arXiv subjects

Rainer Klages

Publications and source records attributed to Rainer Klages.

At least 19 recordsLinked to original sources

arXiv2026

An active Lorentz gas: walking droplets in periodic media

The Lorentz gas is a paradigmatic model in dynamical systems theory for understanding the origin of nonequilibrium transport in terms of microscopic deterministic chaos. In the periodic setting, a point particle scatters elastically off disks arranged on a two-dimensional lattice. Here we replace the disks by smooth potentials and the particle by the widely studied walking droplet, which propels itself on a vertically vibrating fluid. In the low-memory limit, this droplet reduces to a particle with nonlinear active friction. We call this system an active Lorentz gas. Using extensive numerical simulations, we analyze how dissipation generated by the active deterministic dynamics alters the phase-space structure of the corresponding conservative Lorentz gas. We find that islands of stability collapse into attracting and repelling sets. To characterize these structures, we introduce an energy-variance filtering method that distinguishes localized periodic, quasi-ballistic periodic, and chaotic trajectories, enabling the construction of bifurcation diagrams in a non-conservative setting. We identify parameter regimes exhibiting strong bifurcation cascades, anomalous diffusion, and significant phase-space contraction. Our results establish the active Lorentz gas as a rich framework for studying transport in dissipative dynamical systems and provide a bridge between active matter and classical models of chaotic transport, with potential implications for hydrodynamic quantum analogs in periodic media.

nlin.CD↗

arXiv2026

Active movement of foraging sea turtles generates anomalous looping

Animals inhabiting diverse environments by moving across different spatial scales, from insects to birds, marine predators, mammals and even humans, often display apparently random movement paths. Over the past decade, novel biologging technologies have recorded these patterns in increasing detail, generating a wealth of experimental data. A central challenge is to understand such complex patterns by constructing data-driven mathematical models. Many animal movements depart from Brownian motion, as described by correlated random walks, L\'evy walks, or active particle dynamics. Yet, these movement models do not incorporate long-term non-Markovian memory extracted from experimental trajectories. Here, we construct a stochastic generalised Langevin equation from satellite tracking data for loggerhead sea turtles (Caretta caretta) foraging off the coast of West Africa. We find that these turtles exhibit active movement characterised by large-scale loops that are not explained by ocean currents or chirality. These loops maintain movement within a specific foraging region and, over intermediate timescales, generate superdiffusion similar to L\'evy walks. We thus identify a loop-based form of active anomalous search related to foraging patterns observed across a wide range of animal species, which may inspire robotic search strategies and AI-based metaheuristic optimisation algorithms.

cond-mat.stat-mech↗

arXiv2026

Quasi-stationary and quasi-ergodic distributions in the Pelikan random map

In this paper we present a concrete example of a substochastic discrete-time Markov chain on a countable state space producing a spectrum of infinitely many quasi-stationary distributions (QSDs) for generic parameter values, with each QSD supporting a distinct escape rate. Our system is motivated by an open variant of the Pelikan dynamical system, a random map introduced in the 1980s. These QSDs, and their stability to perturbative random noise, are tested in numerical simulations. The existence of unique QEDs is also established for some parameter values.

nlin.CD↗

arXiv2025

Modelling the movements of organisms by stochastic theory in a comoving frame

Imagine you walk in a plane. You move by making a step of a certain length per time interval in a chosen direction. Repeating this process by randomly sampling step length and turning angle defines a two-dimensional random walk in what we call comoving frame coordinates. This is precisely how Ross and Pearson proposed to model the movements of organisms more than a century ago. Decades later their concept was generalised by including persistence leading to a correlated random walk, which became a popular model in Movement Ecology. In contrast, Langevin equations describing cell migration and used in active matter theory are typically formulated by position and velocity in a fixed Cartesian frame. In this article, we explore the transformation of stochastic Langevin dynamics from the Cartesian into the comoving frame. We show that the Ornstein-Uhlenbeck process for the Cartesian velocity of a walker can be transformed exactly into a stochastic process that is defined self-consistently in the comoving frame, thereby profoundly generalising correlated random walk models. This approach yields a general conceptual framework how to transform stochastic processes from the Cartesian into the comoving frame. Our theory paves the way to derive, invent and explore novel stochastic processes in the comoving frame for modelling the movements of organisms. It can also be applied to design novel stochastic dynamics for autonomously moving robots and drones.

physics.bio-ph↗

arXiv2025

Does an intermittent dynamical system remain (weakly) chaotic after drilling in a hole?

