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Jürgen Vollmer

Publications and source records attributed to Jürgen Vollmer.

At least 19 recordsLinked to original sources
arXiv2024

Universal hyper-scaling relations, power-law tails, and data analysis for strong anomalous diffusion

Strong anomalous diffusion is {often} characterized by a piecewise-linear spectrum of the moments of displacement. The spectrum is characterized by slopes $ξ$ and $ζ$ for small and large moments, respectively, and by the critical moment $α$ of the crossover. The exponents $ξ$ and $ζ$ characterize the asymptotic scaling of the bulk and the tails of the probability distribution function of displacements, respectively. Here, we adopt asymptotic theory to match the behaviors at intermediate scales. The resulting constraint explains how distributions with algebraic tails imply strong anomalous diffusion, and it relates $α$ to the corresponding power law. Our theory provides novel relations between exponents characterizing strong anomalous diffusion, and it yields explicit expressions for the leading-order corrections to the asymptotic power-law behavior of the moments of displacement. They provide the time scale that must be surpassed to clearly discriminate the leading-order power law from its sub-leading corrections. This insight allows us to point out sources of systematic errors in their numerical estimates. Rather than separately fitting an exponent for each moment we devise a robust scheme to determine $ξ$, $ζ$ and $α$. The findings are supported by numerical and analytical results on five different models exhibiting strong anomalous diffusion.

math-ph↗
arXiv2024

Nonlinear spectral analysis extracts harmonics from land-atmosphere fluxes

Understanding the dynamics of the land-atmosphere exchange of CO$_2$ is key to advance our predictive capacities of the coupled climate-carbon feedback system. In essence, the net vegetation flux is the difference of the uptake of CO$_2$ via photosynthesis and the release of CO$_2$ via respiration, while the system is driven by periodic processes at different time-scales. The complexity of the underlying dynamics poses challenges to classical decomposition methods focused on maximizing data variance, such as singular spectrum analysis. Here, we explore whether nonlinear data-driven methods can better separate periodic patterns and their harmonics from noise and stochastic variability. We find that Nonlinear Laplacian Spectral Analysis (NLSA) outperforms the linear method and detects multiple relevant harmonics. However, these harmonics are not detected in the presence of substantial measurement irregularities. In summary, the NLSA approach can be used to both extract the seasonal cycle more accurately than linear methods, but likewise detect irregular signals resulting from irregular land-atmosphere interactions or measurement failures. Improving the detection capabilities of time-series decomposition is essential for improving land-atmosphere interactions models that should operate accurately on any time scale.

math.DS↗
arXiv2024

Constraining safe and unsafe overshoots in saddle-node bifurcations

We consider a dynamical system undergoing a saddle-node bifurcation with an explicitly time dependent parameter~$p(t)$. The combined dynamics can be considered as a dynamical systems where $p$ is a slowly evolving parameter. Here, we investigate settings where the parameter features an overshoot. It crosses the bifurcation threshold for some finite duration $t_e$ and up to an amplitude $R$, before it returns to its initial value. We denote the overshoot as safe when the dynamical system returns to its initial state. Otherwise, one encounters runaway trajectories (tipping), and the overshoot is unsafe. For shallow overshoots (small $R$) safe and unsafe overshoots are discriminated by an inverse square-root border, $t_e \propto R^{-1/2}$, as reported in earlier literature. However, for larger overshoots we here establish a crossover to another power law with an exponent that depends on the asymptotics of $p(t)$. For overshoots with a finite support we find that $t_e \propto R^{-1}$, and we provide examples for overshoots with exponents in the range $[-1, -1/2]$. All results are substantiated by numerical simulations, and it is discussed how the analytic and numeric results pave the way towards improved risks assessments separating safe from unsafe overshoots in climate, ecology and nonlinear dynamics.

