arXiv subjects

Heiko Rieger

Publications and source records attributed to Heiko Rieger.

At least 19 recordsLinked to original sources

arXiv2026

Extensive Spatio-Temporal Chaos in Non-reciprocal Flocking

Non-reciprocal interactions in active matter give rise to a multitude of fascinating phenomena among which are collective oscillatory states without intrinsic particle chirality and active turbulence. Here we show that in a paradigmatic model for non-reciprocal flocking, the two species Vicsek model, these two states coexist: chiral order for small flocks, and extensive spatiotemporal chaos for large flocks, both separated by a finite-wavelength instability whose scale is set by the rotation radius of the chiral orbits. For system sizes larger than this length scale extensive spatiotemporal chaos unfolds, as manifested by an extensive number of positive Lyapunov exponents as well as of Floquet exponents, a finite correlation and chaotic length and a broad energy spectrum. Our results suggest that complex, turbulent behavior is a generic possibility in systems where particles or fields interact asymmetrically, and may have significant implications for understanding how non-reciprocal interactions could drive chaotic, fluid-like behavior in active matter.

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arXiv2026

Field-controlled interfacial transport and pinning in an active spin system

Field control provides a practical route to programmable active matter, yet how weak fields modify non-equilibrium coexistence and interfaces remains unclear. To address this, we study a minimal flocking model of active Potts particles coupled to an external field and show that even weak fields can reconfigure phase behavior and interfacial dynamics. For a homogeneous unidirectional field, the flocking phase is reshaped: the coexistence regime between an apolar gas and a polar liquid is replaced by a phase separation between two field-aligned polar phases: a low-density, weakly polarized background and a high-density, strongly polarized band, both moving along the field. When the system forms a dense longitudinal lane oriented transverse to the field, it executes a slow treadmilling motion against the field, driven by the weakly polarized background. If the system is divided into regions with opposite field directions, particles accumulate at the interface, leading to field-induced interface pinning with flocks performing back-and-forth oscillatory motion. In the presence of quenched random field orientations, this pinning favors a disordered state in which global order diminishes with increasing system size, consistent with Imry-Ma arguments, while the quenched disorder smoothens sharp first-order signatures, in line with the Aizenman-Wehr theorem, with activity modifying the scaling. A coarse-grained hydrodynamic theory supports these observations and is consistent with microscopic simulations.

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arXiv2025

Nonreciprocal yet Symmetric Multi-Species Active Matter: Emergence of Chirality and Species Separation

Nonreciprocal active matter systems typically feature an asymmetric role among interacting agents, such as a pursuer-evader relationship. We propose a multi-species nonreciprocal active matter model that is invariant under permutations of the particle species. The nonreciprocal, yet symmetric, interactions emerge from a constant phase shift in the velocity alignment interactions, rather than from an asymmetric coupling matrix. This system possessing permutation symmetry displays rich collective behaviors, including a species-mixed chiral phase with quasi-long-range polar order and a species separation phase characterized by vortex cells. The system also displays a coexistence phase of the chiral and the species separation phases, in which intriguing dynamic patterns emerge. These rich collective behaviors are a consequence of the interplay between nonreciprocity and permutation symmetry.

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arXiv2025

Collective behavior in the nonreciprocal multi-species Vicsek model

We investigate collective behavior in a $Q$-species Vicsek model with a nonreciprocal velocity alignment interaction. This system is characterized by a constant phase shift $\alpha$ in the inter-species velocity alignment rule. While the phase shift renders the interaction nonreciprocal, the system is globally invariant under any permutations of particle species, possessing Potts symmetry. The combination of Potts symmetry and nonreciprocity gives rise to a rich phase diagram. The nonreciprocal phase shift generates either counter-clockwise or clockwise chirality. Potts symmetry can be broken spontaneously. Consequently, the system exhibits four distinct phases: A species-mixed chiral phase where particles perform counter-clockwise chiral motion with quasi-long-range order, a species separation phase where Potts symmetry is broken and species-separated particles form vortex cells with clockwise chirality, a coexistence phase, and a disordered phase, for $0<\alpha<\pi$. We derive a Boltzmann equation and a hydrodynamic equation describing the system in the continuum limit, and present analytic arguments for the emergence of chirality and species separation.

