arXiv subjects

Rüdiger Kürsten

Publications and source records attributed to Rüdiger Kürsten.

17 recordsLinked to original sources

arXiv2026

Asymptotically exact scattering theory of the Kuramoto-Vicsek model

We consider the Kuramoto-Vicsek model of self-propelled particles with velocity-alignment interactions. Starting from the exact $N$-particle Liouville equation, a kinetic equation for the one-particle distribution function is obtained in a self-consistent manner. We show that the usual mean-field assumption of molecular chaos leads to qualitatively wrong predictions such as an infinite coefficient of self-diffusion. Going beyond mean-field and applying the refined assumption of \emph{one-sided molecular chaos} where the two-particle-correlations during binary interactions are explicitly taken into account, we analytically calculate the scattering of particles in the limit of low density and obtain explicit expressions for the dynamical noise of an effective one-particle Langevin-equation and the corresponding self-diffusion. The theory is developed in detail for anti-aligning couplings, where exact analytical results are obtainable. In this calculation, the superposition principle of traditional kinetic theory is modified to handle a system with non-Hamiltonian dynamics involving phase-space compression. The predicted theoretical expressions for the relaxation of hydrodynamic modes and the self-diffusion coefficient are in excellent, quantitative agreement with agent-based simulations. At large particle densities, a given particle is constantly approached and abandoned by different collision partners. Modeling this switching by a random telegraph process and exactly solving a self-consistent integral equation, we obtain explicit expressions for the noise correlations of the effective one-particle Langevin-equation. We also consider the effect of frozen disorder in the particle speeds and show how this can be used to calculate the exact Boltzmann collision operator for positive alignment strengths.

cond-mat.stat-mech↗

arXiv2024

Universal Scaling of Clustering Instability for Interacting Active Brownian Particles

Clustering is one of the mayor collective phenomena observed in active matter. We study the overdamped motion of interacting active Brownian particles in two dimensions. An instability in the pair correlation function causes the onset of clustering. This clustering mechanism depends mainly on the self-propulsion properties of the active particles and details of the interactions do not effect the scaling of the clustering instability. Theoretical predictions from repeated ring-kinetic theory are confirmed by agent-based simulations.

cond-mat.soft↗

arXiv2023

Flocking in Binary Mixtures of Anti-aligning Self-propelled Particles

We consider two species of self-propelled point particles: A-particles and B-particles. The orientations between nearby particles are subject to pair interactions of different strength for A-A-, A-B-(=B-A-) and B-B-interactions, respectively. Even if all interactions involved are repelling, that is, if they locally favor anti-alignment between each pair of particles, we find global polar order of both A-particles and B-particles We find qualitative agreement between agent-based simulations and mean field theory. Beyond mean field, we develop a Boltzmann-scattering theory based on one-sided molecular chaos that yields excellent quantitative agreement with simulations for dilute systems. For large systems, we find, depending on parameters, either micro-phase-separation or static patterns with either patches or stripes that carry different polarization orientations.

cond-mat.soft↗

arXiv2023

Scattering theory of Non-Brownian active particles with social distancing

We consider deterministic self-propelled particles with anti-alignment interactions. An asymptotically exact kinetic theory for particle scattering at low densities is constructed by a non-local closure of the BBGKY-hierarchy, involving pair correlations. We show that the mean-field assumption of molecular chaos yields unphysical predictions, whereas the scattering theory shows excellent agreement with agent-based simulations. To extend the theory to high densities, a self-consistent mapping to a random-telegraph process is performed. The approach is used to derive a one-particle Langevin-equation and leads to analytical expressions for the correlations of its effective noise.

cond-mat.stat-mech↗

arXiv2023

Emergent States in Systems of Chiral Self-Propelled Rods

We study inherently chiral self-propelled particles, self-rotating at a fixed frequency, in two dimensions, subjected to nematic alignment interactions and rotational noise. By means of both, homogeneous and spatially resolved mean field kinetic theory, we identify various different flocking states. We confirm the presence of the predicted phases using agent-based simulations, in particular, an homogeneous nematic phase at low frequencies, followed by a microflock pattern phase at larger frequencies, characterized by finite-size nematic clusters. We emphasize that special care has to be taken within the simulations in order to avoid artifacts, and present a non-standard simulation technique in order to avoid them.

