Energetics of synchronization in coupled oscillators rotating on circular trajectories

We derive a concise and general expression of the energy dissipation rate for coupled oscillators rotating on circular trajectories by unifying the nonequilibrium aspects with the nonlinear dynamics via stochastic thermodynamics. In the framework of phase oscillator models, it is known that the even and odd parts of the coupling function express the effect on collective and relative dynamics, respectively. We reveal that the odd part always decreases the dissipation upon synchronization, while the even part yields a characteristic square-root change of the dissipation near the bifurcation point whose sign depends on the specific system parameters. We apply our theory to hydrodynamically coupled Stokes spheres rotating on circular trajectories that can be interpreted as a simple model of synchronization of coupled oscillators in a biophysical system. We show that the coupled Stokes spheres gain the ability to do more work on the surrounding fluid as the degree of phase synchronization increases.

Figure 1

Schematic illustration of the model.

Figure 2

(a) Schematic illustration of hydrodynamically coupled Stokes spheres on circular trajectories described by Eqs. (26) and (27). (b) The mean frequency ${\mathrm{\Omega }}_i$ and (c) the energy dissipation rate $P$ normalized by $\gamma l^2$ as a function of the hydrodynamic coupling parameter $G$. The numerical data obtained by Eqs. (26) and (27) (original), Eqs. (33) and (34) (cycle-averaged), and Eqs. (33) and (34) with $\epsilon =0$ with other parameters being unchanged (noiseless) are compared, where the theoretical bifurcation point $G^{*}=\left|\frac{\mathrm{\Delta }\omega }{\overline{\omega }A}\right|\simeq 0.0664$. The theoretical expression of $P$ in Eq. (21) (theory) is also compared with its numerical counterpart (cycle-averaged) in (c). The dotted line in (c) denotes the contribution from the (normalized) uncoupled part ${\omega }_1^2+{\omega }_2^2$ in Eq. (21).

Figure 3

The decomposition of the energy dissipation rate (theory) from Fig. 2: (a) The odd part $P_{\mathrm{o}}$ as a function of the hydrodynamic coupling parameter $G$ with the (normalized) theoretical value in Eq. (36). (b) The even part $P_{\mathrm{e}}$ as a function of the hydrodynamic coupling parameter $G$ with the (normalized) theoretical curve after the frequency synchronization in Eq. (37) with ${\theta }^{*}=-\frac{\pi }{2}$ and $G^{*}=\left|\frac{\mathrm{\Delta }\omega }{\overline{\omega }A}\right|\simeq 0.0664$.