arXiv subjects

Anders Levermann

Publications and source records attributed to Anders Levermann.

13 recordsLinked to original sources

arXiv2022

The folding of art: avoiding one's past in finite space

Through-out human history the new generations have sought to create their own artistic style while trying to avoid repeating, for example, earlier generations' music. If we assume that this search occurs in a multi-dimensional but confined space of creativity, this gives rise to highly complex dynamics. We present a very simple mathematical model with two parameters which can serve as a generic representation for past avoidance in finite space and is qualitatively distinct from earlier dynamical systems. In the presented radially confined form, the trajectory preserves its complexity while retracting to the vicinity of a hypersurface of constant radius when considered in higher dimension.

nlin.AO↗

arXiv2016

Clustered marginalization of minorities during social transitions induced by co-evolution of behaviour and network structure

Large-scale transitions in societies are associated with both individual behavioural change and restructuring of the social network. These two factors have often been considered independently, yet recent advances in social network research challenge this view. Here we show that common features of societal marginalization and clustering emerge naturally during transitions in a co-evolutionary adaptive network model. This is achieved by explicitly considering the interplay between individual interaction and a dynamic network structure in behavioural selection. We exemplify this mechanism by simulating how smoking behaviour and the network structure get reconfigured by changing social norms. Our results are consistent with empirical findings: The prevalence of smoking was reduced, remaining smokers were preferentially connected among each other and formed increasingly marginalised clusters. We propose that self-amplifying feedbacks between individual behaviour and dynamic restructuring of the network are main drivers of the transition. This generative mechanism for co-evolution of individual behaviour and social network structure may apply to a wide range of examples beyond smoking.

physics.soc-ph↗

arXiv2004

Dynamic sea level changes following changes in the thermohaline circulation

Using the coupled climate model CLIMBER-3a, we investigate changes in sea surface elevation due to a weakening of the thermohaline circulation (THC). In addition to a global sea level rise due to a warming of the deep sea, this leads to a regional dynamic sea level change which follows quasi-instantaneously any change in the ocean circulation. We show that the magnitude of this dynamic effect can locally reach up to ~1m, depending on the initial THC strength. In some regions the rate of change can be up to 20-25 mm/yr. The emerging patterns are discussed with respect to the oceanic circulation changes. Most prominent is a south-north gradient reflecting the changes in geostrophic surface currents. Our results suggest that an analysis of observed sea level change patterns could be useful for monitoring the THC strength.

physics.ao-ph↗

arXiv2004

Solution of a Model for the Oceanic Pycnocline Depth: Scaling of Overturning Strength and Meridional Pressure Difference

We present an analysis of the model by Gnanadesikan [1999] for the pycnocline depth in the ocean. An analytic solution for the overturning strength as a function of the meridional pressure difference is derived and used to discuss their mutual scaling. We show that scaling occurs only in two unphysical regimes of the model. In the absence of the Southern Ocean (SO) processes, i.e. for a northern overturning cell, the volume transport is proportional to the square root of the pressure difference. Linear scaling is seen when the overturning is restricted entirely to the SO, i.e. when no northern downwelling exists. For comparison, we present simulations with the coupled climate model CLIMBER-3$α$ which show linear scaling over a large regime of pressure differences in the North Atlantic (NA). We conclude that the pycnocline model is not able to reproduce the linear scaling between its two central variables, pressure and volume transport.

physics.ao-ph↗

arXiv2003

New Algorithm for Parallel Laplacian Growth by Iterated Conformal Maps

We report a new algorithm to generate Laplacian Growth Patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth patterns are compared to those obtained by the best algorithms of direct numerical solutions. The fractal dimension of the patterns is discussed.

cond-mat.stat-mech↗

arXiv2002

Bi-Laplacian Growth Patterns in Disordered Media

Experiments in quasi 2-dimensional geometry (Hele Shaw cells) in which a fluid is injected into a visco-elastic medium (foam, clay or associating-polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a pre-requisite to describing the fracture of elastic material is the solution of the bi-Laplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of bi-Laplacian growth patterns based on the method of iterated conformal maps.

cond-mat.stat-mech↗

arXiv2002

Quasi-Static Brittle Fracture in Inhomogeneous Media and Iterated Conformal Maps: Modes I, II and III

The method of iterated conformal maps is developed for quasi-static fracture of brittle materials, for all modes of fracture. Previous theory, that was relevant for mode III only, is extended here to mode I and II. The latter require solution of the bi-Laplace rather than the Laplace equation. For all cases we can consider quenched randomness in the brittle material itself, as well as randomness in the succession of fracture events. While mode III calls for the advance (in time) of one analytic function, mode I and II call for the advance of two analytic functions. This fundamental difference creates different stress distribution around the cracks. As a result the geometric characteristics of the cracks differ, putting mode III in a different class compared to modes I and II.