nlin.CD↗
arXiv2023

Measuring tropical rainforest resilience under non-Gaussian disturbances

The Amazon rainforest is considered one of the Earth's tipping elements and may lose stability under ongoing climate change. Recently a decrease in tropical rainforest resilience has been identified globally from remotely sensed vegetation data. However, the underlying theory assumes a Gaussian distribution of forest disturbances, which is different from most observed forest stressors such as fires, deforestation, or windthrow. Those stressors often occur in power-law-like distributions and can be approximated by $α$-stable Lévy noise. Here, we show that classical critical slowing down indicators to measure changes in forest resilience are robust under such power-law disturbances. To assess the robustness of critical slowing down indicators, we simulate pulse-like perturbations in an adapted and conceptual model of a tropical rainforest. We find few missed early warnings and few false alarms are achievable simultaneously if the following steps are carried out carefully: First, the model must be known to resolve the timescales of the perturbation. Second, perturbations need to be filtered according to their absolute temporal autocorrelation. Third, critical slowing down has to be assessed using the non-parametric Kendall-$τ$ slope. These prerequisites allow for an increase in the sensitivity of early warning signals. Hence, our findings imply improved reliability of the interpretation of empirically estimated rainforest resilience through critical slowing down indicators.

physics.ao-ph↗
arXiv2021

Dose-response functions and surrogate models for exploring social contagion in the Copenhagen Networks Study

Spreading dynamics and complex contagion processes on networks are important mechanisms underlying the emergence of critical transitions, tipping points and other nonlinear phenomena in complex human and natural systems. Increasing amounts of temporal network data are now becoming available to study such spreading processes of behaviours, opinions, ideas, diseases and innovations to test hypotheses regarding their specific properties. To this end, we here present a methodology based on dose-response functions and hypothesis testing using surrogate data models that randomise most aspects of the empirical data while conserving certain structures relevant to contagion, group or homophily dynamics. We demonstrate this methodology for synthetic temporal network data of spreading processes generated by the adaptive voter model. Furthermore, we apply it to empirical temporal network data from the Copenhagen Networks Study. This data set provides a physically-close-contact network between several hundreds of university students participating in the study over the course of three months. We study the potential spreading dynamics of the health-related behaviour "regularly going to the fitness studio" on this network. Based on a hierarchy of surrogate data models, we find that our method neither provides significant evidence for an influence of a dose-response-type network spreading process in this data set, nor significant evidence for homophily. The empirical dynamics in exercise behaviour are likely better described by individual features such as the disposition towards the behaviour, and the persistence to maintain it, as well as external influences affecting the whole group, and the non-trivial network structure. The proposed methodology is generic and promising also for applications to other temporal network data sets and traits of interest.

physics.soc-ph↗
arXiv2021

Complex networks of interacting stochastic tipping elements: cooperativity of phase separation in the large-system limit

Tipping elements in the Earth System receive increased scientific attention over the recent years due to their nonlinear behavior and the risks of abrupt state changes. While being stable over a large range of parameters, a tipping element undergoes a drastic shift in its state upon an additional small parameter change when close to its tipping point. Recently, the focus of research broadened towards emergent behavior in networks of tipping elements, like global tipping cascades triggered by local perturbations. Here, we analyze the response to the perturbation of a single node in a system that initially resides in an unstable equilibrium. The evolution is described in terms of coupled nonlinear equations for the cumulants of the distribution of the elements. We show that drift terms acting on individual elements and offsets in the coupling strength are sub-dominant in the limit of large networks, and we derive an analytical prediction for the evolution of the expectation (i.e., the first cumulant). It behaves like a single aggregated tipping element characterized by a dimensionless parameter that accounts for the network size, its overall connectivity, and the average coupling strength. The resulting predictions are in excellent agreement with numerical data for Erdös-Rényi, Barabási-Albert and Watts-Strogatz networks of different size and with different coupling parameters.

physics.soc-ph↗
arXiv2021

Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling

Strong anomalous diffusion is characterized by asymptotic power-law growth of the moments of displacement, with exponents that do not depend linearly on the order of the moment. The exponents concerning small-order moments are dominated by random motion, while higher-order exponents grow by faster trajectories, such as ballistic excursions or "light fronts". Often such a situation is characterized by two linear dependencies of the exponents on their order. Here, we introduce a simple exactly solvable model, the Fly-and-Die (FnD) model, that sheds light on this behavior and on the consequences of light fronts on displacement autocorrelation functions in transport processes. We present analytical expressions for the moments and derive a scaling form that expresses the long-time asymptotics of the autocorrelation function $\langle x(t_1)\,x(t_2)\rangle$ in terms of the dimensionless time difference $(t_2-t_1)/t_1$. The scaling form provides a faithful collapse of numerical data for vastly different systems. This is demonstrated here for the Lorentz gas with infinite horizon, polygonal billiards with finite and infinite horizon, the Lévy-Lorentz gas, the Slicer Map, and Lévy walks. Our analysis also captures the system-specific corrections to scaling.