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arXiv2025

Flocking with random non-reciprocal interactions

Flocking is ubiquitous in nature and emerges due to short- or long-range alignment interactions among self-propelled agents. Two unfriendly species that antialign or even interact nonreciprocally show more complex collective phenomena, ranging from parallel and antiparallel flocking over run-and-chase behavior to chiral phases. Whether flocking or any of these collective phenomena can survive in the presence of a large number of species with random nonreciprocal interactions remained elusive so far. As a first step here, the extreme case of a Vicsek-like model with fully random nonreciprocal interactions between the individual particles is considered. For infinite-range interaction, as soon as the alignment bias is of the same order as the random interactions, the ordered flocking phase occurs, but deep within this phase, the random nonreciprocal interactions can still support global chiral and oscillating states in which the collective movement direction rotates or oscillates slowly. For short-range interactions, moreover, even without alignment bias self-organized cliques emerge, in which medium-size clusters of particles that have predominantly aligning interactions meet accidentally and stay together for macroscopic times. These results may serve as a starting point for the study of multispecies flocking models with nonrandom but complex nonreciprocal interspecies interactions.

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arXiv2025

Stability of discrete-symmetry flocks: sandwich state, traveling domains and motility-induced pinning

Polar flocks in discrete active systems are often assumed to be robust, yet recent studies reveal their fragility under both imposed and spontaneous fluctuations. Here, we revisit the four-state active Potts model (APM) and show that its globally ordered phase is metastable across a broad swath of parameter space. Small counter-propagating droplets disrupt the flocking phase by inducing a persistent sandwich state, where the droplet-induced opposite-polarity lane remains embedded within the original flock, particularly pronounced at low noise, influenced by spatial anisotropy. In contrast, small transversely propagating droplets, when introduced into the flock, can trigger complete phase reversal due to their alignment orthogonal to the dominant flow and their enhanced persistence. At low diffusion and strong self-propulsion, such transverse droplets also emerge spontaneously, fragmenting the flock into multiple traveling domains and giving rise to a short-range order (SRO) regime. We further identify a motility-induced pinning (MIP) transition in small diffusion and low-temperature regimes when particles of opposite polarity interact, flip their state, hop, and pin an interface. Our comprehensive phase diagrams, encompassing full reversal, sandwich coexistence, stripe bands, SRO, and MIP, delineate how thermal fluctuations, self-propulsion strength, and diffusion govern flock stability in discrete active matter systems.

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arXiv2025

Minimum Action Principle for Entropy Production Rate of Far-From-Equilibrium Systems

The Boltzmann distribution connects the energetics of an equilibrium system with its statistical properties, and it is desirable to have a similar principle for non-equilibrium systems. Here, we derive a variational principle for the entropy production rate (EPR) of far-from-equilibrium discrete state systems, relating it to the action for the transition probability measure of discrete state processes. This principle leads to a tighter, non-quadratic formulation of the dissipation function, speed limits, the thermodynamic-kinetic uncertainty relation, the large deviation rate functional, and the fluctuation relation, all within a unified framework of the thermodynamic length. Additionally, the optimal control of non-conservative transition affinities using the underlying geodesic structure is explored, and the corresponding slow-driving and finite-time optimal driving exact protocols are analytically computed. We demonstrate that discontinuous endpoint jumps in optimal protocols are a generic, model-independent physical mechanism that reduces entropy production during finite-time driving of far-from-equilibrium systems.