Chaotic dynamical systems are often characterised by a positive Lyapunov exponent, which signifies an exponential rate of separation of nearby trajectories. However, in a wide range of so-called weakly chaotic systems, the separation of nearby trajectories is sub-exponential in time, and the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the positive Lyapunov exponents on the system's fractal repeller can be related to the generation of metric entropy and the escape rate from the system. The escape rate, in turn, cross-links these two chaos properties to important statistical-physical quantities like the diffusion coefficient. However, no suitable generalisation of this escape rate formalism exists for weakly chaotic systems. In our paper we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, a generalisation of its Lyapunov exponent (which we call `stretching') is completely suppressed in the presence of a hole. This result is based on numerical evidence and a corresponding stochastic model. The correspondence between map and model is tested via a related partially absorbing map. We examine the structure of the map's fractal repeller, which we reconstruct via a simple algorithm. Our findings are in line with rigorous mathematical results concerning the collapse of the system's density as it evolves in time. We also examine the generation of entropy in the open map, which is shown to be consistent with the collapsed stretching. As a result, we conclude that no suitable generalisation of the escape rate formalism to weakly chaotic systems can exist.

nlin.CD↗

arXiv2025

Diffusion in the Inverted Triangular Soft Lorentz Gas

We investigate diffusion in a two-dimensional inverted soft Lorentz gas, where attractive Fermi-type potential wells are arranged in a triangular lattice. This configuration contrasts with earlier studies of soft Lorentz gases involving repulsive scatterers. By systematically varying the gap width and softness of the potential, we explore a rich landscape of diffusive behaviors. We present numerical simulations of the mean squared displacement and compute diffusion coefficients, identifying tongue-like structures in parameter space associated with quasiballistic transport. Furthermore, we develop an extension to the Machta-Zwanzig approximation that incorporates correlated multi-hop trajectories and correct for the influence of localized periodic orbits. Our findings highlight the qualitative and quantitative differences between inverted and repulsive soft Lorentz gases and offer new insights into transport phenomena in smooth periodic potentials.

nlin.CD↗

arXiv2025

Individual particle persistence antagonizes global ordering in populations of nematically-aligning self-propelled particles

The transition from individual to collective motion plays a significant role in many biological processes. While the implications of different types of particle-particle interactions for the emergence of particular modes of collective motion have been well studied, it is unclear how particular types of individual migration patterns influence collective motion. Here, motivated by swarming bacteria Myxococcus xanthus, we investigate the combined effects of the individual pattern of migration and of particle-particle interactions, on the emergence of collective migration. We analyze the effects of a feature of individual pattern migration, the persistence of motion, on the collective properties of the system that emerge from interactions among individuals; in particular, when nematic velocity alignment interaction mediates collective dynamics. We find, through computer simulations and mathematical analysis, that an initially disordered migratory state can become globally ordered by increasing either, the particle-particle alignment interaction strength or the persistence of individual migration. In contrast, we find that persistence prevents the emergence of global nematic order when both persistence and nematic alignment are comparatively high. We conclude that behavior at the population level does not only depend on interactions between individuals but also on the individuals' own intrinsic behavior.

physics.bio-ph↗

arXiv2024

Anomalous Dynamics of Superparamagnetic Colloidal Microrobots with Tailored Statistics

Living organisms have developed advanced motion strategies for efficient space exploration, serving as inspiration for the movements of microrobots. These real-life strategies often involve anomalous dynamics displaying random movement patterns that deviate from Brownian motion. Despite their biological inspiration, autonomous stochastic navigation strategies of current microrobots remain much less versatile than those of their living counterparts. Supported by theoretical reasoning, this work demonstrates superparamagnetic colloidal microrobots with fully customizable stochastic dynamics displaying the entire spectrum of anomalous diffusion, from subdiffusion to superdiffusion, across statistically significant spatial and temporal scales (covering at least two decades). By simultaneously tuning microrobots' step-length distribution and, critically, their velocity autocorrelation function with magnetic fields, fundamental anomalous dynamics are reproduced with tailored properties mimicking L\'evy walks and fractional Brownian motion. These findings pave the way for programmable microrobotic systems that replicate optimal stochastic navigation strategies found in nature for applications in medical robotics and environmental remediation.

