Systematic Density Expansion of the Lyapunov Exponents for a Two-dimensional Random Lorentz Gas

We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form $A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}$, where $A_{0}$ and $B_{0}$ are known constants and $\tilde{n}$ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order $(\tilde{n}^{2})$, the positive Lyapunov exponent is of the form $A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}+A_{1}\tilde{n}^{2}\ln\tilde{n} +B_{1}\tilde{n}^{2}$. Explicit numerical values of the new constants $A_{1}$ and $B_{1}$ are obtained by means of a systematic analysis. This takes into account, up to $O(\tilde{n}^{2})$, the effects of {\it all\/} possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.Comment: 12 pages, 9 figures, minor changes in this version, to appear in J. Stat. Phy

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

Protocol-Safe Workflow Support for Santa Claus

Practical software analysis techniques exploit a form a process description, mostly in some \ud avour of state diagram. Unlike typing information, these process structures are usually not passed down to the implementation level, and neither are they exploited in any form of consistency check. It is our belief that the information in most designs suffices to perform all sorts of consistency checks. This workshop paper studies a simple case where work\ud ow processes interact with `actual' objects at the implementation level, and demonstrates how useful protocol checking can be in making and keeping these processes consistent with each other

Measurement of the inelastic proton-proton cross section at s\sqrt{s} = 13 TeV

A measurement of the inelastic proton-proton cross section at a centre-of-mass energy of $\sqrt{s}$ = 13 TeV is presented. The analysis is performed using the CMS detector, in particular with information from forward calorimetry at pseudorapidities of 3.0 < {\eta} < 5.2 and -6.6 < {\eta} < -3.0. A visible cross section is measured in two different detector acceptances and finally extrapolated to the full inelastic phase space domain. The results are compared with those of other experiments, and with models used to describe high-energy hadronic interactions.Comment: 5 pages, 2 figures, proceedings of the XXIV International Workshop on Deep-Inelastic Scattering and Related Subjects, 11-15 April 2016, DESY Hamburg, German

On Practical Verification of Processes

The integration of a formal process theory with a practically usable notation is not straightforward, but it is necessary for practical verification of process specifications. Given such an intermediate language, a verification process that gives useful feedback is not trivial either: Model checkers are not powerful enough to deal with object models, and theorem provers provide insu#cient feedback and are not certain to find a proof

Non-abelian Littlewood-Offord inequalities

In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.Comment: 14 pages Second version. Dependence of the upper bound on the matrix size in the main results has been remove