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Fractals, critical phenomena

Percolation and the Metal-Insulator Transition in the Quantum-Hall Regime

The metal-insulator transition observed in two-dimensional electron systems is an intriguing example of a phase transition in solid state physics that despite an intensive interest is poorly understood. Based on the percolation picture of the metal-insulator transition at high magnetic fields, we have under rather general assumptions shown that the inverse thermodynamic density of states vanishes at the critical filling factor. We have compared the results with experimental observations for high mobility samples.

P. Alstrøm, P.J. Jørgensen and H. Nielsen

Threshold Variation in Neural Networks

Threshold variation with firing rate is a well-established phenomenon in biological neural networks. Despite this fact, the use of threshold variation as a control system has almost been neglected in computer studies of neural networks. One advantage of threshold control is the locking of the outgoing to the incoming firing rate, which is important for the noise sensitivity of the firing patterns. Another advantage is that the probing regime (low firing rates) is separated from the regime, where stable firing patterns are formed. We have carried out detailed numerical computations to describe the effect of threshold variation in neural networks.

P. Alstrøm and N.K. Petersen

Machine Learning and Neural Networks in DNA Classification

We have performed a comparison between standard machine learning and neural network algorithms for the DNA splicing site prediction problem, where the goal is to separate protein coding sequences from non-coding sequences. Also Bayesian classifiers have been applied. We have shown that machine learning algorithms are outperformed by Bayesian classifiers with sufficient structure and by back-propagation neural networks with a sufficient number of hidden units.

P. Alstrøm and A.V. Gomez

Dynamic Programming and Wood Cutting

Based on methods known from dynamic programming, we have constructed a program, optimizing the way, wood planks are to be cut, in order to minimize waste and maximize outcome. The program is now industrially implemented.

P. Alstrøm and J. Nordfalk

Power Spectra of Sea-level Elevations

We have analyzed a 106 year time record of hourly sea-level elevations for the port of Esbjerg, Denmark. In addition to the periodic tidal components, the power spectrum has a low-frequency broadband structure. We propose a generic model for the broadband spectrum by considering a damped wave equation which is driven by a spatially extended source and whose field is measured at a point. We have also studied shorter records at other locations which display similar spectra, and compared the spatial correlations with the model predictions.

(H. Svensmark, J.D. Pietrzak) and P. Dimon

Morphological Instabilities in a growing Yeast Colony: Experiment and Theory

We study the growth of colonies of the yeast Pichia membranaefaciens on agarose film. The growth conditions are controlled in a setup where nutrients are supplied through an agarose film suspended over a solution of nutrients. As the thickness of the agarose film is varied, the morphology of the front of the colony changes. The growth of the front is modeled by coupling it to a diffusive field of inhibitory metabolites. Qualitative agreement with experiments suggests that such a coupling is responsible for the observed instability of the front.

T. Sams, K. Sneppen, M.H. Jensen, C. Ellegaard, (B. Christensen and U. Thrane)

String Dynamics in a Quenched Random Landscape

We consider the dynamical and spatial properties of a string moving in a quenched random landscape. The string is subjected to a surface tension constraint similar to the one used in the Sneppen model of interface dynamics. Because the string moves in a quenched random landscape it may get stuck for a long time in various valleys. This influences the dynamical roughening which appears to follow a law similar to a random walker in a quenched path. The spatial roughening appears to define a new class similar but different from the directed polymer class.

K. Sneppen, M.H. Jensen and (Y.-C. Zhang)

Persistence in the Voter Model

Recently a new quantity has been widely studied in connection with critical and non-equilbrium systems - the persistence probability. For example, in a kinetic Ising Model this quantity corresponds to the fraction of spins which have never flipped. In our project we study the Voter Model (a varaint of the q-state Potts Model) in general dimension and ask the question, what fraction of the voters have always held the same opinion? By mapping this problem onto a reaction-diffusion system, and using finite-size scaling, we have been able to analytically extract the time decay of this probability and its dependence on the number of possible opinions q.

