All entomological traps have a capturing bias, and amber, viewed as a trap, is no exception. Thus the fauna trapped in amber does not represent the total existing fauna of the former amber forest, rather the fauna living in and around the resin producing tree. In this paper we compare arthropods from a forest very similar to the reconstruction of the Miocene Mexican amber forest, and determine the bias of different trapping methods, including amber. We also show, using cluster analyses, measurements of the trapped arthropods, and guild distribution, that the amber trap is a complex entomological trap not comparable with a single artificial trap. At the order level, the most similar trap to amber is the sticky trap. However, in the case of Diptera, at the family level, the Malaise trap is also very similar to amber. Amber captured a higher diversity of arthropods than each of the artificial traps, based on our study of Mexican amber from the Middle Miocene, a time of climate optimum, where temperature and humidity were probably higher than in modern Central America. We conclude that the size bias is qualitatively independent of the kind of trap for non–extreme values. We suggest that frequent specimens in amber were not necessarily the most frequent arthropods in the former amber forest. Selected taxa with higher numbers of specimens appear in amber because of their ecology and behavior, usually closely related with a tree–inhabiting life. Finally, changes of diversity from the Middle Miocene to Recent time in Central and South America can be analyzed by comparing the rich amber faunas from Mexico and the Dominican Republic with the fauna trapped using sticky and Malaise traps in Central America.

We introduce a construction to "periodize" a quasiperiodic lattice of obstacles, i.e., embed it into a unit cell in a higher-dimensional space, reversing the projection method used to form quasilattices. This gives an algorithm for simulating dynamics, as well as a natural notion of uniform distribution, in quasiperiodic structures. It also shows the generic existence of channels, where particles travel without colliding, up to a critical obstacle radius, which we calculate for a Penrose tiling. As an application, we find superdiffusion in the presence of channels, and a subdiffusive regime when obstacles overlap.

We study the structure of quasiperiodic Lorentz gases, i.e., particles bouncing elastically off fixed obstacles arranged in quasiperiodic lattices. By employing a construction to embed such structures into a higher-dimensional periodic hyperlattice, we give a simple and efficient algorithm for numerical simulation of the dynamics of these systems. This same construction shows that quasiperiodic Lorentz gases generically exhibit a regime with infinite horizon, that is, empty channels through which the particles move without colliding, when the obstacles are small enough; in this case, the distribution of free paths is asymptotically a power law with exponent -3, as expected from infinite-horizon periodic Lorentz gases. For the critical radius at which these channels disappear, however, a new regime with locally finite horizon arises, where this distribution has an unexpected exponent of -5, previously observed only in a Lorentz gas formed by superposing three incommensurable periodic lattices in the Boltzmann-Grad limit where the radius of the obstacles tends to zero.

We introduce an algorithm based on generalized dual method (GDM) to efficiently study the dynamics of a particle in quasiperiodic environments without the need to use periodic approximations or to save the information of the vertices that make up the quasiperiodic lattice. We show that the computation time and the memory required to find the tile in which a particle is located as a function of the distance R to the center of symmetry remains constant in our algorithm, while using the GDM directly both quantities go like R 2. This allows us to perform realistic simulations with low consumption of computational resources. The algorithm can be used to study any quasiperiodic lattice that can be produced by the cut-and-project method. Using this algorithm, we have calculated the free path length distribution in quasiperiodic Lorentz gases reproducing previous results and for systems with high symmetries at the Boltzmann–Grad limit. We have found for the Boltzmann–Grad limit, that the distribution of free paths depends on the rank r of the quasiperiodic system and not on its symmetry. The distribution as a function of the free path length l appears to be a combination of exponential decay and a power-law behavior. The latter seems to become important only for probabilities less than ( 2 r − 2 r ( r + 1 ) ) − 1 , showing an exponential decaying free-path length distribution for r → ∞, similar to what is observed in disordered systems.

In this work, we introduce the idea of cage formation probability, defined by considering the angular space needed by a particle in order to leave a cage given an average distance to its neighbors. Considering extreme fluctuations, two phases appear as a function of the number of neighbors and their distances to a central one: Solid and fluid. This allows us to construct an approximated phase diagram based on a geometrical approach. As an example, we apply this probability concept to hard disks in two dimensions and hard spheres in three dimensions. The results are compared with numerical simulations using a Monte Carlo method.