cond-mat.soft↗

arXiv2024

Singularity of Levy walks in the lifted Pomeau-Manneville map

Since groundbreaking works in the 1980s it is well-known that simple deterministic dynamical systems can display intermittent dynamics and weak chaos leading to anomalous diffusion. A paradigmatic example is the Pomeau-Manneville (PM) map which, suitably lifted onto the whole real line, was shown to generate superdiffusion that can be reproduced by stochastic Levy walks (LWs). Here we report that this matching only holds for parameter values of the PM map that are of Lebesgue measure zero in its two-dimensional parameter space. This is due to a bifurcation scenario that the map exhibits under variation of one parameter. Constraining this parameter to specific singular values at which the map generates superdiffusion by varying the second one, as has been done in previous literature, we find quantitative deviations between deterministic diffusion and diffusion generated by stochastic LWs in a particular range of parameter values, which cannot be cured by simple LW modifcations. We also explore the effect of aging on superdiffusion in the PM map and show that this yields a profound change of the diffusive properties under variation of the aging time, which should be important for experiments. Our findings demonstrate that even in this simplest well-studied setting, a matching of deterministic and stochastic diffusive properties is non-trivial.

nlin.CD↗

arXiv2024

Anomalous Diffusion in the Square Soft Lorentz Gas

We demonstrate and analyze anomalous diffusion properties of point-like particles in a two-dimensional system with circular scatterers arranged in a square lattice and governed by smooth potentials, referred to as the square soft Lorentz gas. Our numerical simulations reveal a rich interplay of normal and anomalous diffusion depending on the system parameters. To describe diffusion in normal regimes, we develop a unit cell hopping model that, in the single-hop limit, recovers the Machta-Zwanzig approximation and converges toward the numerical diffusion coefficient as the number of hops increases. Anomalous diffusion is characterized by quasiballistic orbits forming Kolmogorov-Arnold-Moser islands in phase space, alongside a complex tongue structure in parameter space defined by the interscatterer distance and potential softness. The distributions of the particle displacement vector show notable similarities to both analytical and numerical results for a hard-wall square Lorentz gas, exhibiting Gaussian behavior in normal diffusion and long tails due to quasiballistic orbits in anomalous regimes. Our work thus provides a catalog of key dynamical system properties that characterize the intricate changes in diffusion when transitioning from hard billiards to soft potentials.

nlin.CD↗

arXiv2024

Cell migration: Beyond Brownian motion

This brief `New & Notable' (perspectives-type) article contains a mini-review on stochastic modelling of cell migration before elaborating on the article by Klimek et al., arXiv:2311.16753 [Biophys. J. 123, 1173-1183 (2024)].

cond-mat.stat-mech↗

arXiv2024

Transition to anomalous dynamics in a simple random map

The famous Bernoulli shift (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability $p$, and the contracting one with probability $1-p$, gives a prototype of a random dynamical system. Here we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of $p$. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability $p_c$, defined by a zero Lyapunov exponent. This anomalous dynamics is characterised by an infinite invariant density, weak ergodicity breaking and power law correlation decay.

nlin.CD↗

arXiv2023

Stochastic hydrodynamic velocity field and the representation of Langevin equations

The fluctuation-dissipation theorem, in the Kubo original formulation, is based on the decomposition of the thermal agitation forces into a dissipative contribution and a stochastically fluctuating term. This decomposition can be avoided by introducing a stochastic velocity field, with correlation properties deriving from linear response theory. Here, we adopt this field as the comprehensive hydrodynamic/fluctuational driver of the kinematic equations of motion. With this description, we show that the Langevin equations for a Brownian particle interacting with a solvent fluid become particularly simple and can be applied even in those cases in which the classical approach, based on the concept of a stochastic thermal force, displays intrinsic difficulties e.g., in the presence of the Basset force. We show that a convenient way for describing hydrodynamic/thermal fluctuations is by expressing them in the form of Extended Poisson-Kac Processes possessing prescribed correlation properties and a continuous velocity density function. We further highlight the importance of higher-order correlation functions in the description of the stochastic hydrodynamic velocity field with special reference to short-time properties of Brownian motion. We conclude by outlining some practical implications in connection with the statistical description of particle motion in confined geometries.

cond-mat.stat-mech↗

arXiv2022

In the folds of the Central Limit Theorem: Lévy walks, large deviations and higher-order anomalous diffusion

This article considers the statistical properties of Lévy walks possessing a regular long-term linear scaling of the mean square displacement with time, for which the conditions of the classical Central Limit Theorem apply. Notwithstanding this property, their higher-order moments display anomalous scaling properties, whenever the statistics of the transition times possesses power-law tails. This phenomenon is perfectly consistent with the classical Central Limit Theorem, as it involves the convergence properties towards the normal distribution. It is closely related to the property that the higher order moments of normalized sums of $N$ independent random variables possessing finite variance may deviate, for $N$ tending to infinity, to those of the normal distribution. The thermodynamic implications of these results are thoroughly analyzed by motivating the concept of higher-order anomalous diffusion.

