Notation and Derivations

1. The standard Kuramoto model

N phase oscillators with natural frequencies ω_i drawn from a distribution g(ω). All-to-all coupling:

dθ_i/dt = ω_i + (K/N) Σ_j sin(θ_j − θ_i)

The order parameter r measures coherence:

r e^iψ = (1/N) Σ_j e^iθ_j

r ≈ 0 = incoherent, r ≈ 1 = fully phase-locked.

2. Ring topology with noise

What I actually simulate: nearest-neighbour coupling on a ring, with additive white noise on each node:

dθ_i/dt = ω_i + K[sin(θ_i+1 − θ_i) + sin(θ_i−1 − θ_i)] + σ ξ_i(t)

Where ξ_i(t) is unit white noise, indices mod N. The ω_i are drawn from a Gaussian with mean 0 and spread Δω.

3. Phase-slip heuristic

When K is too small to maintain lock, individual oscillators drift relative to the mean field. A phase slip is a 2π jump in the phase difference between an oscillator and the order parameter ψ.

For the homogeneous case (Δω = 0, σ = 0), the critical coupling for ring topology can be shown to be:

K_c = (π/2) Δω_max

where Δω_max is the maximum frequency difference between adjacent nodes. With noise, the boundary smears.

4. Open questions

• Does the slip rate follow an Arrhenius-like law in K − K_c? Preliminary runs suggest yes, but the exponent doesn't match the Kramers escape-rate prediction.
• Odd N vs even N: there's a persistent parity effect in the slip distribution that I can't explain from the equations.
• What happens with long-range (power-law decaying) coupling instead of nearest-neighbour? Not simulated yet.

Personal research notes. Expect errors.