Geometric approach to Hamiltonian dynamics and statistical mechanics

This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of freedom of interest for statistical mechanics. The first part of the paper concerns the applications of methods used in classical differential geometry to study the chaotic dynamics of Hamiltonian systems. Starting from the identity between the trajectories of a dynamical system and the geodesics in its configuration space, a geometric theory of chaotic dynamics can be developed, which sheds new light on the origin of chaos in Hamiltonian systems. In fact, it appears that chaos can be induced not only by negative curvatures, as was originally surmised, but also by positive curvatures, provided the curvatures are fluctuating along the geodesics. In the case of a system with a large number of degrees of freedom it is possible to give an analytical estimate of the largest Lyapunov exponent by means of a geometric model independent of the dynamics. In the second part of the paper the phenomenon of phase transitions is addressed and it is here that topology comes into play. In fact, when a system undergoes a phase transition, the fluctuations of the configuration-space curvature exhibit a singular behavior at the phase transition point, which can be qualitatively reproduced using geometric models. In these models the origin of the singular behavior of the curvature fluctuations appears to be caused by a topological transition in configuration space. This leads us to put forward a Topological Hypothesis (TH). The content of the TH is that phase transitions would be related at a deeper level to a change in the topology of the configuration space of the system.Comment: REVTeX, 81 pages, 36 ps/eps figures (some low-quality figures to save space); review article submitted to Physics Report

A Spinorial Formulation of the Maximum Clique Problem of a Graph

We present a new formulation of the maximum clique problem of a graph in complex space. We start observing that the adjacency matrix A of a graph can always be written in the form A = B B where B is a complex, symmetric matrix formed by vectors of zero length (null vectors) and the maximum clique problem can be transformed in a geometrical problem for these vectors. This problem, in turn, is translated in spinorial language and we show that each graph uniquely identifies a set of pure spinors, that is vectors of the endomorphism space of Clifford algebras, and the maximum clique problem is formalized in this setting so that, this much studied problem, may take advantage from recent progresses of pure spinor geometry

Carbon nanotubes as target for directional detection of light WIMP

In this paper I will briefly introduce the idea of using Carbon Nanotubes (CNT) as target for the detection of low mass WIMPs with the additional information of directionality. I will also present the experimental efforts of developing a Time Projection Chamber with a CNT target inside and the results of a test beam at the Beam Test Facility of INFN-LNF.Comment: 3 figures, IFAE2017 poster session proceeding

Checklist of the Aphodiini of Mexico, Central and South America (Coleoptera: Scarabaeidae: Aphodiinae)

This preliminary checklist of Aphodiini south of the United States is prepared to provide published data for a future web-based checklist of all New World Aphodiinae. All species names are used in combination with their currently accepted generic name, creating many new combinations. A few genus-species combinations are discussed. New synonymies based on recent studies of type specimens are made: Aphodius azteca Harold = Aphodius multimaculosus Hinton; Aphodius ornatus Schmidt = Aphodius magnopunctatus Hinton; Aphodius caracaensis Petrovitz = Aphodius brasilicola Balthasar; Aphodius guatemalensis Bates = Aphodius striatipennis Petrovitz; Aphodius kuntzeni Schmidt = Aphodius amplinotum Gordon and Howden = Aphodius michiliensis Deloya; Aphodius bimaculosus Schmidt = Aphodius xalapensis Galante et al.; Aphodius caracanus Balthasar = Aphodius martinsi Petrovitz; Aphodius volxemi Harold = Aphodius squamifer Petrovitz

Homologous self-organising scale-invariant properties characterise long range species spread and cancer invasion

The invariance of some system properties over a range of temporal and/or spatial scales is an attribute of many processes in nature1, often characterised by power law functions and fractal geometry2. In particular, there is growing consensus in that fat-tailed functions like the power law adequately describe long-distance dispersal (LDD) spread of organisms 3,4. Here we show that the spatial spread of individuals governed by a power law dispersal function is represented by a clear and unique signature, characterised by two properties: A fractal geometry of the boundaries of patches generated by dispersal with a fractal dimension D displaying universal features, and a disrupted patch size distribution characterised by two different power laws. Analysing patterns obtained by simulations and real patterns from species dispersal and cell spread in cancer invasion we show that both pattern properties are a direct result of LDD and localised dispersal and recruitment, reflecting population self-organisation