Why is the dynamics O(N) if the textbook has a sum over j?

Because Σ_j sin(θ_j − θ_i) = Im(e^{-iθ_i} Σ_j e^{iθ_j}) = N Im(Z e^{-iθ_i}) = N r sin(ψ − θ_i). The entire population enters only through the two numbers r and ψ, so each timestep is linear in N.

Is r = 1 perfect global synchronization?

Usually r ≈ 1 means almost all oscillators are locked into a common rotating cluster; exact r = 1 requires identical phases modulo 2π, which finite spread in ω_i typically prevents unless coupling is very strong relative to disorder.

What does the Lorentzian K_c = 2γ note mean?

For the Kuramoto model in the infinite-N limit with Cauchy-distributed ω of half-width γ, the incoherent state loses stability at K_c = 2γ. Real finite boxes of uniform frequencies behave similarly in spirit but not identically; treat it as intuition, not a fitted critical point here.

Kuramoto Oscillators

All-to-all coupled phases θ_i with intrinsic frequencies ω_i: order parameter r = |N⁻¹ Σ e^{iθ_i}| grows as coupling K crosses the synchronization window—classic mean-field rhythm transition.

About this model

The Kuramoto model is the canonical mean-field example of collective synchronization: N coupled phase oscillators with natural frequencies ω_i interact through an all-to-all sine coupling of strength K. Writing the complex order parameter Z = r e^{iψ} = N⁻¹ Σ_j e^{iθ_j}, each phase obeys θ̇_i = ω_i + K r sin(ψ − θ_i)—the same law as the full sum, but computed in O(N) time. This page integrates the ODEs with RK4, draws each oscillator as a colored dot on the unit circle (hue encodes ω_i), shows the mean-field pointer Z in gold, and plots r(t) as a simple coherence diagnostic. You can sweep K and the frequency spread to watch partial and nearly complete locking on a finite system; quantitative critical coupling depends on the frequency distribution and the N → ∞ limit.

Who it's for: Students in nonlinear dynamics, statistical physics, or networks who want a clean bridge between coupled oscillators and order-parameter thinking.

Key terms

Kuramoto model
Synchronization
Order parameter
Mean-field coupling
RK4
Phase oscillator

How it works

The Kuramoto model couples N phase oscillators with natural frequencies ω_i and all-to-all interaction strength K. It is the standard playground for synchronization: beyond a coupling threshold, a macroscopic fraction of oscillators locks to a common rhythm, tracked by the complex order parameter Z = N⁻¹ Σ e^{iθ_i}.

Key equations

θ̇_i = ω_i + (K/N) Σ_j sin(θ_j − θ_i). With Z = r e^{iψ} = N⁻¹ Σ_j e^{iθ_j}, the sum becomes K r sin(ψ − θ_i), so each oscillator feels the mean field of the population.

Frequently asked questions

Why is the dynamics O(N) if the textbook has a sum over j?: Because Σ_j sin(θ_j − θ_i) = Im(e^{-iθ_i} Σ_j e^{iθ_j}) = N Im(Z e^{-iθ_i}) = N r sin(ψ − θ_i). The entire population enters only through the two numbers r and ψ, so each timestep is linear in N.
Is r = 1 perfect global synchronization?: Usually r ≈ 1 means almost all oscillators are locked into a common rotating cluster; exact r = 1 requires identical phases modulo 2π, which finite spread in ω_i typically prevents unless coupling is very strong relative to disorder.
What does the Lorentzian K_c = 2γ note mean?: For the Kuramoto model in the infinite-N limit with Cauchy-distributed ω of half-width γ, the incoherent state loses stability at K_c = 2γ. Real finite boxes of uniform frequencies behave similarly in spirit but not identically; treat it as intuition, not a fitted critical point here.

Similar topics nearby (including related fields) — or browse all 46 in Thermodynamics.

View category →

NewSchool

Huygens Pendulum Sync

Launch Simulator

Two pendula + κ(θ₁−θ₂) on shared beam; phases drift toward lock.

NewSchool

Coupled FitzHugh–Nagumo Neurons

Launch Simulator

Two excitable FHN units with diffusive coupling: phase plots, time traces, and synchronization vs coupling strength.

NewUniversity / research

2D Ising Model

Launch Simulator

Square lattice Metropolis: kT/J vs |m| and E; T_c ≈ 2.27; optional field h.

School

Phase Diagram

Launch Simulator

Temperature-pressure diagram with phase transitions.

NewSchool

Leidenfrost Effect (Toy)

Launch Simulator

T_plate vs ~200 °C threshold: vapor gap and lifetime curves — pedagogical, not measured boiling data.

NewSchool

Vicsek Model (Active Matter)

Launch Simulator

Self-propelled particles on a torus align headings with neighbors within radius R, add noise η, then drift at v₀; polarization V_a = |⟨e^{iθ}⟩| captures the flocking crossover with density ρ = N/L².

Kuramoto Oscillators

About this model

How it works

Key equations

Frequently asked questions

Related simulators

Huygens Pendulum Sync

Coupled FitzHugh–Nagumo Neurons

2D Ising Model

Phase Diagram

Leidenfrost Effect (Toy)

Vicsek Model (Active Matter)

Kuramoto Oscillators

About this model

How it works

Key equations

Frequently asked questions