- Titel
- Effects of time delay in motile active matter
- Untertitel
- From microscopic to collective behavior
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-841517
- Datum der Einreichung
- 22.08.2022
- Datum der Verteidigung
- 17.02.2023
- Abstract (EN)
- Groups of living entities that exhibit well-organized collective behavior, such as artistic flight formations of bird flocks or the complex job sharing in ant colonies, have always exerted a fascination on people. Understanding the basic principles of organization as well as the microscopic details of locomotion represent central questions of the interdisciplinary research field of motile active matter [27]. From a physical perspective, assemblies of interacting, energy-consuming self-propelling atoms form a new class of non-equilibrium systems that give rise to a broad range of new interesting phenomena, such as motility-induced phase separations or self-organized criticality. The appearance of retardations in the dynamics of those systems is rather natural. On a single-particle level, e.g., inertia or embedding in a viscous solvent, may give rise to memory. This thesis may be regarded as one of the first major attempts to systematically study the effects of (discrete) time delays in the interactions of multiple active particles – resulting, e.g., from finite reaction times or feedback – and to place them in a broader context. Corresponding delays have already been studied intensively in traffic models, where they play an important role in the formation of congestion, while they have been mostly neglected in other active systems. We provide an elaborate literature review of existing theoretical and experimental studies in which delays occur in a wide range of active systems. A variety of different effects become apparent, ranging from limiting effects such as oscillations or inaccuracies to new possibilities for control tasks. To give further expression to this general discussion, we examine paradigmatic theoretical models that illustrate the effects of delay at both the microscopic level and on collective phenomena. The mathematical structures of delayed systems are explained and comparisons to real measurements are made. Based on modern micromanipulation techniques, synthetic swimmers can even be implemented at the microscale, with a time delay occurring as a result of feedback control. Inspired by the experiments of Khadka et al. [237], we present an analytical paradigmatic model of N Brownian particles coupled via time-delayed harmonic interactions. The problem is formulated in terms of stochastic delay differential equations, where the delay renders the dynamics non-Markovian. In general, only few closed analytical solutions for delayed systems are known, so approximation methods or numerical treatment are often required. In our case, the equations of motion can be transformed into time-local Langevin equations [167], which allow an elaborate analytical treatment of the stochastic motion. For small systems of N ≤ 3 particles, we derive the time-dependent probability distributions, while evaluating larger systems via Brownian dynamics simulations. The particles form molecule-like non-equilibrium structures where the delay implies the appearance of different dynamical regimes, including damped, oscillatory and diverging motion. To quantitatively characterize their stability, we determine the underlying entropy fluxes caused by the feedback control. Assuming distinguishable particles, differ- ent isomers of the same molecule exist. The noise-driven transitions between the different spatial configurations are investigated by an approximate time-dependent transition-state theory [194]. We find that the corresponding effects of a delay can be mapped to an increased effective temperature. The second part of this work is dedicated to the popular Vicsek model [467] describing (dry) motile active matter with alignment interactions, which we further generalize by including a time delay. In the exchange of information time delays play a central role. On the one hand, a finite speed of information propagation can be a source of delays; on the other hand, finite reaction and decision times have a strong impact on the transmission dynamics. Hence, we numerically study the transfer of directional changes based on exemplary leadership scenarios in a polarized flock and draw analogies to models of heat transfer. While the breaking of symmetries such as reciprocity and spin conservation seems to play a minor role, increasing particle velocities lead to strong directional-dependent convection. In the opposite direction of the leaders motion the information spreading stays diffusive. Similar to high velocities, long delays favor ballistic information transfer. Previous work [65] has found that adding inertia can improve the system’s ability to perform collective turns. In contrast, this ability is diminished by the presence of a delay, which is often likened to inertia. On the other hand, an increased persistence tends to make it more stable against random perturbations. Moreover, we investigate the linear response of highly ordered systems to a weak orientational bias applied to a subgroup of particles. For sufficiently small time delays and sufficiently weak orientational bias, the total orientation change of the system is shown to be a linear function of the perturbation, scaling with the inverse delay, while on the other hand its variance increases with the delay. The linear response is valid in a parameter regime that preserves the average orientation. We present a simplistic analytical model from which the delay-dependence of the response and the variance can be derived. Empirical studies have shown that biological systems often exhibit scale-free correlations, indicating that corresponding systems operate close to criticality [69]. By varying the density or the noise, the time-delay Vicsek model undergoes a phase transition from order to disorder. We analyze both the static and dynamical finite-size scaling behavior of the correlations of the velocity fluctuations at the transition. We find that the overall strength of the equal-time correlations decreases as the delay increases and the dynamical correlation functions exhibit oscillations with a period determined by the delay. The critical exponents and order parameter show a strong dependence on the delay. In particular, the critical order-parameter is found to be a non-monotonic function indicating that small delays increase the order while large delays are disruptive when compared to a situation without delay. In accordance with empirical observations in swarms of midges [63] which have been attributed to inertial effects, the dynamical critical exponent converges for large delays to ∼1.1 and the time-correlation functions are initially flat. In addition, analytical results for the correlation function are derived based on a linearized version of the model.