cond-mat.stat-mech↗
arXiv2019

Hybrid sideways/longitudinal swimming in the monoflagellate Shewanella oneidensis: from aerotactic band to biofilm

Shewanella oneidensis MR-1 are facultative aerobic electroactive bacteria, with an appealing potential for sustainable energy production and bioremediation. They gather around air sources, forming aerotactic bands and biofilms. Though accumulation is crucial to technological exploitation, their collective behaviour remains poorly reported. Here we establish a comprehensive framework for the study of aerotaxis, unveiling a novel hybrid locomotion pattern. Despite having only one flagellum, MR-1 combine motility features of mono- and multiflagellate bacteria, alternating longitudinal fast and sideways slow swimming. The adaptive tuning of the resulting bimodal velocity distributions fulfils different biological functions, such as aerotaxis and confinement. Overall, we reveal the mechanisms underlying the aerotactic collective behaviour of MR-1, in the process leading from accumulation to biofilm formation

physics.bio-ph↗
arXiv2015

Fluctuating Currents in Stochastic Thermodynamics I. Gauge Invariance of Asymptotic Statistics

Stochastic Thermodynamics uses Markovian jump processes to model random transitions between observable mesoscopic states. Physical currents are obtained from anti-symmetric jump observables defined on the edges of the graph representing the network of states. The asymptotic statistics of such currents are characterized by scaled cumulants. In the present work, we use the algebraic and topological structure of Markovian models to prove a gauge invariance of the scaled cumulant-generating function. Exploiting this invariance yields an efficient algorithm for practical calculations of asymptotic averages and correlation integrals. We discuss how our approach generalizes the Schnakenberg decomposition of the average entropy-production rate, and how it unifies previous work. The application of our results to concrete models is presented in an accompanying publication.

cond-mat.stat-mech↗
arXiv2015

Fluctuating Currents in Stochastic Thermodynamics II. Energy Conversion and Nonequilibrium Response in Kinesin Models

Unlike macroscopic engines, the molecular machinery of living cells is strongly affected by fluctuations. Stochastic Thermodynamics uses Markovian jump processes to model the random transitions between the chemical and configurational states of these biological macromolecules. A recently developed theoretical framework [Wachtel, Vollmer, Altaner: "Fluctuating Currents in Stochastic Thermodynamics I. Gauge Invariance of Asymptotic Statistics"] provides a simple algorithm for the determination of macroscopic currents and correlation integrals of arbitrary fluctuating currents. Here, we use it to discuss energy conversion and nonequilibrium response in different models for the molecular motor kinesin. Methodologically, our results demonstrate the effectiveness of the algorithm in dealing with parameter-dependent stochastic models. For the concrete biophysical problem our results reveal two interesting features in experimentally accessible parameter regions: The validity of a non-equilibrium Green--Kubo relation at mechanical stalling as well as negative differential mobility for superstalling forces.

cond-mat.stat-mech↗
arXiv2015

A microscopic perspective on stochastic thermodynamics

We consider stochastic thermodynamics as a theory of statistical inference for experimentally observed fluctuating time-series. To that end, we introduce a general framework for quantifying the knowledge about the dynamical state of the system on two scales: a fine-grained or microscopic, deterministic and a coarse-grained or mesoscopic, stochastic level of description. For a generic model dynamics, we show how the mathematical expressions for fluctuating entropy changes used in Markovian stochastic thermodynamics emerge naturally. Our ideas are conceptional approaches towards (i) connecting entropy production and its fluctuation relations in deterministic and stochastic systems and (ii) providing a complementary information-theoretic picture to notions of entropy and entropy production in stochastic thermodynamics.

