Kim Christensen

Complexity and criticality

Percolation: Percolating Phase Transition In One and Two Dimensions, and in the Bethe Lattice Geometric Properties of Clusters Scaling Ansatz, Scaling Functions and Scaling Relations Universality Real-Space Renormalisation Group Ising Model: Review of Thermodynamics and Statistical Mechanics Symmetry Breaking Ferromagnetic Phase Transition In One and Two Dimensions, and in the Mean-Field Landau Theory of Continuous Phase Transitions Scaling Ansatz, Scaling Functions and Scaling Relations Universality Real-Space Renormalisation Group Self-Organised Criticality: BTW Model in One and Two Dimensions, and in the Mean-Field A Rice Pile Experiment and the Oslo Model Earthquakes and the OFC Model Rainfall Self-Organised Criticality as a Unifying Principle.

A complexity view of rainfall.

We show that rain events are analogous to a variety of nonequilibrium relaxation processes in Nature such as earthquakes and avalanches. Analysis of high-resolution rain data reveals that power laws describe the number of rain events versus size and number of droughts versus duration. In addition, the accumulated water column displays scale-less fluctuations. These statistical properties are the fingerprints of a self-organized critical process and may serve as a benchmark for models of precipitation and atmospheric processes.

Universal fluctuations in correlated systems

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D XY model. We demonstrate that this function describes the fluctuations of global quantities in other correlated equilibrium and nonequilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sandpiles, avalanches, and granular media in a self-organized critical state. We discuss the relationship with both Gaussian and extremal statistics.

Rain: Relaxations in the sky

We demonstrate how, from the point of view of energy flow through an open system, rain is analogous to many other relaxational processes in nature such as earthquakes. By identifying rain events as the basic entities of the phenomenon, we show that the number density of rain events per year is inversely proportional to the released water column raised to the power of 1.4. This is the rain equivalent of the Gutenberg-Richter law for earthquakes. The event durations and the waiting times between events are also characterized by scaling regions, where no typical time scale exists. The Hurst exponent of the rain intensity signal H=0.76>0.5. It is valid in the temporal range from minutes up to the full duration of the signal of half a year. All of our findings are consistent with the concept of self-organized criticality, which refers to the tendency of slowly driven nonequilibrium systems towards a state of scale-free behavior.

Variation of the Gutenberg‐Richter b values and nontrivial temporal correlations in a Spring‐Block Model for earthquakes

We show that a two-dimensional spring-block model for earthquakes is equivalent to a continuous, nonconservative cellular automaton model. The level of conservation is a function of the relevant elastic parameters describing the model. The model exhibits power law distributions for the energy released during an earthquake. The corresponding exponent is not universal. It is a function of the level of conservation. Thus the observed variation in the b value in the Gutenberg-Richter law could be explained by a variation in the elastic parameters. We address the problem of the boundary conditions and display results for two extreme possibilities. Furthermore, we discuss the correlation in the interoccurrence time of earthquakes. The model exhibits the features of real earthquakes: the occurrence of small earthquakes is random, while the larger earthquakes seem to be bunched. Primarily, the results of our work indicate that the dynamic of earthquakes is intimately related to the nonconservative nature of the model, which gives birth to both the change in the exponent and the correlations in interoccurrence time.

TRACER DISPERSION IN A SELF-ORGANIZED CRITICAL SYSTEM

We have studied experimentally transport properties in a slowly driven granular system which recently was shown to display self-organized criticality [Frette et al., Nature (London) 379, 49 (1996)]. Tracer particles were added to a pile and their transit times measured. The distribution of transit times is a constant with a crossover to a decaying power law. The average transport velocity decreases with system size. This is due to an increase in the active zone depth with system size. The relaxation processes generate coherently moving regions of grains mixed with convection. This picture is supported by considering transport in a 1D cellular automaton modeling the experiment.

Self-similar correlation function in brain resting-state functional magnetic resonance imaging.

Adaptive behaviour, cognition and emotion are the result of a bewildering variety of brain spatio-temporal activity patterns. An important problem in neuroscience is to understand the mechanism by which the human brains 100 billion neurons and 100 trillion synapses manage to produce this large repertoire of cortical configurations in a flexible manner. In addition, it is recognized that temporal correlations across such configurations cannot be arbitrary, but they need to meet two conflicting demands: while diverse cortical areas should remain functionally segregated from each other, they must still perform as a collective, i.e. they are functionally integrated. Here, we investigate these large-scale dynamical properties by inspecting the character of the spatio-temporal correlations of brain resting-state activity. In physical systems, these correlations in space and time are captured by measuring the correlation coefficient between a signal recorded at two different points in space at two different times. We show that this two-point correlation function extracted from resting-state functional magnetic resonance imaging data exhibits self-similarity in space and time. In space, self-similarity is revealed by considering three successive spatial coarse-graining steps while in time it is revealed by the 1/f frequency behaviour of the power spectrum. The uncovered dynamical self-similarity implies that the brain is spontaneously at a continuously changing (in space and time) intermediate state between two extremes, one of excessive cortical integration and the other of complete segregation. This dynamical property may be seen as an important marker of brain well-being in both health and disease.

Dynamical and spatial aspects of sandpile cellular automata

The Bak, Tang, and Wiesenfeld cellular automaton is simulated in 1, 2, 3, 4, and 5 dimensions. We define a (new) set of scaling exponents by introducing the concept of conditional expectation values. Scaling relations are derived and checked numerically and the critical dimension is discussed. We address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2. The scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation. Primarily, the results of our work show that the flow of sand down the slope does not have a 1/f power spectrum in any dimension, although the model does show clear critical behavior with scaling exponents depending on the dimension. The power spectrum behaves as 1/f2 in all the dimensions considered.

ON SELF-ORGANIZED CRITICALITY AND SYNCHRONIZATION IN LATTICE MODELS OF COUPLED DYNAMICAL SYSTEMS

Lattice models of coupled dynamical systems lead to a variety of complex behaviors. Between the individual motion of independent units and the collective behavior of members of a population evolving synchronously, there exist more complicated attractors. In some cases, these states are identified with self-organized critical phenomena. In other situations, they are identified with clusterization or phase-locking. The conditions leading to such different behaviors in models of integrate-and-fire oscillators and stick-slip processes are reviewed.

Range-based estimation of quadratic variation

This paper proposes using realized range-based estimators to draw inference about the quadratic variation of jump-diffusion processes. We also construct a range-based test of the hypothesis that an asset price has a continuous sample path. Simulated data shows that our approach is efficient, the test is well-sized and more powerful than a return-based t-statistic for sampling frequencies normally used in empirical work. Applied to equity data, we show that the intensity of the jump process is not as high as previously reported.