- Verweis
- Brownian molecules formed by delayed harmonic interactions
DOI: 10.1088/1367-2630/ab3d76 - Brownian thermometry beyond equilibrium
DOI: 10.1002/syst.201900041 - Finite-size scaling at the edge of disorder in a time-delay Vicsek model
DOI: 10.1103/PhysRevLett.127.258001 - Information conduction and convection in noiseless Vicsek flocks
DOI: 10.1103/PhysRevE.106.014609 - Signal propagation and linear response in the delay Vicsek model
DOI: 10.1103/PhysRevE.106.054612 - Delay in motile active matter: the role of perceptionreaction delays
in preparation - Freie Schlagwörter (EN)
- Non-equilibrium statistical physics, (active) Brownian motion, stochastic delay differential equations, transition rate theory, phase transition
- Klassifikation (DDC)
- 530
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-841517
- Veröffentlichungsdatum Qucosa
- 20.03.2023
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY 4.0- Inhaltsverzeichnis
1 Introduction 1.1 Motile active matter 1.2 Common types and origins of delay 1.2.1 Delays due to “coarse-graining” or “reduction” 1.2.2 Delays in biological and “intelligent” systems 1.3 Paradigmatic examples of motile active matter systems perception-reaction time delay 1.3.1 Microswimmers 1.3.2 Microflyers 1.3.3 Swarms and flocks 1.3.4 Road traffic 1.4 Scope and structure of the thesis 2 Theoretical foundations 2.1 Langevin equation 2.2 Fluctuation-dissipation theorem 2.3 Fokker-Planck equation 2.4 Transition state theory 2.5 Phase transitions 3 Delayed Brownian molecules 3.1 Mathematical concepts for delay problems 3.1.1 (Stochastic) delay differential equations 3.1.2 Distributed delays 3.1.3 Fokker-Planck equations with delay 3.1.4 Approximation schemes 3.2 Molecular dynamics 3.2.1 Center of mass 3.2.2 Dimer 3.2.3 Trimer 3.3 Structure formation 3.4 Entropy fluxes 3.5 Transition rates for isomer transformations 3.5.1 Dimer 3.5.2 Trimer 3.6 Extensions to other memory kernels 3.7 Conclusion Appendices 3.A Solution of the noiseless problem 3.B Solution with noise 3.B.1 One dimension 3.B.2 Higher dimensions 3.C Time-correlation matrix and stationary covariance matrix 3.D Comparison to small delay expansion 4 Information spreading 4.1 The Vicsek model 4.1.1 Standard formulation 4.1.2 Limiting cases and extensions 4.1.3 Order-disorder phase transition 4.2 Mechanisms of information transport 4.2.1 Zero-velocity limit of the VM 4.2.2 The motile case 4.3 Response in the time delay Vicsek model 4.3.1 Collective maneuver 4.3.2 Perturbation 4.3.3 Signal propagation 4.3.4 Linear response to leadership 4.4 Conclusion Appendices 4.A Breaking of information conservation in the linearized VM 4.B Exactly solvable model for linear response 5 Criticality 5.1 Correlation function 5.2 Static correlations 5.2.1 Static correlation functions and susceptibility. 5.2.2 Static scaling 5.3 Dynamical correlations 5.3.1 Dynamical correlation functions 5.3.2 Relaxation time 5.3.3 Dynamical scaling 5.4 Long-delay limit 5.5 Conclusion Appendices 5.A Exactly solvable linearized delay Vicsek model. 5.A.1 Continuous-time delay Vicsek model 5.A.2 Spin-wave expansion 5.A.3 Green’s function 5.A.4 Correlation functions 6 Conclusion and outlook