PHYSICAL REVIEW E 107, 024114 (2023) Weak ergodicity breaking and anomalous diffusion in collective motion of active particles under spatiotemporal disorder Hongda Shi, 1 Luchun Du , 2 Feijie Huang, 1 and Wei Guo 1, 3, 4 , * 1 Key Laboratory of Artificial Microstructures in Yunnan Higher Education Institutions, School of Physical Science and Technology, Kunming University, Kunming 650214, China 2 Department of Physics, Yunnan University, Kunming 650091, China 3 Yunnan Key Laboratory of Metal-Organic Molecular Materials and Devices, Kunming University, Kunming 650214, China 4 National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, China (Received 17 August 2021; revised 25 July 2022; accepted 12 January 2023; published 10 February 2023) The effects of spatiotemporal disorder, i.e., both the noise and quenched disorder, on the dynamics of active particles in two dimensions are investigated. We demonstrate that within the tailored parameter regime, noner- godic superdiffusion and nonergodic subdiffusion occur in the system, identified by the observable quantities (the mean squared displacement and ergodicity-breaking parameter) averaged over both the noise and realizations of quenched disorder. Their origins are attributed to the competition effects between the neighbor alignment and spatiotemporal disorder on the collective motion of active particles. These results may be helpful for further understanding the nonequilibrium transport process of active particles, as well as for detection of the transport of self-propelled particles in complex and crowded environments. DOI: 10.1103/PhysRevE.107.024114 I. INTRODUCTION The collective motion of active particles has been exten- sively explored heretofore [1–5], ranging from bacteria [6,7] and active colloid [8] to animals such as fish [9] and birds [10]. The active particles can transform the internal energy into the energy of motion that drives themselves out of equilibrium. Generally, the magnitude of self-propelled speed is consid- ered as keeping constant, while the direction of speed suffers stochastic fluctuations (space-dependent, termed disorder, or time-dependent, termed noise) from surroundings [1,2]. These stochastic fluctuations exert significant effects on their collec- tive motion [1,2], e.g., forming the traveling bands [11] and lattices and polar lanes [12], to mention but a few. Over the past couple decades, the anomalous diffu- sion and the ergodicity breaking in the disorder systems have been of wide concerned [3,13,14–20]. The anoma- lous diffusion refers to a situation different from the normal diffusion (α = 1), which is characterized by the ensemble- averaged mean squared displacement (MSD): = [〈X 2 (t )〉− 〈X (t )〉 2 ] ∝ t α , where α< 1 is subdiffusion and α> 1 is superdiffusion [14,15,19,20]. Expressly, α = 2 and α = 0, respectively, correspond to ballistic diffusion and localiza- tion [14,18,19]. Recently, interesting work has been to decode anomalous diffusion using machine learning [21]. Three major tasks have been accomplished: (1) inference of the diffusion exponent (α), (2) model classification, and (3) tra- jectory segmentation. More significantly, the Bayesian deep learning technique has been employed to obtain high predic- tion accuracies in the analysis of anomalous diffusion [22]. * guoweiphys@163.com The anomalous diffusion is usually accompanied by ergodic- ity breaking [13,14,18,19]. The ergodicity breaking includes the following: (1) there are noncontinuous regions in the states or phase spaces of systems, i.e., strong ergodicity breaking, and (2) there exist continuous regions but the as- sociated trajectories cannot sample completely them even over an infinite period of time, i.e., weak ergodicity break- ing [13,14,23]. Generally speaking, since the state spaces of systems are not known, one practically adopts the equal- ity between the ensemble average and time average for an observable, e.g., the MSD characteristic, to identify the ergod- icity of stochastic systems. Practically, the ergodicity is vital for interpreting information provided by the experiments of single-particle tracking [23]: whether or not it is representa- tive of the ensemble average. (It is true only for the ergodic processes.) The ergodicity breaking has been discovered both in many theoretical models, e.g., the continuous-time ran- dom walk processes [24,25], the heterogeneous diffusion process [16,26], the scaled Brownian motion [27], and dis- order systems [15,18,23], and in numerous experiments, e.g., in the blinking dynamics of quantum dots [28] and living cells [29–31]. Motivated by the works concentrating on abundant phase diagrams induced by the memory of noise [12] and localiza- tion induced by quenched disorder [32], here we would like to explore the anomalous diffusion and ergodicity breaking in a fundamentally conceptual paradigm—the two-dimensional collective motion of active particles under spatiotemporal dis- order (i.e., under both the noise and quenched disorder). As noted in Ref. [32], the behaviors of system with pure disorder may depend on either the initial condition or the specific disorder (the ergodicity lost); namely, either the different initial conditions or different realizations of the quenched 2470-0045/2023/107(2)/024114(11) 024114-1 ©2023 American Physical Society