Per Bak

A forest-fire model and some thoughts on turbulence

Abstract In the context of a forest-fire model we demonstrate critical scaling behavior in a “turbulent” non-equilibrium system. Energy is injected uniformly, and dissipated on a fractal. Critical exponents are estimated by means of a Monte Carlo renormalization- group calculation.

The Devil's Staircase

In the 17th century the Dutch physicist Christian Huyghens observed that two clocks hanging back to back on the wall tend to synchronize their motion. This phenomenon is known as phase locking, frequency locking or resonance, and is generally present in dynamical systems with two competing frequencies. The two frequencies may arise dynamically within the system, as with Huyghenss coupled clocks, or through the coupling of an oscillator to an external periodic force, as with the swing and attendant shown in figure 1. If some parameter is varied—the length of a pendulum or the frequency of the force that drives it, for instance—the system will pass through regimes that are phase locked and regimes that are not. When systems are phase locked the ratio between their frequencies is a rational number. For weak coupling the phase‐locked intervals are narrow, so that even if there is an infinity of intervals, the motion is quasiperiodic for most driving frequencies; that is, the ratio between the two frequencies...

Self-organized criticality

Abstract Dissipative dynamical systems with many degrees of freedom naturally evolve to a critical state with fluctuations extending over all length- and time-scales. It is suggested, and supported by simulations on simple toy-model systems, that turbulence, earthquakes, “1/f” noise, and economics may operate at the self-organized critical state.

Adaptive learning by extremal dynamics and negative feedback.

We describe a mechanism for biological learning and adaptation based on two simple principles: (i) Neuronal activity propagates only through the networks strongest synaptic connections (extremal dynamics), and (ii) the strengths of active synapses are reduced if mistakes are made, otherwise no changes occur (negative feedback). The balancing of those two tendencies typically shapes a synaptic landscape with configurations which are barely stable, and therefore highly flexible. This allows for swift adaptation to new situations. Recollection of past successes is achieved by punishing synapses which have once participated in activity associated with successful outputs much less than neurons that have never been successful. Despite its simplicity, the model can readily learn to solve complicated nonlinear tasks, even in the presence of noise. In particular, the learning time for the benchmark parity problem scales algebraically with the problem size N, with an exponent k approximately 1.4.

Mode-Locking and the Transition to Chaos in Dissipative Systems

Dissipative systems with two competing frequencies exhibit transitions to chaos. We have investigated the transition through a study of discrete maps of the circle onto itself, and by constructing and analyzing return maps of differential equations representing some physical systems. The transition is caused by interaction and overlap of mode-locked resonances and takes place at a critical line where the map looses invertibility. At this line the mode-locked intervals trace up a complete Devils Staircase whose complementary set is a Cantor set with universal fractal dimension D ~ 0.87. Below criticality there is room for quasiperiodic orbits, whose measure is given by an exponent β ~ 0.34 which can be related to D through a scaling relation, just as for second order phase transitions. The Lebesgue measure serves as an order parameter for the transition to chaos. The resistively shunted Josephson junction, and charge density waves (CDWs) in r.f. electric fields are usually described by the differential equation of the damped driven pendulum. The 2d return map for this equation collapses to 1d circle map at and below the transition to chaos. The theoretical results on universal behavior, derived here and elsewhere, can thus readily be checked experimentally by studying real physical systems. Recent experiments on Josephson junctions and CDWs indicating the predicted fractal scaling of mode-locking at criticality are reviewed.

Field Theory for a Model of Self-organized Criticality.

The specific mechanism of self-organization to a critical state is identified for the Bak-Sneppen evolution model. This model is mapped exactly to an underlying branching process. Theoretical arguments, supported by numerical simulations, indicate that the resulting critical behavior is in the same universality class as Reggeon field theory.

A physicist's sandbox

We discuss some recent results suggesting that certain spatially extended dynamical systems naturally evolve toward a state characterized by domains of all length scales. The analogy with second-order phase transitions has prompted the name “self-organized criticality” specific results are available for cellular automaton models, which can be thought of as caricatures of a sandpile undergoing avalances. The potential generality of the results stems from the very simple (nonlinear) diffusion dynamics governing the system.

Self-organized criticality in layered, lacustrine sediments formed by landsliding

Landsliding is the dominant mass-wasting process in humid-temperate uplands and an important regulator of sediment yield from steep-land drainage basins. Information about the magnitude and frequency distribution of landslides has been derived from aerial photography, but it has proved difficult to set limits on the long-term scaling behavior of landsliding because the requirements of spatial and temporal coherence and the large number of observations necessary to undertake magnitude versus frequency analyses are not easy to fulfill. We use a 2250-yr-long record of hillslope erosion associated with extreme hydrologic events preserved in sediments from Lake Tutira, New Zealand, to investigate scaling in landslide deposits. Both the magnitude versus frequency distribution of sediment layers attributed to landsliding and the distribution of time intervals between landsliding events take the form of power laws, the former with an exponent b = 2.06 and the latter with an exponent b = 1.4. These results suggest that the erosional events originate from a self-organized critical process, and are in agreement with observations of scaling in turbidite deposits and grain flows in controlled laboratory experiments. The implications are that the aggregate behavior of landsliding at the catchment scale is orderly and that the stratigraphic record preserves a unique, long-term perspective on a fundamental geomorphic process and the extreme hydrologic events that trigger it.

Fractals and Self-Organized Criticality

Many objects in nature are best described geometrically as fractals, with self-similar features on all length scales. The universe consists of clusters of galaxies, organized in clusters of clusters of galaxies and so on [2.1]. Mountain landscapes have peaks of all sizes, from kilometers down to millimeters. River networks consist of streams of all sizes. Turbulent fluids have vortices over a wide range of sizes. Earthquakes occur on structures of faults ranging from thousands of kilometers to centimeters. Fractals are scale-free in the sense that in viewing a picture of a part of a fractal one cannot deduce its actual size if a yardstick is not shown in the same picture.

Self-organized criticality and punctuated equilibria

Abstract Many natural phenomena evolve intermittently, with periods of tranquillity interrupted by bursts of activity, rather than following a smooth gradual path. Examples include earthquakes, volcanic eruptions, solar flares, gamma-ray bursts, and biological evolution. Stephen Jay Gould and Niles Eldredge have coined the term “punctuated equilibria” for this behavior. We argue that punctuated equilibria reflects the tendency of dynamical systems to evolve towards a critical state, and review recent work on simple models. A good metaphoric picture is one where the systems are temporarily trapped in valleys of deformable, interacting landscapes. Similarities with spin glasses are pointed out. Punctuated equilibria are essential for the emergence of complex phenomena. The periods of stasis allow the system to remember its past history; yet the intermittent events permit further change.