Biological systems in Nature are intrinsically out-of-equilibrium to maintain their structural complexity and functional diversity. Similarly, out-of-equilibrium dissipative colloidal systems subjected to an external energy injection often develop nontrivial collective dynamics and self-organize into large scale structures, which are far more complex than their equilibrium counterparts [1-17]. The main sources of such emergent behavior are the many-body dissipative interactions between colloids (e. g. steric, electrostatic, magnetic), the external energy injection, and the coupling of particles dynamics through the fluid flow around them. Collective dynamics and self-organization in out-of-equilibrium colloidal systems (often termed as active colloids) is a rapidly growing area of research which led to the discovery of novel dynamic architectures and functionalities that are not generally available at equilibrium.

 Colloidal systems have been the subject of intense research for a long time due to their ubiquitous technological applications. Colloidal particles display Brownian motion, size in the visible wavelength and dynamics in experimentally accessible timeframes (milliseconds to seconds) making them an attractive platform for the experiments and the computational modeling. The pair interactions between particles can be easily adjusted in strength and range by applying relatively small external fields. When driven by external forces or an internal energy source, colloids can mimic motile biological entities and can serve as a testbed for exploring the rich and complex physics of out-of-equilibrium systems. These dissipative colloidal structures utilize energy to generate and maintain structural complexity. Experiments and numerical simulations along this line of research have often revealed nontrivial collective dynamics and emergent large-scale structures [1-17]. With the proposed workshop we would like to provide a platform for discussing several new and important trends in this field of active colloidal materials, that is, chirality, non-reciprocity, and memory.

A recent hot trend in the field of active colloids explores the emergence of coherent motion and self-organization in systems with chirality [5-11]. Chirality is an intrinsic fundamental property of many natural and synthetic systems. Colloidal particles driven by external torques [12-18] constitute an ideal model system to investigate these phenomena since they avoid the inherent complexity of biological active matter. Spinning   particles dispersed in a fluid represent a special class of artificial active systems that inject vorticity at the microscopic level [19-25]. Dense collections of interacting spinning particles represent a chiral fluid [26], which breaks parity and time-reversal symmetries, and displays a novel viscosity feature called the odd viscosity and elasticity [27, 28]. The odd viscosity has been identified in interacting chiral spinners [29], and it led to remarkable effects such as production of flow perpendicular to the pressure [27], topological waves [30], or the emergence of edge currents [29]. Magnetic rollers dynamically assemble into a vortex under harmonic confinement, that spontaneously selects a sense of rotation and is capable of chirality switching [31,32]. Multiple motile vortices unbound from any confinement have been revealed in ensembles of magnetic rollers powered by a uniaxial field [33]. Oscillating chiral flows were generated when a roller liquid was coupled to fixed obstacles [34]. There has been an increasing effort to investigate collective phenomena in systems composed of    chiral active units [11, 35-40]. Synchronized self-assembled magnetic spinners at the liquid interface revealed structural transitions from liquid to nearly crystalline states and demonstrated reconfigurability coupled to a self-healing behavior [41]. Activity-induced synchronization leading to a mutual flocking, and chiral self- sorting has been observed in modeled ensembles of self-propelled circle swimmers [42]. Shape anisotropic particles powered by the Quincke phenomenon led to the realization of chiral rollers (similar to circle swimmers) with spontaneously selected handedness of their motion and activity-dependent curvature of trajectories [43].