cond-mat.stat-mech↗

arXiv2002

Transition in the Fractal Properties from Diffusion Limited Aggregation to Laplacian Growth via their Generalization

We study the fractal and multifractal properties (i.e. the generalized dimensions of the harmonic measure) of a 2-parameter family of growth patterns that result from a growth model that interpolates between Diffusion Limited Aggregation (DLA) and Laplacian Growth Patterns in 2-dimensions. The two parameters are βwhich determines the size of particles accreted to the interface, and C which measures the degree of coverage of the interface by each layer accreted to the growth pattern at every growth step. DLA and Laplacian Growth are obtained at β=0, C=0 and β=2, C=1, respectively. The main purpose of this paper is to show that there exists a line in the β-C phase diagram that separates fractal (D<2) from non-fractal (D=2) growth patterns. Moreover, Laplacian Growth is argued to lie in the non-fractal part of the phase diagram. Some of our arguments are not rigorous, but together with the numerics they indicate this result rather strongly. We first consider the family of models obtained for β=0, C>0, and derive for them a scaling relation D=2 * D_3. We then propose that this family has growth patterns for which D=2 for some C>C_{cr}, where C_{cr} may be zero. Next we consider the whole β-C phase diagram and define a line that separates 2-dimensional growth patterns from fractal patterns with D<2. We explain that Laplacian Growth lies in the region belonging to 2-dimensional growth patterns, motivating the main conjecture of this paper, i.e. that Laplacian Growth patterns are 2-dimensional. The meaning of this result is that the branches of Laplacian Growth patterns have finite (and growing) area on scales much larger than any ultra-violet cut-off length.

cond-mat.stat-mech↗

arXiv2001

Multifractal Structure of the Harmonic Measure of Diffusion Limited Aggregates

The method of iterated conformal maps allows to study the harmonic measure of Diffusion Limited Aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as $10^{-35}$, and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions $D_q$ are infinite for $q<q^*$, where $q^*$ is of the order of -0.2. In the language of $f(α)$ this means that $α_{max}$ is finite. The $f(α)$ curve loses analyticity (the phenomenon of "phase transition") at $α_{max}$ and a finite value of $f(α_{max})$. We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of $q^*$ and $f(α_{max})$. We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.

cond-mat.stat-mech↗

arXiv2001

Quasi-Static Fractures in Disordered Media and Iterated Conformal Maps

We study the geometrical characteristic of quasi-static fractures in disordered media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern; it crosses over from about 0.5 for small scales to about 0.75 for large scales, in excellent agreement with experiments. We propose that this cross-over reflects the increased ramification of the fracture pattern.

cond-mat.stat-mech↗

arXiv2001

Thermodynamic Formalism of the Harmonic Measure of Diffusion Limited Aggregates: Phase Transition and Converged $f(α)$

We study the nature of the phase transition in the multifractal formalism of the harmonic measure of Diffusion Limited Aggregates (DLA). Contrary to previous work that relied on random walk simulations or ad-hoc models to estimate the low probability events of deep fjord penetration, we employ the method of iterated conformal maps to obtain an accurate computation of the probability of the rarest events. We resolve probabilities as small as $10^{-70}$. We show that the generalized dimensions $D_q$ are infinite for $q<q^*$, where $q^*= -0.17\pm 0.02$. In the language of $f(α)$ this means that $α_{max}$ is finite. We present a converged $f(α)$ curve.

cond-mat.stat-mech↗

arXiv2001

Laplacian Growth and Diffusion Limited Aggregation: different universality classes

It had been conjectured that Diffusion Limited Aggregates and Laplacian Growth patterns (with small surface tension) are in the same universality class. Using iterated conformal maps we construct a 1-parameter family of fractal growth patterns with a continuously varying fractal dimension. This family can be used to bound the dimension of Laplacian Growth patterns from below. The bound value is higher than the dimension of Diffusion Limited Aggregates, showing that the two problems belong to two different universality classes.

cond-mat.stat-mech↗

arXiv2000

Convergent Calculation of the Asymptotic Dimension of Diffusion Limited Aggregates: Scaling and Renormalization of Small Clusters

Diffusion Limited Aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (2-3 particles) to asymptotically large (n \to \infty) clusters. The computed dimension is D=1.713\pm 0.003.

cond-mat.stat-mech↗