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arXiv2025

Generalized Finite-time Optimal Control Framework in Stochastic Thermodynamics

Optimal processes in stochastic thermodynamics represent a frontier for understanding the control and design of non-equilibrium systems, with broad practical applications in biology, chemistry, and nanoscale/mesoscale systems. Optimal transport theory and thermodynamic geometry have emerged as leading optimal control methodologies, but both rely on slow--driving and close-to-equilibrium assumptions. An optimal control framework in stochastic thermodynamics for finite--time driving remains elusive. Here, we solve an optimal control problem for driving the control parameters of a discrete-state far-from-equilibrium process from an initial to a final value in finite time. Optimal driving protocols are derived that minimize the total finite--time dissipation cost of the driving process. Our framework reveals that discontinuous endpoint jumps are a generic, model-independent physical mechanism that minimizes the optimal driving entropy production --- `geometric thermodynamic far-from-equilibrium shortcuts in swift state-to-state transformations' --- whose importance is further amplified in far-from-equilibrium systems. The thermodynamic and dynamical interpretation of discontinuous endpoint jumps is formulated. An exact mapping between the finite--time and slow--driving optimal control formulations is elucidated, advancing the state of the art in optimal transport theory and thermodynamic geometry, which has been the prevailing paradigm for studying optimal processes in stochastic thermodynamics under slow--driving assumptions. Our framework opens up a broad range of applications to the thermodynamically efficient control of far-from-equilibrium systems in finite time, and thereby a route towards their efficient design principles.

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arXiv2025

Thermodynamic Length in Stochastic Thermodynamics of Far-From-Equilibrium Systems: Unification of Fluctuation Relation and Thermodynamic Uncertainty Relation

The Boltzmann distribution for an equilibrium system constrains the statistics of the system by the energetics. Despite the non-equilibrium generalization of the Boltzmann distribution being studied extensively, a unified framework valid for far-from-equilibrium discrete state systems is lacking. Here, we derive an exact path-integral representation for discrete state processes and represent it using the exponential of the action for stochastic transition dynamics. Solving the variational problem, the effective action is shown to be equal to the inferred entropy production rate (a thermodynamic quantity) and a non-quadratic dissipation function of the thermodynamic length (TL) defined for microscopic stochastic currents (a dynamic quantity). This formulates a far-from-equilibrium analog of the Boltzmann distribution, namely, the minimum action principle. The non-quadratic dissipation function is physically attributed to incorporating non-Gaussian fluctuations or far-from-equilibrium non-conservative driving. Further, an exact large deviation dynamical rate functional is derived. The equivalence of the variational formulation with the information geometric formulation is proved. The non-quadratic TL recovers the non-quadratic thermodynamic-kinetic uncertainty relation (TKUR) and the speed limits, which are tighter than the close-to-equilibrium quadratic formulations. Moreover, if the transition affinities are known, the non-quadratic TL recovers the fluctuation relation (FR). The minimum action principle manifests the non-quadratic TKUR and FR as two faces corresponding to the thermodynamic inference and partial control descriptions, respectively. In addition, the validity of these results is extended to coarse-grained observable currents, strengthening the experimental/numerical applicability of them.

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arXiv2025

Thermodynamically Consistent Coarse-graining: from Interacting Particles to Fields via Second Quantization

We systematically derive an exact coarse-grained description for interacting particles with thermodynamically consistent stochastic dynamics, applicable across different observation scales, the mesoscopic and macroscopic scales. We implement the coarse-graining procedure using the Doi--Peliti field theory, which preserves microscopic noise effects on the mesoscopic/macroscopic scale. The exact mapping reveals the key role played by the Poissonian particle occupancy statistics. We show the implications of the exact coarse-graining method using a prototypical flocking model, namely the active Ising model, which exhibits a mismatch between the microscopic and macroscopic mean-field coarse-grained descriptions. Our analysis shows that the high- and low-density regimes are governed by two different coarse-grained equations. In the low-density regime, noise effects play a prominent role, leading to a first-order phase transition. In contrast, a second-order phase transition occurs in the high-density regime. Due to the exact coarse-graining method, our work also opens up the applicability to systematically analyze noise-induced phase transitions in other models of reciprocally and non-reciprocally interacting particles.

