$, the incoherent branch has order parameter $r=0$ and the local partially coherent branch has $r>0$. It meets $r=0$ at ${\sigma }_c=\pi K{g}_0(0)\mathrm{/}2$ and lies on the side $\sigma <{\sigma }_c$ sufficiently near the threshold. Here $r$ is the modulus of the mean phase phasor; it measures the magnitude of the population's mean unit complex phase.

The first paragraph records the article's introductory attribution, not a proof of a finite-population limit. The calculation in [F3] is a stationary self-consistency input with its own cited source and hypotheses; it does not establish an interchange of $n\to \mathrm{\infty }$ and long-time limits, nonlinear stability, or the article's full unqualified transition claim.