Fractals, critical phenomena

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

Fractals, critical phenomena


Dynamic Phases in a Spring-block System 0.01truecm

When a block is pulled via a spring across a surface, there appear to be several different dynamic phases. These can be characterized by the pulling velocity and a dynamic velocity where is the acceleration of gravity, is the mass of the block, and is the spring constant. For sufficiently small , the block displays stick-slip (relaxation) motion. Then as a function of decreasing , this stick-slip motion is first nearly periodic, then aperiodic with approximately an exponential slip size distribution, then aperiodic with possibly a power-law slip size distribution. When is increased, the block eventually ceases to stick and just slides across the surface. The motion can then be adequately described by a Langevin model. -0.1truecm A. Johansen, P. Dimon, C. Ellegaard, J.S. Larsen, and H.H. Rugh 0.3truecm

Studies of Granular Flow 0.01truecm

When small beads flow through a narrow channel, there will be fluctuations in the flow rate around the mean. We are studying the nature of the these fluctuations as a function of the various parameters which we can control, such as the ratio of the bead size to the channel width, and the tilt of the channel. -0.1truecm C. Veje and P. Dimon 0.3truecm

Study of Fragmentation 0.01truecm

The measured mass distributions of fragments from various objects show evidence of obeying a scaling law. The scaling exponent is found to depend on the dimensionality of the object but otherwise not on its morphology or composition. A cube, for example, has the same exponent as a sphere. -0.1truecm (L. Oddershede), (J. Bohr), and P. Dimon 0.3truecm

Colored Activity in Self Organized Critical Interface Dynamics 0.01truecm

We study roughening interfaces that become self organized critical by a rule similar to that of invasion percolation. We demonstrate that there is a fundamental difference between transient and critical dynamical exponents. The exponents break the Galilean invariance and temporal multiscaling is observed. We show that the activity along the interface exhibits non trivial power law correlations in both space and time even though only quenched Gaussian noise is applied. The results are compared with simulations where spatial power law correlated noise are used as input. -0.1truecm K. Sneppen and M.H. Jensen 0.3truecm

Multidiffusion in Critical Dynamics of Strings and Membranes 0.01truecm

We study dynamical roughening of strings which do not break the symmetry neither parallel nor vertical to the overall suspension. The suggested non local dynamics enforces a build up of long range correlations and the system becomes self organized critical with a new universality class which exhibits nontrivial temporal scalings and power law correlated activity along the string, even though only localized Gaussian noise is applied. We observe multiscaling of the temporal behaviour both perpendicular and parallel (multidiffusion) to the evolving string at criticality. -0.1truecm K. Sneppen and M.H. Jensen 0.3truecm

Random Fractals, Phase Transitions, and Negative Dimension Spectra 0.01truecm

We introduce an exactly soluble model of a random fractal where the corresponding dimension spectrum , as a consequence of finite size effects in ensemble averaging, exhibits negative values of corresponding to the strongest singularities of the probability measure, i.e. at the left part where . The right part of the spectrum, which corresponds to the regular part of the measure, is not well defined as the scaling length . These two effects are related to the fact that the generalized dimensions exhibit: 1) a first order phase transition at and 2) a non-existing thermodynamic limit for . We show that the scaling properties of the model can be understood as the thermodynamic limit of versus . The connections to fractal aggregates is briefly discussed. -0.1truecm M.H. Jensen, (G. Paladin), (A. Vulpiani), and (W. v.d. Water) 0.3truecm

Punctuated Equilibrium and the Critical Edge of an Evolving Ecosystem 0.01truecm

We study a dynamic model for an evolving ecology of interacting species. The species are constrained by barriers in an underlying fitness landscape that parametrize the reproduction stability of each species and its potential mutants in the environment. A central element of the model is a variability in the couplings between species, which form a metric of the ecosystem that is self organized and subject to dynamic changes. This is obtained by a fine graining of the ecosystem into possible dependencies between species. In the limit of very rarely mutating species, the ecology expands its genetic space always keeping the system on the edge of generic scale invariance. Furthermore, an ecosystem governed by local punctuations within individual species morphology, will during its evolution show collective behaviour in terms of global punctuations, involving many simultaneous local punctuations. The expanding ecology increases the diversity of life which in turn opens for an enhanced complexity of the individual species. -0.1truecm K. Sneppen, M.H. Jensen, (P. Bak), and (H. Flyvbjerg) 0.3truecm

Inflation Rule for Canonical-cell Tilings 0.01truecm

Despite the fact that quasicrystals were first discovered a decade ago, the atomic structures for all but a handful of them have yet to be determined. In other words, physicists have been measuring the properties of a class of solid-state material for ten years, but they still do not know what it is, exactly, they have been measuring the properties of. This must set some sort of record. I am presently trying to formulate the geometrical framework of an atomic model for icosahedral quasicrystals. This endeavour involves the search for an inflation rule, whereby quasiperiodic ``canonical-cell'' tilings can be generated. Such tilings, once decorated with atoms, specify the atomic structures of model quasicrystals; these structures should accurately approximate the structures of real quasicrystals. I have not yet found any such inflation rule. -0.1truecm M. Oxborrow 0.3truecm

Self-Organized Critical Dynamics of Fronts: Intermittency and Multiscaling. 0.01truecm