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arXiv2025

Stability of flocking in the reciprocal two-species Vicsek model: Effects of relative population, motility, and noise

Natural flocks need to cope with various forms of heterogeneities, for instance, their composition, motility, interaction, or environmental factors. Here, we study the effects of such heterogeneities on the flocking dynamics of the reciprocal two-species Vicsek model [Phys. Rev. E 107, 024607 (2023)], which comprises two groups of self-propelled agents with anti-aligning inter-species interactions and exhibits either parallel or anti-parallel flocking states. The parallel and anti-parallel flocking states vanish upon reducing the size of one group, and the system transitions to a single-species flock of the majority species. At sufficiently low noise (or high density), the minority species can exhibit collective behavior, anti-aligning with the liquid state of the majority species. Unequal self-propulsion speeds of the two species strongly encourage anti-parallel flocking over parallel flocking. However, when activity landscapes with region-dependent motilities are introduced, parallel flocking is retained if the faster region is given more space, highlighting the role of environmental constraints. Under noise heterogeneity, the colder species (subjected to lower noise) attain higher band velocity compared to the hotter one, temporarily disrupting any parallel flocking, which is subsequently restored. These findings collectively reveal how different forms of heterogeneity, both intrinsic and environmental, can qualitatively reshape flocking behavior in this class of reciprocal two-species models.

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arXiv2025

Emergent complex phases in a discrete flocking model with reciprocal and non-reciprocal interactions

There is growing interest in multi-species active matter systems with reciprocal and non-reciprocal interactions. While such interactions have been explored in continuous symmetry models, less is known about multi-species discrete-symmetry systems. To address this, we study the two-species active Ising model (TSAIM), a discrete counterpart of the two-species Vicsek model. Our investigation explores both inter-species reciprocal and non-reciprocal interactions, along with the possibility of species interconversion. In the reciprocal TSAIM, we observe the emergence of a high-density parallel flocking state, a feature not seen in previous flocking models. With species interconversion, the TSAIM corresponds to an active extension of the Ashkin-Teller model and exhibits rich state diagrams. In the non-reciprocal TSAIM, a run-and-chase dynamics emerge. We also find that the system is metastable due to droplet excitation and exhibits spontaneous motility-induced interface pinning. A hydrodynamic theory validates our numerical simulations and confirms the phase diagrams.

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arXiv2025

Stochastic Thermodynamics of Non-reciprocally Interacting Particles and Fields

Nonreciprocal interactions that violate Newton's law 'actio=reactio' are ubiquitous in nature and are currently intensively investigated in active matter, chemical reaction networks, population dynamics, and many other fields. An outstanding challenge is the thermodynamically consistent formulation of the underlying stochastic dynamics that obeys local detailed balance and allows for a rigorous analysis of the stochastic thermodynamics of non-reciprocally interacting particles. Here, we present such a framework for a broad class of active systems and derive by systematic coarse-graining exact expressions for the macroscopic entropy production. Four independent contributions to the thermodynamic dissipation can be identified, among which the energy flux sustaining vorticity currents manifests the presence of non-reciprocal interactions. Then, Onsager's non-reciprocal relations, the fluctuation-response relation, the fluctuation relation and the thermodynamic uncertainty relations for non-reciprocal systems are derived. Finally, we demonstrate that our general framework is applicable to a plethora of active matter systems and chemical reaction networks and opens new paths to understand the stochastic thermodynamics of non-reciprocally interacting many-body systems.

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arXiv2025

Optimal number of agents in a collective search, and when to launch them

Search processes often involve multiple agents that collectively search a randomly located target. While increasing the number of agents usually decreases the time at which the first agent finds the target, it also requires resources to create and sustain more agents. In this manuscript, we raise the question of the optimal timing for launching multiple agents in a search in order to reach the best compromise between minimizing the overall search time and minimizing the costs associated with launching and sustaining agents. After introducing a general formalism for independent agents in which we allow them to be launched at arbitrary times, we investigate by means of analytical calculations and numerical optimization the optimal launch strategies to optimize the quantiles of the search cost and its mean. Finally, we compare our results with the case of stochastic resetting and study the conditions under which it is preferable to launch new searchers rather than resetting the first one to its initial position.