cond-mat.soft↗

arXiv2022

Aligning Active Particles Py Package

The package performs molecular-dynamics-like agent-based simulations for models of aligning self-propelled particles in two dimensions such as e.g. the seminal Vicsek model or variants of it. In one class of the covered models, the microscopic dynamics is determined by certain time discrete interaction rules. Thus, it is no Hamiltonian dynamics and quantities such as energy are not defined. In the other class of considered models (that are generally believed to behave qualitatively the same) Brownian dynamics is considered. However, also there, the forces are not derived from a Hamiltonian. Furthermore, in most cases, the forces depend on the state of all particles and can not be decomposed into a sum of forces that only depend on the states of pairs of particles. Due to the above specified features of the microscopic dynamics of such models, they are not implemented in major molecular dynamics simulation frameworks to the best of the authors knowledge. Models that are covered by this package have been studied with agent-based simulations by dozens of papers. However, no simulation framework of such models seems to be openly available. The program is provided as a Python package. The simulation code is written in C. In the current version, parallelization is not implemented.

physics.comp-ph↗

arXiv2021

A Quantitative Kinetic Theory of Flocking with Three-Particle-Closure

We consider aligning self-propelled particles in two dimensions. Their motion is given by generalized Langevin equations and includes non-additive N-particle interactions. The qualitative behavior is as for the famous Vicsek model. We develop a kinetic theory of flocking beyond mean field. In particular, we self-consistently take into account the full pair correlation function. We find excellent quantitative agreement of the pair correlations with direct agent-based simulations within the disordered regime. Furthermore we use a closure relation to incorporate spatial correlations of three particles. In that way we achieve good quantitative agreement of the onset of flocking with direct simulations. Compared to mean field theory, the flocking transition is shifted significantly towards lower noise because directional correlations favor disorder. We compare our theory with a recently developed Landau-kinetic theory.

cond-mat.soft↗

arXiv2020

Dry Active Matter exhibits a self-organized 'Cross Sea' Phase

The Vicsek model of self-propelled particles is known in three different phases: (i) a polar ordered homogeneous phase also called Toner-Tu phase, (iii) a phase of polar ordered regularly arranged high density bands (waves) with surrounding low density regions without polar order and (iv) a homogeneous phase without polar order. It has been questioned whether the band phase (iii) should be divided into two parts [Chaté2020]: one with periodically arranged and one with strongly interacting but not ordered bands. We answer this question by showing that the standard Vicsek model has a fourth phase for large system sizes: (ii) a polar ordered cross sea phase. Close to the transition towards (i) this phase becomes unstable and looks like strongly interacting bands. We demonstrate that the cross sea phase is not just a superposition of two waves, but it is an independent complex pattern. Furthermore we show that there is a non-zero mass flow through the structure of the cross sea pattern within its co-moving frame.

cond-mat.soft↗

arXiv2019

On the Count Probability of Many Correlated Symmetric Events

We consider $N$ events that are defined on a common probability space. Those events shell have a common probability function that is symmetric with respect to interchanging the events. We ask for the probability distribution of the number of events that occur. If the probability of a single event is proportional to $1/N$ the resulting count probability is Poisson distributed in the limit of $N\rightarrow \infty$ for independent events. In this paper we calculate the characteristic function of the limiting count probability distribution for events that are correlated up to an arbitrary but finite order.

math.PR↗

arXiv2019

Multiple Particle Correlation Analysis of Many-Particle Systems: Formalism and Application to Active Matter

We introduce a fast spatial point pattern analysis technique which is suitable for systems of many identical particles giving rise to multi-particle correlations up to arbitrary order. The obtained correlation parameters allow to quantify the quality of mean field assumptions or theories that incorporate correlations of limited order. We study the Vicsek model of self-propelled particles and create a correlation map marking the required correlation order for each point in phase space incorporating up to ten-particle correlations. We find that multi-particle correlations are important even in a large part of the disordered phase. Furthermore, the two-particle correlation parameter serves as an excellent order parameter to locate both phase transitions of the system, whereas two different order parameters were required before.

cond-mat.soft↗

arXiv2017

Giant Kovacs-Like Memory Effect for Active Particles

Dynamical properties of a Vicsek-like gas of self-propelled particles are investigated by means of kinetic theory and agent based simulations. While memory effects have been observed in disordered systems, we show that they also occur in active matter systems. In particular, we find that the system exhibits a giant Kovacs-like memory effect that is much larger than predicted by a generic linear theory. Based on a separation of time scales we develop a nonlinear theory to explain this effect. We apply this theory to driven granular gases and propose further applications to spin glasses.

cond-mat.stat-mech↗

arXiv2016

Critical assessment of von Mises distribution and an infinite series ansatz for self-propelled particles

