Effective attractive and repulsive interactions behind lift synchronization

Abstract

Synchronization is a ubiquitous phenomenon in nonequilibrium systems. One intriguing example found in every-day life is lifts installed next to each other, that move closely and arrive almost simultaneously during a busy time. However, the basic mechanism behind this lift synchronization is yet to be elucidated. Here, we investigate the effective interaction acting between the lifts quantitatively. Through the analysis on the time-series data obtained by numerically solving a rule-based discrete model of lifts, in which passengers at each floor show up stochastically and call a lift that is expected to arrive first, we find that the effective interaction acting between the lifts consists of not only attraction but also repulsion. By changing the parameters of the rule-based model, we are successful to tune the ratio of these competing interactions and to control the dynamics of lifts, realising the transition between in-phase and anti-phase synchronizations. Our strategy is applicable to the data of real lifts, and thus it is expected to help controlling lift systems. We believe that this study provides a novel approach to design optimal transportation, which is of great importance in improving sustainability of social systems.

Figures (3)

• Figure 1: Schematics of the model and resulting trajectories. (a) The rule-based discrete model with two lifts and (b) the definition of phase $\varphi_{\ell}$. (c--f) Trajectories of two lifts (blue and magenta lines) with $\gamma = 10$ for (c) $\mu=0.1$ and (d) $\mu=1$, (e) with $\gamma = 3$ for $\mu = 1$, and (f) with $\gamma=8$ for $\mu=1$. On each floor the gray horizontal lines indicate the existence of waiting passengers and the gray vertical bars represents appearance of new passengers. The black arrowheads indicate the departure times $\hat{t}_i$.
• Figure 2: Effective interaction of lifts for $\mu = 1$ and $\gamma=10$. (a) The measured coupling function $G(\varphi_{12})$, as well as (b) that for co-descent period $G^\prime(\varphi_{12})$ and (c) that for non-co-descent period $G^{\prime\prime}(\varphi_{12})$ are displayed against phase difference $\varphi_{12}$. The red filled dots and blue open circles represent stable and unstable fixed points, respectively. The stable fixed point of $G(\varphi_{12})$ and $G^\prime(\varphi_{12})$ at $\varphi_{12} = 0$ demonstrate the existence of effective attractive interaction overall that originates from the co-descent period, while the unstable fixed point $\varphi_{12} = 0$ of $G^{\prime\prime}(\varphi_{12})$ indicate the existence of effective repulsive interaction during non-co-descent period. The errorbars are for the ensemble average over 1000 data sets.
• Figure 3: Transition of in-phase and anti-phase synchronization states. (a) Phase diagram. The colour represents the value of $\tilde{\Phi}_K = \Phi_K \,{\rm sign}{(\cos{\Theta_K})}$, and the black bars indicates the value of $\Theta_K$. The thick green and magenta region correspond to in-phase and anti-phase synchronization states, respectively. (b,d) Dependence of the synchronization state on (b) the stopping time $\gamma$ and (d) the rate $x$ of the stochastic stopping time. Order parameters $\Phi_K$ and $\Theta_K$, round trip time $T$, and the mean times for waiting $\tau_{\rm w}$, riding $\tau_{\rm r}$, and travelling $\tau_{\rm t}$ are displayed. The stochastic stopping time takes a value unity with the rate $1-x$ and 10 with the rate $x$. Ensemble average is calculated over 100 samples and the error bars represent the standard deviation. In the third panels for $T$, the dashed and dotted gray lines indicate the round trip time when a lift stops at all floors ($T^S$) and half of them ($T^{S/2}$), respectively. (c) Distribution of the departure time intervals $\theta = 2\pi \mathit{\Delta}\hat{t} / T$ for $\mu=1$ and $\gamma = 1$ (left), 4 (middle), and 10 (right). The purple arrow shows its average, i.e., $\Phi_K \exp{\left[ i \Theta_K \right]}$. The black dotted line represents $\theta^\dag = 2\pi S/T$. In the right panel for $\gamma=10$, its complement $2\pi -\theta^\dag$ is also plotted.