Probing the Role of Mobility in the Collective Motion of Nonequilibrium Systems

By systematically varying the mobility of self-propelled particles in a 2D lattice, we experimentally study the influence of particle mobility on system’s collective motion. Our system is intrinsically nonequilibrium due to the lack of energy equipartition. By constructing the covariance matrix of spatial fluctuations and solving for its eigenmodes, we obtain the collective motions of the system with various magnitudes. Interestingly, our structurally ordered nonequilibrium system exhibits properties almost identical to disordered glassy systems under thermal equilibrium: the modes with large overall motions are spatially correlated and quasilocalized while the modes with small collective motions are highly localized, resembling the low- and high-frequency modes in glass. More surprisingly, a peak similar to the boson peak forms in our nonequilibrium system as the number of mobile particles increases, revealing the possible origin of the boson peak from a dynamic aspect. We further illustrate that the spatially correlated large-movement modes can be produced by the cooperation of highly active particles above a threshold fraction, while the localized small-movement modes can be created by adding individual inactive particles. Our study clarifies the role of mobility in collective motions, and further suggests a promising possibility of extending the powerful mode analysis approach to nonequilibrium systems.

Figure 1

(a) The image of our system. Identical metal spheres driven by independent motors are connected by springs to form a 2D square lattice. (b) The spatial fluctuations in both $x$ and $y$ directions for all particles and frames. The data for each particle are renormalized by the local variance $\sigma$. The red line is the standard Gaussian function.

Figure 2

(a) Three typical eigenmodes at small, intermediate, and large $\widehat{\omega }$. (b) The spatial correlation function in direction for the three typical modes. (c) The participation ratio $P$ of all the modes.

Figure 3

(a) The $D(\widehat{\omega })/\widehat{\omega }$ spectra of four typical samples, $A$ to $D$, demonstrate the formation of a low-$\widehat{\omega }$ peak. (b) The distribution of time-averaged local displacement, $⟨\delta r_i⟩$, for samples $A$ to $D$. To achieve such distributions, the four samples are driven with different voltages: $A$ (2.5 V), $B$ (normally 2.5 V with 16.5% of 2 V), $C$ (normally 2.5 V with 20.7% of 3.2 V), and $D$ (normally 2.5 V with 20.7% of 3.2 V and 20.7% of 3.6 V). $⟨\delta r_i⟩$’s are renormalized to make the main peak locate at 1. (c) Tuning the active particles in sample $D$ back to normal activities eliminates the peak. Inset shows the $⟨\delta r_i⟩$ distribution with and without active particles. (d) The relative importance of small-, medium-, and large-mobility groups throughout all $\widehat{\omega }$. The hatched area where all groups are comparable corresponds to the peak in the participation ratio.

Figure 4

(a) The variation in cumulative number of modes $N(\widehat{\omega })$ as $N_a$ active particles are added. Inset: the number of low-$\widehat{\omega }$ quasilocalized modes ($P<0.1$) versus $N_a$ increases abruptly around the threshold value of $N_a=6$ (i.e., 5%). (b) Large motions occur at the active-particle sites (labeled with circles) in the low-$\widehat{\omega }$ modes. (c) and (d) compare the participation ratio in systems with and without active particles: panel (c) shows the situation of $N_a<6$ and panel (d) the situation of $N_a>6$.

Figure 5

(a) The cumulative number of modes $N(1/\widehat{\omega })$ for different numbers of inactive particles. Inset: the number of high-$\widehat{\omega }$ localized modes ($P<0.05$) versus $N_i$. (b) Motions are concentrated at inactive particles (labeled with circles) in high-$\widehat{\omega }$ modes. (c) As $N_i$ increases, the participation ratio in the large-$\widehat{\omega }$ region decreases significantly.