Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases

The kinetic theory of gases provides methods for calculating Lyapunov exponents and other quantities, such as Kolmogorov-Sinai entropies, that characterize the chaotic behavior of hard-ball gases. Here we illustrate the use of these methods for calculating the Kolmogorov-Sinai entropy, and the largest positive Lyapunov exponent, for dilute hard-ball gases in equilibrium. The calculation of the largest Lyapunov exponent makes interesting connections with the theory of propagation of hydrodynamic fronts. Calculations are also presented for the Lyapunov spectrum of dilute, random Lorentz gases in two and three dimensions, which are considerably simpler than the corresponding calculations for hard-ball gases. The article concludes with a brief discussion of some interesting open problems.Comment: 41 pages (REVTEX); 7 figs., 4 of which are included in LaTeX source. (Fig.7 doesn't print well on some printers) This revised paper will appear in "Hard Ball Systems and the Lorentz Gas", D. Szasz ed., Encyclopaedia of Mathematical Sciences, Springe

Optimization of the composition of crop collections for ex situ conservation

Many crop genetic resources collections have been established without a clearly defined conservation goal or mandate, which has resulted in collections of considerable size, unbalanced composition and high levels of duplication. Attempts to improve the composition of collections are hampered by the fact that conceptual views to optimize collection composition are very rare. An optimization strategy is proposed herein, which largely builds on the concepts of core collection and core selection. The proposed strategy relies on hierarchically structuring the crop gene pool and assigning a relative importance to each of its different components. Comparison of the resulting optimized distribution of the number of accessions with the actual distribution allows identification of under- and over-representation within a collection. Application of this strategy is illustrated by an example using potato. The proposed optimization strategy is applicable not only to individual genebanks, but also to consortia of cooperating genebanks, which makes it relevant for ongoing activities within projects that aim at sharing responsibilities among institutions on the basis of rational conservation, such as a European genebank integrated system and the global cacao genetic resources network CacaoNet

Front propagation techniques to calculate the largest Lyapunov exponent of dilute hard disk gases

A kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point, measured by the largest Lyapunov exponent, for a dilute system of hard disks, both in equilibrium and in a uniform shear flow. We derive a generalized Boltzmann equation for an extended one-particle distribution that includes deviations from the reference phase space point. The equation is valid for very low densities n, and requires an unusual expansion in powers of 1/|ln n|. It reproduces and extends results from the earlier, more heuristic clock model and may be interpreted as describing a front propagating into an unstable state. The asymptotic speed of propagation of the front is proportional to the largest Lyapunov exponent of the system. Its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand. For the equilibrium case, an explicit expression for the largest Lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest Lyapunov exponent.Comment: 26 pages REVTeX, 1 eps figure. Added remarks, a reference and corrected some typo

Variable-mesh method of solving differential equations

Multistep predictor-corrector method for numerical solution of ordinary differential equations retains high local accuracy and convergence properties. In addition, the method was developed in a form conducive to the generation of effective criteria for the selection of subsequent step sizes in step-by-step solution of differential equations

Marker-assisted optimization of an expert-based strategy for the acquisition of modern lettuce varieties to improve a genebank collection

To regularly improve the composition of the lettuce collection of the Centre for Genetic Resources, the Netherlands (CGN) with modern varieties, feedback from crop experts is used to select approximately 10% of the new material for incorporation in the collection. In the present study, assessments of six experts were compared to microsatellite data of 414 new varieties and 1408 existing accessions. Based on the microsatellite data, the extent to which the genetic diversity of the collection would be enriched (added value) was calculated for specific sets of new varieties. When individual assessments of experts were evaluated, the total added value of expert-based selections was not significantly higher compared to randomly chosen groups, except for a single expert. Unfamiliarity with new varieties was shown to be a crucial factor in the assessment of crop experts. According to the current acquisition protocol that seeks for consensus among experts, varieties are selected based on recommendations from at least three experts. This protocol also did not perform better than randomly chosen groups of new varieties. However, significantly better results were obtained with alternative protocols. It was concluded that breeding value was a more decisive criterion in the current acquisition protocol than maximal extension of the genetic diversity within the collection. A modified protocol addressing both commercial and diversity aspects was suggested in order to meet the demands of plant breeders as well as conservationist