cond-mat.stat-mech↗

arXiv2022

On the stochastic thermodynamics of fractional Brownian motion

This paper is concerned with the stochastic thermodynamics of non-equilibrium Gaussian processes that can exhibit anomalous diffusion. In the systems considered, the noise correlation function is not necessarily related to friction. Thus, there is no conventional fluctuation-dissipation relation (FDR) of the second kind and no unique way to define a temperature. We start from a Markovian process with time-dependent diffusivity (an example being scaled Brownian motion). It turns out that standard stochastic thermodynamic notions can be applied rather straightforwardly by introducing a time-dependent temperature, yielding the integral fluctuation relation. We then proceed to our focal system, that is, a particle undergoing fractional Brownian motion (FBM). In this system, the noise is Gaussian but the noise correlation function is nonlocal in time, defining a non- Markovian process. We analyse in detail the consequences when using the conventional notions of stochastic thermodynamics with a constant medium temperature. In particular, the heat calculated from dissipation into the medium differs from the log ratio of path probabilities of forward and backward motion, yielding a deviation from the standard integral fluctuation relation for the total entropy production if the latter is defined via system entropy and heat exchange. These apparent inconsistencies can be circumvented by formally defining a time-nonlocal temperature that fulfils a generalized FDR. To shed light on the rather abstract quantities resulting from the latter approach we perform a perturbation expansion. This allows us to calculate analytically, up to linear order, the generalized temperature and the corresponding heat exchange. By this, we provide explicit expressions and a physical interpretation for the leading corrections induced by non-Markovianity.

cond-mat.stat-mech↗

arXiv2022

Dynamical transitions and aging in the superdiffusive Pomeau-Manneville map

The Pomeau-Manneville map is a paradigmatic intermittent dynamical system exhibiting weak chaos and anomalous dynamics. In this paper we analyse the parameter dependence of superdiffusion for the map lifted periodically onto the real line. From numerical simulations we compute the generalised diffusion coefficient (GDC) of this model as a function of the map's nonlinearity parameter. We identify two singular dynamical transitions in the GDC, one where it diverges to infinity, and a second one where it is fully suppressed. Using the continuous-time random walk theory of L\'evy walks we calculate an analytic expression for the GDC and show that it qualitatively reproduces these two transitions. Quantitatively it systematically deviates from the deterministic dynamics for small parameter values, which we explain by slow decay of velocity correlations. Interestingly, imposing aging onto the dynamics in simulations eliminates the dynamical transition that led to suppression of the GDC, thus yielding a non-trivial change in the parameter dependence of superdiffusion. This also applies to a respective intermittent model of subdiffusive dynamics displaying a related transition.

nlin.CD↗

arXiv2022

Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity

Stochastic processes play a key role for modeling a huge variety of transport problems out of equilibrium, with manifold applications throughout the natural and social sciences. To formulate models of stochastic dynamics the conventional approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or Lévy. While these distributions are motivated by (generalised) central limit theorems they are nevertheless \textit{unbounded}, meaning that arbitrarily large fluctuations can be obtained with finite probability. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. Here we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing physically realistic finite propagation velocity. Our approach is motivated by the theory of Lévy walks, which we embed into an extension of conventional Poisson-Kac processes. The resulting extended theory employs generalised transition rates to model subtle microscopic dynamics, which reproduces non-trivial spatio-temporal correlations on macroscopic scales. It thus enables the modelling of many different kinds of dynamical features, as we demonstrate by three physically and biologically motivated examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking `Brownian yet non Gaussian' diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory can therefore be used to model a wide range of finite velocity dynamical phenomena that are observed experimentally.

cond-mat.stat-mech↗

arXiv2022

Spectral properties of stochastic processes possessing finite propagation velocity

This article investigates the spectral structure of the evolution operators associated with the statistical description of stochastic processes possessing finite propagation velocity. Generalized Poisson-Kac processes and Lévy walks are explicitly considered as paradigmatic examples of regular and anomalous dynamics. A generic spectral feature of these processes is the lower-boundedness of the real part of the eigenvalue spectrum, corresponding to an upper limit for the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We analyze also Generalized Poisson-Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there exists a critical value of the wavevector above which the point spectrum ceases to exist, and the relaxation dynamics becomes controlled by the essential part of the spectrum. This model can be extended to the quantum case and, in point of fact, it represents a simple and highlighting example of a sub-quantum dynamics with hidden variables.

cond-mat.stat-mech↗