Another fast-developing direction in the field of non-equilibrium active and driven colloids is the realization of systems characterized by non-reciprocity of interactions or memory effects and how they can lead to emerging collective phenomena. Due to the intrinsic nonequilibrium nature of active systems, the couplings between particles often deviate from the standard form derivable from a Hamiltonian. One intriguing example is a time-delayed coupling involving a discrete delay time (or a distribution of such times). Such a situation arises, for example, through a delay in communication or sensing, and can be artificially created via a feedback loop [44]. Another topic attracting a lot of attention in the community is based on active systems with nonreciprocal couplings that can arise, for example, through chemotaxis or phoretic interactions between self-propelling colloids [45], or through predator-prey or vision-cone interactions [46,47] in macroscopic active systems. On the collective level, is now well established that non-reciprocity can induce new types of phase transitions [48] and patterns with broken time- and parity symmetry, including travelling patterns [49,50] and globally chiral motion without chirality of the individual constituents [51]. While many of these studies have been pursued only at a mean field-theoretical level, there is also an increasing interest in understanding corresponding particle-scale effects, that can only be accessed by numerical simulations [52] or corresponding experiments. For example, non-reciprocal interactions may generate new types of self-assembled systems able to learn and to produce transition between different shapes [53]. Establishing the precise connection between the different length and time scales is still an important challenge. Here, computer simulations are an indispensable tool.

Many standard models of active motion implicitly assume an inert (equilibrium) environment yielding instantaneous friction and noise. In contrast, several recent studies [54,19] explore the impact of retarded friction as it arises in viscoelastic environments made, e.g., of polymers, liquid crystals, or biological tissues [55-57]. An extreme case is time-delay [44]. From a theoretical and computational perspective, retarded friction or, more generally, non-Markovian dynamics, still provides a severe challenge. This concerns, e.g., the extraction (or modelling) of memory kernels, but also the actual solution of the coupled equations of motion, each being subject to history effects. As a consequence, only few studies on the emerging collective behavior of active particles with memory are currently available, including collective effects in systems of feedback-driven colloids [58] and pattern formation in a non-Newtonian active system [59]. Advancing numerical methods capable of treating memory effects will become more and more important in view of the recent experimental progress in this field. Experimentally, the memory effects in the system can be induced, e.g., by temporal activity modulations at intermediate timescales of the interactions in the colloidal ensemble [60]. Such modulations generate active particles with partial memory (at the particle level) of their motion from the previous activity cycles (either through partial depolarization or remnant hydrodynamic flows induced by the particle motion). Novel dynamic patterns (such as localized multiple vortices, flocks, pulsating lattices) has been revealed in ensembles of Quinke rollers [60,61]. When coupled to the fluid flows, active particle with memory can produce activity shockwaves [62]. Also, it has been recently demonstrated that active colloidal ensembles realized by Quinke rollers can effectively develop “ensemble memory”, where the information about the dynamic state of the system is distributed over the whole ensemble [63]. This information can be effectively exploited to command subsequent collective polar states of the active colloidal ensemble through activity cycling [63] and can pave the way toward direct applications in different technological fields related to microfluidics and microrobotics.

Developing fundamental understanding of the complex colloidal dynamics in systems driven out-of-equilibrium by external fields represents a significant theoretical and computational challenge as it involves multi-body interactions, the overlapping of length- and timescales, and the coupling of particle interactions with the fluid flow. Some of the features may be understood using phenomenological using continuum descriptions [21-23] Nevertheless, the microscopic mechanisms leading   to the dynamic self-assembly and their relations to the emergent behavior in active colloidal fluids with chirality, non-reciprocal interactions, and memory often remain unclear. Computer simulations are practically the only method to theoretically investigate such questions. However, modeling of the nonequilibrium dynamics presents a formidable computational challenge due to the complex many- body interactions and collective dynamics at different time and lengths scales. One of the main challenges is to properly account for the particle-fluid coupling. On a coarse-grained level, the fluid flow around colloids is modeled by molecular dynamics methods like Lattice-Boltzmann [64] and Multi Particle Collision Dynamics [65,66]. An alternative approach is to describe the colloidal dynamics by molecular dynamics simulation, or an amplitude equation (Ginzburg-Landau type equation) coupled to the Navier-Stokes equations describing large-scale time- averaged hydrodynamic flows induced by the colloids [67,68].

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