cond-mat.stat-mech↗
arXiv2014

Crossing the bottleneck of rain formation

The demixing of a binary fluid mixture, under gravity, is a two stage process. Initially droplets, or in general aggregates, grow diffusively by collecting supersaturation from the bulk phase. Subsequently, when the droplets have grown to a size, where their Peclet number is of order unity, buoyancy substantially enhances droplet growth. The dynamics approaches a finite-time singularity where the droplets are removed from the system by precipitation. The two growth regimes are separated by a bottleneck of minimal droplet growth. Here, we present a low-dimensional model addressing the time span required to cross the bottleneck, and we hence determine the time, Δt, from initial droplet growth to rainfall. Our prediction faithfully captures the dependence of Δt on the ramp rate of the droplet volume fraction, ξ, the droplet number density, the interfacial tension, the mass diffusion coefficient, the mass density contrast of the coexisting phases, and the viscosity of the bulk phase. The agreement of observations and the prediction is demonstrated for methanol/hexane and isobutoxyethanol/water mixtures where we determined Δt for a vast range of ramp rates, ξ, and temperatures. The very good quantitative agreement demonstrates that it is sufficient for binary mixtures to consider (i) droplet growth by diffusive accretion that relaxes supersaturation, and (ii) growth by collisions of sedimenting droplets. An analytical solution of the resulting model provides a quantitative description of the dependence of Δt on the ramp rate and the material constants. Extensions of the model that will admit a quantitative prediction of Δt in other settings are addressed.

cond-mat.stat-mech↗
arXiv2014

Unidirectionally Coupled Map Lattices with Non-Linear Coupling: Unbinding Transitions and Super-Long Transients

Recently, highly resolved experiments and simulations have provided detailed insight into the dynamics of turbulent pipe flow. This has revived the interest to identify mechanisms that generate chaotic transients with super-exponential growth of lifetime as a function of a control parameter, the Reynolds number for pipe flow, and with transitions from bounded chaotic patches to an invasion of space of irregular motion. Dynamical systems models are unique tools in this respect because they can provide insight into the origin of the very long life time of puffs, and the dynamical mechanism leading to the transition from puffs to slugs in pipe flow. The present paper contributes to this enterprise by introducing a unidirectionally coupled map lattice. It mimics three of the salient features of pipe-flow turbulence: (i) the transition from laminar flow to puffs, (ii) a super-exponential scaling of puff lifetime, and (iii) the transition from puffs to slugs by an unbinding transition in an intermittency scenario. In our model all transitions and scalings are theoretically described from a dynamical systems point of view.

math.DS↗
arXiv2013

Flow-induced shear instabilities of cohesive granulates

In this work we use a multi-scale framework to calculate the fluidization threshold of three-dimensional cohesive granulates under shear forces exerted by a creeping flow. A continuum model of flow through porous media provides an analytical expression for the average drag force on a single grain. The balance equation for the forces and a force propagation model are then used to investigate the effects of porosity and packing structure on the stability of the pile. We obtain a closed-form expression for the instability threshold of a regular packing of mono-disperse frictionless cohesive spherical grains in a planar fracture. Our result quantifies the compound effect of structural (packing orientation and porosity) and dynamical properties of the system on its stability.

physics.flu-dyn↗
arXiv2013

Granular Brownian Motors: role of gas anisotropy and inelasticity

We investigate the motion of a wedge-shaped object (a granular Brownian motor), which is restricted to move along the x-axis and cannot rotate, as gas particles collide with it. We show that its steady-state drift, resulting from inelastic gas-motor collisions, is dramatically affected by anisotropy in the velocity distribution of the gas. We identify the dimensionless parameter providing the dependence of this drift on shape, masses, inelasticity, and anisotropy: the anisotropy leads to a dramatic breaking of equipartition, which should easily be visible in experimental realizations.

cond-mat.stat-mech↗
arXiv2012

Arrest of three-dimensional gravity-confined shear flow of wet granular matter

We study the arrest of three-dimensional flow in wet granular matter subject to a sinusoidal external force and a gravitational field confining the flow in the vertical direction. The minimal strength of the external force that is required to keep the system in motion is determined by considering the balance of injected and dissipated power. This provides a prediction whose excellent quality is demonstrated by a data collapse for an extensive set of event-driven molecular dynamics simulations where we varied the system size, particle number, the energy dissipated upon rupturing capillary bridges, and the bridge length where rupture occurs. The three parameters of the theoretical prediction all lie within narrow margins of theoretical estimates.

cond-mat.soft↗
arXiv2012

The journey of hydrogen to quantized vortex cores

Nanoscale hydrogen particles in superfluid helium track the motions of quantized vortices. This provides a way to visualize turbulence in the superfluid. Here, we trace the evolution of the hydrogen from a gas to frozen particles migrating toward the cores of quantized vortices. Not only are the intervening processes interesting in their own right, but understanding them better leads to more revealing experiments.

cond-mat.stat-mech↗