We present recent work on punctuated dynamics of fronts. This kind of dynamics drives the evolving front to a critical state with temporal multiscaling of the interface profile due intermittent activity characterized by avalanches which connect regions of recent local activity. This activity pattern exhibits non trivial power law correlations in both space and time. We present numerical results in both 1+1 and 2+1 dimensions and generalize a theory for the critical exponents to all dimensions. -0.1truecm (J. Falk), K. Sneppen, and M.H. Jensen 0.3truecm

Punctuated Equilibrium and Criticality in a Simple Model of Evolution 0.01truecm

A simple and robust model of biological evolution of an ecology of interacting species is introduced. The model self-organizes into a critical steady state with intermittent co- evolutionary avalanches of all sizes, i. e. it exhibits "punctuated equilibrium" behavior. This collaborative evolution is much faster than non-cooperative scenarios since no large and coordinated, and hence prohibitively unlikely, mutations are involved. -0.1truecm K. Sneppen and (P. Bak) 0.3truecm

Mean Field Theory for a Simple Model of Evolution 0.01truecm

A simple dynamical model for Darwinian evolution on its slowest time-scale is analyzed. Its mean-field theory is formulated and solved. A random neighbor version of the model is simulated, as is a one-dimensional version. In one dimension, the dynamics can be described in terms of a "repetitious random walker" and anomalous diffusion with exponent 0.4. In all cases the model self-organizes to a robust critical attractor. -0.1truecm K. Sneppen, (H. Flyvbjerg), and (P. Bak) 0.3truecm

Self-Organization in Simple Models of the Brain 0.01truecm

One of the most important goals in neuroscience is to understand the way the brain manage to undertake a diversity of tasks. The processing units are the neurons and their connections. On the neural level the underlying network dynamics is not task specific, and a realistic attempt to explain the brains ability to function must take this into account. In sharp contrast, the traditional artificial neural networks are heavily constrained, and the a priori knowledge of the task is crucial. Based on this, the system is trained to construct an appropriate input-output function, which is then used in different but similar situations. The brain is not hard-wired at birth, with all its connections coded into the DNA. Evolution is efficient, but not that efficient. The amount of information contained in DNA is vastly insufficient to specify all neural connections. The structure has to be self-organized rather than by design. Thus to understand the brain, we must understand the principles by which is organizes itself through interactions with the environment.

Recently, we have introduced a new class of adaptive networks that by construction are able to function in an changeable environment. Their performance is intimately connected with the adaptive nature, where information flows through a variety of ever changing paths. We denote this adaptive performance to emphasize that the dynamics cannot be divided into a learning mode and a retrieving mode as known from traditional artificial neural networks. Studies on the performing networks can follow one of two directions. Either a study on the relations to physiology and phycology, or a study on industrial applications. On the medical side, the point is to extract some important principles for brain functioning associated with the proposed network dynamics. This include a study of the chemistry and the `electronics' involved when a neuron fires. On the network level, we would like to understand the processes of conditioning, and the behaviors like habituation and sensitization. Moreover, there is a growing interest in the variety of timescales present in recorded EEG signals. On the industrial side, there are a multitude of possible candidates in the literature, from control and robotics to analysis and diagnosis. Initial tests are carried out on computer, but electronic `neurons' have been build for further testing. -0.1truecm P. Alstrøm, (P. Bak) and (D. Stassinopoulos) 0.3truecm

Intermittent Dynamics and Self Organized Depinning in Propagating Fronts 0.01truecm

We study the roughening dynamics of a 2 dimensional front where advances are made at minimal pinning sites, while slopes of the front are kept finite by additional lateral growth. The interface self organizes toward a critical state with long range correlations in space and time. The dynamics is governed by intermittent burst which give rise to a scale invariant avalanche distribution and multiscaling of the temporal roughening. Generalizing a recently proposed theory for the 1-d case, we demonstrate that the multiscaling can be explained by the static roughness exponent alone. The correlation between subsequent deposition activities exhibits the same power law as in the one dimensional case. The implication of such dimensional independence is discussed. -0.1truecm (J. Falk), M.H. Jensen, and K. Sneppen 0.3truecm

Critical Punctuations and Metastability for Evolving Strings, Interfaces 0.01truecm

We study the dynamics of interfaces that become self organized critical by a rule similar to that of invasion percolation. In the critical state we observe temporal multiscaling of the interface profile and an activity pattern that exhibits non trivial power law correlations in both space and time even though only quenched Gaussian noise is applied. One way to characterize this intermittent activity is through avalanches consistent of connected regions of recent local activity. These avalanches display scale invariance, thus reflecting both small and large scale properties of the dynamics. An extension of the presented concepts to evolving strings leads to a new universality class with other scalings. Possible systems governed by the investigated algorithms are discussed. -0.1truecm K. Sneppen and M.H. Jensen 0.3truecm

Scaling Behaviour in Daily Air Humidity Fluctuations 0.01truecm

We show that the daily average air humidity fluctuations exhibit non-trivial behaviour which is different from the spectral properties of other meteorological quantities. This feature and the fractal spatial structure found in clouds make it plausible to regard air humidity fluctuations as a manifestation of self-organized criticality. We give arguments why the dynamics of air humidity can be similar to dynamics of sandpile models of SOC. -0.1truecm G. Vattay and A. Harnos

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

Nikolaj Berntsen
Mon Feb 6 10:24:27 MET 1995
14. Feb. 1995
Nikolaj Berntsen,