M.J. Howard, (C. Godrèche and I. Dornic)

Fractal Patterns in 1D Domain Growth

We consider domain growth in a zero temperature, 1-d version of the q state ising model and investigate the spatial distribution of spins which have never flipped. We find that for low q this distribution is completely determined by the Derrida, Hakim-Pasquier relation, whereas for intermediate q one needs new exponents to describe the max distance between non flipped states.

T. Bohr, M. Howard, M.H. Jensen and K. Sneppen

Hierarchical Model for Self Assembly: The "Mirror-Effect" in Proteins

Motivated by recent experiments on pathway in protein dynamics we study a hierarchical model of self assembly. The model exhibits a first order phase transition and by quenching from high to low temperature it follows a dynamics that starts by freezing the upper end of the hierarchy. Quenching instead from low to high temperature the system also starts by melting the upper end of the hierarchy. This models the "mirror-effect" observed in the experiments on protein unfolding: the measured steps occur in the same order both when the protein unfolds and when it folds back again.

M.H. Jensen, K. Sneppen, G. Zocchi and (A. Hansen)

Noisy Kuramoto-Sivashinsky Equation and Ion-Sputtering

We investigate a noisy version of the Kuramoto-Sivashinsky equation which is a prototype equation to study pattern formation. We have shown by a renormalization group calculation that the noisy KS equation exhibits the same critical behavior as the KPZ equation (the RG flow is towards the KPZ fixed point). As a result, the instability renormalizes into a surface tension at a coarse-grained scale. Our results shed new light on the discussions in the literature concerning the scaling behavior of the (deterministic) Kuramoto-Sivashinsky equation and its relation to the KPZ equation. The equation we study has previously been shown to describe the dynamics of ion-sputtering processes for certain parameter values, and the scaling which we obtain is in nice agreement with that observed experimentally. In order to address the ion-sputtering from a different point of view, we have derived the continuum equation for a simple erosion model which incorporates all the features of the ion-sputtering mechanism. We find that the continuum equation reduces to the noisy Kuramoto-Sivashinsky equation when higher order terms are neglected (they are irrelevant in the RG sense for the scaling behavior).

(R. Cuerno and H. Makse) and K.B. Lauritsen

Interface Growth Equations with Conservation Laws

I have carried out dynamic renormalization group calculations and determined the critical behavior and universality classes for a KPZ growth equation with long-range (nonlocal) interactions described by an integral kernel. The results show that there is a competition between noise and long-range correlations which typically lead to a new universality class. The results can be relevant for the understanding of, e.g., electrodeposition experiments where non-KPZ exponents are observed. In a complementary analytical investigation, I have developed a spectral scaling analysis and rederived some of the results. Specifically, in the cases when the noise does not renormalize the scaling analysis becomes exact.

K.B. Lauritsen

Self-Organized Branching Processes

The concept of SOC was introduced as a possible explanation for the ubiquity of fractals, power-law scaling, and temporal ``1/f'' noise in nature and has attracted extensive consideration since its introduction. We have calculated the mean-field (i.e., dimension going to infinity) phase diagram for self-organized critical systems. In that limit, we can obtain rigorous results and discuss the self-organizing process through a dynamical equation for a control parameter. We find that local dissipation destroys the criticality of the avalanche process, and that only when there is no dissipation present, does the fixed point of the dynamics coincide with the critical point for the spreading of avalanches as a branching process. We have analytically calculated the critical exponents describing the criticality and crossover behavior when dissipation is present.

(S. Zapperi and G. Stanley) and K.B. Lauritsen

Directed Percolation with Absorbing Boundary

In this project, we investigate a modification of directed percolation (DP) in order to understand, e.g., how a catalytic process would be influenced if it takes place close to an absorbing wall. DP is a well-studied process and it is important for understanding such different processes as chemical reactions, catalyzers, spatio-temporal intermittency, directed polymers, epidemics, and Reggeon field theory. It turns out that due to the presence of the boundary, a new critical exponent is needed in order to describe the scaling of the cluster lifetime distribution. Rather surprisingly, this exponent seems to be exactly equal to one, in close agreement with numerical series expansions performed by Essam et al. earlier this year [J. Phys. A 29, 1619 (1996)]. On the one hand, we try to understand the emergence of this new exponent by using a mapping of the dynamics to the fluctuations of an interface in a quenched random medium. On the other hand, we are in the process of constructing a field theory for the DP system in order to carry out an tex2html_wrap_inline1034-expansion of the critical exponents. With a similar type of stochastic one-dimensional driven diffusive system, but with dynamically updated (annealed) impurity sites, we have studied the transport properties and the onset of criticality. The system spontaneously orders into a critical state, and we find that the onset of transport coincides with critical behavior in the model.

