Two-dimensional coupled map lattices display, in a specific parameter range, a stable phase (quasi-) periodic in both space and time. With small changes to the model parameters, this stable phase develops spontaneous eruptions of nonperiodic behavior. Although this behavior itself appears irregular, it can be characterized in a systematic fashion. In particular, parameter-independent features of the spontaneous eruptions may allow useful empirical characterizations of other phenomena that are intrinsically hard to predict and reproduce. Specific features of the distributions of lifetimes (...) and emergence rates of irregular states display such parameter-independent properties. Ó 2004 Elsevier Ltd. All rights reserved. (shrink)

Traditional philosophical discourse draws a distinction between ontology and epistemology and generally enforces this distinction by keeping the two subject areas separated and unrelated. In addition, the relationship between the two areas is of central importance to physics and philosophy of physics. For instance, all kinds of measurement-related problems force us to consider both our knowledge of the states and observables of a system (epistemic perspective) and its states and observables independent of such knowledge (ontic perspective). This applies to quantum (...) systems in particular. (shrink)

Cosmology has undergone a revolution in recent years. The exciting interplay between astronomy and fundamental physics has led to dramatic revelations, including the existence of the dark matter and the dark energy that appear to dominate our cosmos. But these discoveries only reveal themselves through small effects in noisy experimental data. Dealing with such observations requires the careful application of probability and statistics. But it is not only in the arcane world of fundamental physics that probability theory plays such an (...) important role. It has an impact in many aspects of our everyday life, from the law courts to the lottery. Why then do so few people understand probability? And why do so few people understand why it is so important for science? Why do so many people think that science is about absolute certainty when, at its core, it is actually dominated by uncertainty? This book attempts to explain the basics of probability theory, and illustrate their application across the entire spectrum of science. (shrink)

Chaos is often explained in terms of random behaviour; and having positive Kolmogorov–Sinai entropy (KSE) is taken to be indicative of randomness. Although seemly plausible, the association of positive KSE with random behaviour needs justification since the definition of the KSE does not make reference to any notion that is connected to randomness. A common way of justifying this use of the KSE is to draw parallels between the KSE and ShannonÕs information theoretic entropy. However, as it stands this no (...) more than a heuristic point, because no rigorous connection between the KSE and ShannonÕs entropy has been established yet. This paper fills this gap by proving that the KSE of a Hamiltonian dynamical system is equivalent to a generalized version of ShannonÕs information theoretic entropy under certain plausible assumptions. Ó 2005 Elsevier Ltd. All rights reserved. (shrink)

In a recent paper [1], O. F. de Alcantara Bonfim, J. Florencio, and F. C. S´ a Barreto claim to have found numerical evidence of chaos in the motion of a Bohmian quantum particle in a double square-well potential, for a wave function that is a superposition of five energy eigenstates. But according to the result proven here, chaos for this motion is impossible. We prove in fact that for a particle on the line in a superposition of n + (...) 1 energy eigenstates, the Bohm motion x(t) is always quasiperiodic, with (at most) n frequencies. This means that there is a function F (y1, . . . , yn) of period 2π in each of its variables and n frequencies ω1, . . . , ωn such that x(t) = F (ω1t, . . . , ωnt). The Bohm motion for a quantum particle of mass m with wave function ψ = ψ(x, t), a solution to Schrödinger’s equation, is defined by.. (shrink)

In his recent book, Explaining Chaos, Peter Smith presents a new problem in the foundations of chaos theory. Specifically, he argues that the standard ways of justifying idealizations in mathematical models fail when it comes to the infinite intricacy found in strange attractors. I argue that Smith's analysis undermines much of the explanatory power of chaos theory. A better approach is developed by drawing analogies from the models found in continuum mechanics.

The notion of (deterministic) chaos is frequently used in an increasing number of scientific (as well as non-scientific) contexts, ranging from mathematics and the physics of dynamical systems to all sorts of complicated time evolutions, e.g., in chemistry, biology, physiology, economy, sociology, and even psychology. Despite (or just because of) these widespread applications, however, there seem to fluctuate around several misunderstandings about the actual impact of deterministic chaos on several problems of philosophical interest, e.g., on matters of prediction and computability, (...) and determinism and the free will. In order to clarify these points a survey of the meaning variance of the concept(s) of deterministic chaos, or the various contexts in which it is applied, is given, and its actual epistemological implications are extracted. In summary, it turns out that the various concepts of deterministic chaos do not constitute a “new science”, or a “revolutionary” change of the “scientific world picture”. Instead, chaos research provides a sort of toolbox of methods which are certainly useful for a more detailed analysis and understanding of such dynamical systems which are, roughly speaking, endowed with the property of exponential sensitivity on initial conditions. Such a property, then, implies merely one, but quantitatively strong type of limitation of long-time computability and predictability, respectively. (shrink)

A physical system has a chaotic dynamics, according to the dictionary, if its behavior depends sensitively on its initial conditions, that is, if systems of the same type starting out with very similar sets of initial conditions can end up in states that are, in some relevant sense, very different. But when science calls a system chaotic, it normally implies two additional claims: that the dynamics of the system is relatively simple, in the sense that it can be expressed in (...) the form of a mathematical expression having relatively few variables, and that the the geometry of the system’s possible trajectories has a certain aspect, often characterized by a strange attractor. (shrink)