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arXiv2024

Collective chemotactic search

We investigate collective search by self-propelled agents that are repelled by their own chemically produced trails, a minimal mechanism that simultaneously generates indirect interactions and memory. Using lattice and off-lattice models, we show that this mechanism enhances search efficiency through two distinct regimes. In a weak-memory regime, chemical cues are short-lived and interactions primarily promote spatial separation between agents. This reduces redundant exploration while preserving mobility, leading to an optimal trade-off between spatial order and persistence. In a strong-memory regime, long-lived chemical trails induce effective self-avoidance, strongly suppressing revisits and long search times. Here optimal search occurs at finite memory strength: permanently persistent trails lead to self-caging, while moderate memory enables efficient exploration. At higher densities, overlapping chemical trails give rise to a collective self-avoidance mechanism that yields substantial cooperative speedup without global spatial order. Together, these results show how chemically mediated memory and interactions can optimize collective search across distinct dynamical regimes.

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arXiv2024

Ordering kinetics in the active Ising model

We undertake a numerical study of the ordering kinetics in the two-dimensional ($2d$) active Ising model (AIM), a discrete flocking model with a conserved density field coupled to a non-conserved magnetization field. We find that for a quench into the liquid-gas coexistence region and in the ordered liquid region, the characteristic length scale of both the density and magnetization domains follows the Lifshitz-Cahn-Allen (LCA) growth law: $R(t) \sim t^{1/2}$, consistent with the growth law of passive systems with scalar order parameter and non-conserved dynamics. The system morphology is analyzed with the two-point correlation function and its Fourier transform, the structure factor, which conforms to the well-known Porod's law, a manifestation of the coarsening of compact domains with smooth boundaries. We also find the domain growth exponent unaffected by different noise strengths and self-propulsion velocities of the active particles. However, transverse diffusion is found to play the most significant role in the growth kinetics of the AIM. We extract the same growth exponent by solving the hydrodynamic equations of the AIM.

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arXiv2024

Consequence of anisotropy on flocking: the discretized Vicsek model

We numerically study a discretized Vicsek model (DVM) with particles orienting in $q$ possible orientations in two dimensions. The study probes the significance of anisotropic orientation and microscopic interaction on the macroscopic behavior. The DVM is an off-lattice flocking model like the active clock model [ACM; EPL {\bf 138}, 41001 (2022)] but the dynamical rules of particle alignment and movement are inspired by the prototypical Vicsek model (VM). The DVM shows qualitatively similar properties as the ACM for intermediate noise strength where a transition from macrophase to microphase separation of the coexistence region is observed as $q$ is increased. But for small $q$ and noise strength, the liquid phase appearing in the ACM at low temperatures is replaced in the DVM by a configuration of multiple clusters with different polarization which does not exhibit any long-range order. We find that the dynamical rules have a profound influence on the overarching features of the flocking phase. We further identify the metastability of the ordered liquid phase subjected to a perturbation.

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arXiv2024

Stationary particle currents in sedimenting active matter wetting a wall

Recently it was predicted, on the basis of a lattice gas model, that scalar active matter in a gravitational field would rise against gravity up a confining wall or inside a thin capillary - in spite of repulsive particle-wall interactions [Phys. Rev. Lett. 124, 048001 (2020)]. In this paper we confirm this prediction with sedimenting active Brownian particles (ABPs) in a box numerically and elucidate the mechanism leading to the formation of a meniscus rising above the bulk of the sedimentation region. The height of the meniscus increases with the activity of the system, algebraically with the Péclet number. The formation of the meniscus is determined by a stationary circular particle current, a vortex, centered at the base of the meniscus, whose size and strength increase with the ABP activity. The origin of these vortices can be traced back to the confinement of the ABPs in a box: already the stationary state of ideal (non-interacting) ABPs without gravitation displays circular currents that arrange in a highly symmetric way in the eight octants of the box. Gravitation distorts this vortex configuration downward, leaving two major vortices at the two side walls, with a strong downward flow along the walls. Repulsive interactions between the ABPs change this situation only as soon as motility induced phase separation (MIPS) sets in and forms a dense, sedimented liquid region at the bottom, which pushes the center of the vortex upwards towards the liquid-gas interface. Self-propelled particles therefore represent an impressive realization of scalar active matter that forms stationary particle currents being able to perform visible work against gravity or any other external field, which we predict to be observable experimentally in active colloids under gravitation.

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