H. Fogedby, M. Howard, P. Fröjdh, M.H. Jensen, K.B. Lauritsen, (M. Markosova) and K. Sneppen

Onset of criticality and transport in a driven diffusive system

We study transport properties in a slowly driven diffusive system where the transport is externally controlled by a parameter p. Three types of behaviour are found: For p<p' the system is not conducting at all. For intermediate p a finite fraction of the external excitations propagate through the system. Third, in the regime p>pc the system becomes completely conducting. For all p>p' the system exhibits self-organized critical behaviour. In the middle of this regime, at pc, the system undergoes a continuous phase transition described by critical exponents.

M. Markosova, K.B. Lauritsen, K. Sneppen and M.H. Jensen.

Random Walks in Quenched Random Environments

The behavior of Brownian particles is an extensively studied subject due to its many applications. Here, we are investigating random walks in random environments with both a quenched and a time-dependent random drift. We study by field-theoretic methods how the correlations in the time-dependent component induces a crossover from anomalous behavior to the usual random walk behavior. We have preliminary results and intend to carry out a two-loop analysis in order to characterize the crossover more precisely. Our next step is to extend the analysis to fluctuating interfaces in quenched environments.

H. Fogedby, M. Howard, P. Fröjdh and K.B. Lauritsen

Stochastic Sandpile Models

In this study we have introduced a novel type of stochastic sandpile models which display self-organized criticality in one dimension. Such a behavior is in contrast to the toy-models original introduced by Bak et al., which show trivial behavior in one dimension. Our model was based on the rice experiment by Jens Feder's group in Oslo where SOC in the rice-pile avalanche distribution was observed, and our model allows for a qualitative comparison with the experiment. The study of granular matter is a fascinating field and the study of simplified models allows one to understand the scaling behavior which is beyond the reach of molecular dynamics simulations. An important feature of the models we have introduced is the fact that the SOC universality class can be understood by mapping the toppling dynamics to an interface dynamics. This creates a link between discrete toy-models, and continuum Langevin equations describing phenomena as fluid flow in porous media, flux lines in superconductors, and pinning/depinning phase transitions. We have carried out a systematic analysis of these stochastic sandpile models and have found that one can group the different behaviors in three universality classes.

(L. Amaral) and K.B. Lauritsen

Scaling in Boundary Driven Systems

I have used the scaling of the return probability for a random walker in a wedge to obtain the avalanche exponent for two-dimensional SOC systems driven at the boundary. My results are in complete accordance with expressions obtained by ``spanning tree'' mappings [Ivashkevick et al., J. Phys. A 27, L585 (1994)]. The next step is to extend the results to higher dimensions, and eventually compare with other mean-field results. The results show that the boundary can affect the scaling strongly in SOC systems, and is thus important for extracting precise values for the exponents from simulations on finite systems.

K.B. Lauritsen

Self-Organized Criticality and Synchronization in Nonconserving Systems

We have developed a novel class of continuously driven SOC models. Our models contain a parameter which measures the amount of nonconservation. As a function of this parameter we have investigated the emergence of either SOC or synchronized behavior. We carried out a mean-field analysis and confirmed our predictions by numerical simulations of the models. We observed a strong tendency for the system to synchronize, as previously observed for other SOC models [Perez et al., Int. J. Mod.\ Phys. B 10, 1111 (1996)], and we will further investigate the connection between synchronized and SOC behavior.

(S. Lise) and K.B. Lauritsen

next up previous contents
Next: Biology and physiology Up: RESEARCH PROJECTS Previous: Turbulence

Klaus Lindemann
Fri Feb 21 15:17:28 MET 1997