Two hundred mechanical metronomes ( ) stand on a shared plywood board that is free to…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Synchronisation, cascades and the resilience of networks

Problem

Two hundred mechanical metronomes (N=200N=200) stand on a shared plywood board that is free to rock gently from side to side; the board's small motion couples every metronome to every other one, so the coupling is well approximated as all-to-all with a single coupling strength KK (units rad s−1\mathrm{rad\,s^{-1}}), as in the Kuramoto model θ˙i=ωi+KN∑j=1Nsin⁡(θj−θi)\dot\theta_i=\omega_i+\dfrac{K}{N}\sum_{j=1}^{N}\sin(\theta_j-\theta_i). The metronomes' natural frequencies ωi\omega_i (deviations from their mean tick rate) are independently drawn from a Lorentzian density g(ω)=1πΔω2+Δ2g(\omega)=\dfrac{1}{\pi}\dfrac{\Delta}{\omega^2+\Delta^2} with half-width Δ=0.20 rad s−1\Delta=0.20\ \mathrm{rad\,s^{-1}}. (a) Starting from the coupling sum above and the complex order parameter reiψ=N−1∑jeiθjre^{i\psi}=N^{-1}\sum_j e^{i\theta_j}, show that the equation of motion of metronome ii can be rewritten as θ˙i=ωi+Krsin⁡(ψ−θi)\dot\theta_i=\omega_i+Kr\sin(\psi-\theta_i): each metronome feels only the board's collective phase and amplitude, not its neighbours individually. (b) Using the synchronisation threshold formula for a general frequency density, compute the coupling KcK_{\rm c} at which the board's metronomes begin to lock. (c) A simulation of these N=200N=200 metronomes (frequencies redrawn once from the stated Lorentzian, seed 1111, fourth-order Runge-Kutta integration with step dt=0.01 s\mathrm{d}t=0.01\ \mathrm{s} to t=500 st=500\ \mathrm{s}, discarding a transient of 100 s100\ \mathrm{s}, averaged over 2020 realisations) gives the measured order parameter r=0.58±0.02r=0.58\pm0.02 at K=0.60 rad s−1K=0.60\ \mathrm{rad\,s^{-1}}, r=0.70±0.02r=0.70\pm0.02 at K=0.80 rad s−1K=0.80\ \mathrm{rad\,s^{-1}}, and r=0.82±0.02r=0.82\pm0.02 at K=1.20 rad s−1K=1.20\ \mathrm{rad\,s^{-1}}. Compute the value predicted by the exact Lorentzian result at each of these three couplings, and state whether the measurements confirm the theory.

Browse the problem library