Information geometric complexity of a trivariate Gaussian statistical model

We evaluate the information geometric complexity of entropic motion on low-dimensional Gaussian statistical manifolds in order to quantify how difficult is making macroscopic predictions about a systems in the presence of limited information. Specifically, we observe that the complexity of such entropic inferences not only depends on the amount of available pieces of information but also on the manner in which such pieces are correlated. Finally, we uncover that for certain correlational structures, the impossibility of reaching the most favorable configuration from an entropic inference viewpoint, seems to lead to an information geometric analog of the well-known frustration effect that occurs in statistical physics.Comment: 16 pages, 1 figur

Information geometric methods for complexity

Research on the use of information geometry (IG) in modern physics has witnessed significant advances recently. In this review article, we report on the utilization of IG methods to define measures of complexity in both classical and, whenever available, quantum physical settings. A paradigmatic example of a dramatic change in complexity is given by phase transitions (PTs). Hence we review both global and local aspects of PTs described in terms of the scalar curvature of the parameter manifold and the components of the metric tensor, respectively. We also report on the behavior of geodesic paths on the parameter manifold used to gain insight into the dynamics of PTs. Going further, we survey measures of complexity arising in the geometric framework. In particular, we quantify complexity of networks in terms of the Riemannian volume of the parameter space of a statistical manifold associated with a given network. We are also concerned with complexity measures that account for the interactions of a given number of parts of a system that cannot be described in terms of a smaller number of parts of the system. Finally, we investigate complexity measures of entropic motion on curved statistical manifolds that arise from a probabilistic description of physical systems in the presence of limited information. The Kullback-Leibler divergence, the distance to an exponential family and volumes of curved parameter manifolds, are examples of essential IG notions exploited in our discussion of complexity. We conclude by discussing strengths, limits, and possible future applications of IG methods to the physics of complexity.Comment: review article, 60 pages, no figure

A note on the factorization conjecture

We give partial results on the factorization conjecture on codes proposed by Schutzenberger. We consider finite maximal codes C over the alphabet A = {a, b} with C \cap a^* = a^p, for a prime number p. Let P, S in Z , with S = S_0 + S_1, supp(S_0) \subset a^* and supp(S_1) \subset a^*b supp(S_0). We prove that if (P,S) is a factorization for C then (P,S) is positive, that is P,S have coefficients 0,1, and we characterize the structure of these codes. As a consequence, we prove that if C is a finite maximal code such that each word in C has at most 4 occurrences of b's and a^p is in C, then each factorization for C is a positive factorization. We also discuss the structure of these codes. The obtained results show once again relations between (positive) factorizations and factorizations of cyclic groups

Relativistic Charged Spheres II: Regularity and Stability

We present new results concerning the existence of static, electrically charged, perfect fluid spheres that have a regular interior and are arbitrarily close to a maximally charged black-hole state. These configurations are described by exact solutions of Einstein's field equations. A family of these solutions had already be found (de Felice et al., 1995) but here we generalize that result to cases with different charge distribution within the spheres and show, in an appropriate parameter space, that the set of such physically reasonable solutions has a non zero measure. We also perform a perturbation analysis and identify the solutions which are stable against adiabatic radial perturbations. We then suggest that the stable configurations can be considered as classic models of charged particles. Finally our results are used to show that a conjecture of Kristiansson et al. (1998) is incorrect.Comment: revtex, 13 pages. five EPS figures. Accepted by CQ

Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models

In the Horndeski's most general scalar-tensor theories with second-order field equations, we derive the conditions for the avoidance of ghosts and Laplacian instabilities associated with scalar, tensor, and vector perturbations in the presence of two perfect fluids on the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background. Our general results are useful for the construction of theoretically consistent models of dark energy. We apply our formulas to extended Galileon models in which a tracker solution with an equation of state smaller than -1 is present. We clarify the allowed parameter space in which the ghosts and Laplacian instabilities are absent and we numerically confirm that such models are indeed cosmologically viable.Comment: 18 pages, 6 figure

The post-Newtonian limit in C-theories of gravitation

C-theory provides a unified framework to study metric, metric-affine and more general theories of gravity. In the vacuum weak-field limit of these theories, the parameterized post-Newtonian (PPN) parameters $\beta$ and $\gamma$ can differ from their general relativistic values. However, there are several classes of models featuring long-distance modifications of gravity but nevertheless passing the Solar system tests. Here it is shown how to compute the PPN parameters in C-theories and also in nonminimally coupled curvature theories, correcting previous results in the literature for the latter.Comment: 5 pages, no figures; To appear in PRD as a rapid communicatio

Tracing a relativistic Milky Way within the RAMOD measurement protocol

Advancement in astronomical observations and technical instrumentation implies taking into account the general relativistic effects due the gravitational fields encountered by the light while propagating from the star to the observer. Therefore, data exploitation for Gaia-like space astrometric mission (ESA, launch 2013) requires a fully relativistic interpretation of the inverse ray-tracing problem, namely the development of a highly accurate astrometric models in accordance with the geometrical environment affecting light propagation itself and the precepts of the theory of measurement. This could open a new rendition of the stellar distances and proper motions, or even an alternative detection perspective of many subtle relativistic effects suffered by light while it is propagating and subsequently recorded in the physical measurements.Comment: Proceeding for "Relativity and Gravitation, 100 Years after Einstein in Prague" to be published by Edition Open Access, revised versio

Singularity problem in f(R) model with non-minimal coupling

We consider the non-minimal coupling between matter and the geometry in the f(R) theory. In the new theory which we established, a new scalar $\psi$ has been defined and we give it a certain stability condition. We intend to take a closer look at the dark energy oscillating behavior in the de-Sitter universe and the matter era, from which we derive the oscillating frequency, and the oscillating condition. More importantly, we present the condition of coupling form that the singularity can be solved. We discuss several specific coupling forms, and find logarithmic coupling with an oscillating period $\Delta T\sim\Delta z$ in the matter era $z>4$, can improve singularity in the early universe. The result of numerical calculation verifies our theoretic calculation about the oscillating frequency. Considering two toy models, we find the cosmic evolution in the coupling model is nearly the same as that in the normal f(R) theory when $lna>4$. We also discuss the local tests of the non-minimal coupling f(R) model, and show the constraint on the coupling form.Comment: 13 pages, 4 figure

Reissner-Nordstrom and charged gas spheres

The main point of this paper is a suggestion about the proper treatment of the photon gas in a theory of stellar structure and other plasmas. This problem arises in the study of polytropic gas spheres, where we have already introduced some innovations. The main idea, already advanced in the contextof neutral, homogeneous, polytropic stellar models, is to base the theory firmly on a variational principle. Another essential novelty is to let mass distribution extend to infinity, the boundary between bulk and atmosphere being defined by an abrupt change in the polytropic index, triggered by the density. The logical next step in this program is to include the effect of radiation, which is a very significant complication since a full treatment would have to include an account of ionization, thus fieldsrepresenting electrons, ions, photons, gravitons and neutral atoms as well. In way of preparation, we consider models that are charged but homogeneous, involving only gravity, electromagnetism and a single scalar field that represents both the mass and the electric charge; in short, anon-neutral plasma. While this work only represents a stage in the development of a theory of stars, without direct application to physical systems, it does shed some light on the meaning of the Reissner-Nordstrom solution of the modified Einstein-Maxwell equations., with an application to a simple system.Comment: 19 pages, plain te