Microscopic theory for random processes in weakly-coupled open systems

In order to relate macroscopic random motion (described e.g. by Langevin-type theories) to microscopic dynamics, we have undertaken the derivation of a Fokker-Planck-type equation from first microscopic principles. Both subsystems are subject to an external force field. Explicit expressions for the diffusion and drift coefficients are obtained, in terms of the field.

The Equiprobability Bias from a Mathematical and Psychological Perspective

The equiprobability bias (EB) is a tendency to believe that every process in which randomness is involved corresponds to a fair distribution, with equal probabilities for any possible outcome. The EB is known to affect both children and adults, and to increase with probability education. Because it results in probability errors resistant to pedagogical interventions, it has been described as a deep misconception about randomness: the erroneous belief that randomness implies uniformity. In the present paper, we show that the EB is actually not the result of a conceptual error about the definition of randomness. On the contrary, the mathematical theory of randomness does imply uniformity. However, the EB is still a bias, because people tend to assume uniformity even in the case of events that are not random. The pervasiveness of the EB reveals a paradox: The combination of random processes is not necessarily random. The link between the EB and this paradox is discussed, and suggestions are made regarding educational design to overcome difficulties encountered by students as a consequence of the EB.

Random walk simulations of fossil proxy data

A wealth of palaeoecological studies (e.g. pollen, diatoms, chironomids and macrofossils from deposits such as lakes or bogs) have revealed major as well as more subtle ecosystem changes over decadal to multimillennial timescales. Such ecosystem changes are usually assumed to have been forced by specific environmental changes. Here, we test if the observed changes in palaeoecological records may be reproduced by random simulations, and we find that simple procedures generate abrupt events, long-term trends, quasi-cyclic behaviour, extinctions and immigrations. Our results highlight the importance of replicated and multiproxy data for reliable reconstructions of past climate and environmental changes.

Monte Carlo simulation of complex cohesive fracture in random heterogeneous quasi-brittle materials

A numerical method is developed to simulate complex two-dimensional crack propagation in quasi-brittle materials considering random heterogeneous fracture properties. Potential cracks are represented by pre-inserted cohesive elements with tension and shear softening constitutive laws modelled by spatially varying Weibull random fields. Monte Carlo simulations of a concrete specimen under uni-axial tension were carried out with extensive investigation of the effects of important numerical algorithms and material properties on numerical efficiency and stability, crack propagation processes and load-carrying capacities. It was found that the homogeneous model led to incorrect crack patterns and load–displacement curves with strong mesh-dependence, whereas the heterogeneous model predicted realistic, complicated fracture processes and load-carrying capacity of little mesh-dependence. Increasing the variance of the tensile strength random fields with increased heterogeneity led to reduction in the mean peak load and increase in the standard deviation. The developed method provides a simple but effective tool for assessment of structural reliability and calculation of characteristic material strength for structural design.