We consider a Vicsek model of self-propelled particles with bounded confidence, where each particle interacts only with neighbors that have a similar direction. Depending on parameters, the system exhibits a continuous or discontinuous polar phase transition from the isotropic phase to a phase with a preferred direction. In a recent paper [1] the von Mises distribution was proposed as an ansatz for polar ordering. In the present system the time evolution of the angular distribution can be solved in Fourier space. We compare the results of the Fourier analysis with the ones obtained by using the von Mises distribution ansatz. In the latter case the qualitative behavior of the system is recovered correctly. However, quantitatively there are serious deviations. We introduce an extended von Mises distribution ansatz such that a second term takes care of the next two Fourier modes. With the extended ansatz we find much better quantitative agreement. As an alternative approach we also use a Gaussian and a geometric series ansatz in Fourier space. The geometric series ansatz is analytically handable but fails for very weak noise, the Gaussian ansatz yields better results but it is not analytically treatable.

cond-mat.stat-mech↗

arXiv2016

Discontinuous transitions in globally coupled potential systems with additive noise

An infinite array of globally coupled overdamped constituents moving in a double-well potential with $n$-th order saturation term under the influence of additive Gaussian white noise is investigated. The system exhibits a continuous phase transition from a symmetric phase to a symmetry-broken phase. The qualitative behavior is independent on $n$. The critical point is calculated for strong and for weak noise, these limits are also bounds for the critical point. Introducing an additional nonlinearity, such that the potential can have up to three minima, leads to richer behavior. There the parameter space divides in three regions, a region with a symmetric phase, a region with a phase of broken symmetry and a region where both phases coexist. The region of coexistence collapses into one of the others via a discontinuous phase transition whereas the transition between the symmetric phase and the phase of broken symmetry is continuous. The tricritical point where the three regions intersect, can be calculated for strong and for weak noise. These limiting values form optimal bounds on the tricritical point. In the region of coexistence simulations of finite systems are performed. One finds that the stationary distribution of finite but large systems differs qualitatively from the one of the infinite system. Hence the limits of stationarity and large system size do not commute.

cond-mat.stat-mech↗

arXiv2015

Random Recursive Trees and the Elephant Random Walk

One class of random walks with infinite memory, so called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We use a coupling between the two models to translate results from elephant random walks to the percolation process. We calculate, besides other quantities, exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation. We further introduce a new model, the skew elephant random walk and calculate the first and second moment of this process.

cond-mat.stat-mech↗

arXiv2015

Patchwork Sampling of Stochastic Differential Equations

We propose a method to sample stationary properties of solutions of stochastic differential equations, which is accurate and efficient if there are rarely visited regions or rare transitions between distinct regions of the state space. The method is based on a complete, non-overlapping partition of the state space into patches on which the stochastic process is ergodic. On each of these patches we run simulations of the process strictly truncated to the corresponding patch, which allows effective simulations also in rarely visited regions. The correct weight for each patch is obtained by counting the attempted transitions between all different patches. The results are patchworked to cover the whole state space. We extend the concept of truncated Markov chains which is originally formulated for processes which obey detailed balance to processes not fulfilling detailed balance. The method is illustrated by three examples, describing the one-dimensional diffusion of an overdamped particle in a double-well potential, a system of many globally coupled overdamped particles in double-well potentials subject to additive Gaussian white noise, and the overdamped motion of a particle on the circle in a periodic potential subject to a deterministic drift and additive noise. In the appendix we explain how other well-known Markov chain Monte Carlo algorithms can be related to truncated Markov chains.

cond-mat.stat-mech↗

arXiv2015

Comment on 'Anomalous diffusion induced by enhancement of memory'

In a recent paper [2] the author introduced and investigated a random walk model similar to a model introduced in [1]. In these models the increment of the random walk depends on the complete past of the process. In this note I will point out that the models considered in [1] and [2] can be mapped onto each other one to one. They can be defined on a common probability space and hence all expectation values of the model [2] with parameter p are equal to the ones of [1] with a corresponding parameter $\tilde{p}$.

physics.data-an↗

arXiv2013

Critical manifold of globally coupled overdamped anharmonic oscillators driven by additive Gaussian white noise

We prove for an infinite array of globally coupled overdamped anharmonic oscillators subject to additive Gaussian white noise the existence of a well-behaved critical manifold in the parameter space which separates a symmetric phase from a symmetry broken phase. Given two of the system parameters there is an unique critical value of the third. The proof exploits that the critical control parameter a_c is bounded by its limit values for weak and for strong noise. In these limits the mechanism of symmetry breaking differs. For weak noise the distribution is Gaussian and the symmetry is broken as the whole distribution is shifted in either the positive or the negative direction. For strong noise there is a symmetric double-peak distribution and the symmetry is broken as the weights of the peaks become different. We derive an ordinary differential equation whose solution describes the critical manifold. Using a series ansatz to solve this differential equation we determine the critical manifold for weak and for strong noise and compare it to numerical results. We derive analytic expressions for the order parameter and the susceptibility close to the critical manifold.

cond-mat.stat-mech↗