L.R. da Silva

Statistical mechanics based on Renyi entropy

In this work we show that it is possible to obtain a generalized statistical mechanics (thermostatistics) based on Renyi entropy, to be maximized with adequate constraints. The equilibrium probability distribution thus obtained has a very interesting property. Indeed, it reminds us the statistical distribution proposed by Tsallis, known to conveniently describe a variety of phenomena in nonextensive systems. Moreover, some examples are worked out in order to illustrate the main features of the herein introduced formalism.

A substellar companion around the intermediate-mass giant star HD 11977

We report the discovery of a substellar companion to the intermediate-mass star HD 11977 (G5 III). Radial velocities of this star have been monitored for five years with FEROS at the 1.52-m ESO and later at the 2.2-m MPG/ESO telescope in of K 1 = Chile. Based on the collected data we calculated an orbital solution with a period of P = 711 days, a semi-amplitude of K 1 = 105 ms -1 , and an eccentricity of e = 0.4. The period of the radial-velocity variation is longer than that of the estimated stellar rotation, rendering it unlikely that rotational modulation is the source of the variation in the radial velocity. This hypothesis is supported by the absence of a correlation between stellar activity indicators and radial-velocity variation. By determining a primary stellar mass of M * = 1.91 M ○. , the best-fit minimum mass of the companion and semi-major axis of the orbit are m 2 sin i = 6.54 M Jup and a 2 = 1.93 AU, respectively. An upper limit for the mass of the companion of m 2 ? 65.5 M Jup has been calculated from HIPPARCOS astrometric measurements. Although the possibility of a brown-dwarf companion cannot be excluded, HD 11977 B is one of the few planet candidates detected around an intermediate-mass star. The progenitor main-sequence star of HD 11977 is probably an A-type star. This discovery gives an indirect evidence for planetary companions around early type main-sequence stars.

Preferential attachment growth model and nonextensive statistical mechanics

We introduce a two-dimensional growth model where every new site is located, at a distance r from the barycenter of the pre-existing graph, according to the probability law 1/r2 + αG (αG > 0), and is attached to (only) one pre-existing site with a probability ki/riαA (αA ≥ 0; ki is the number of links of the i-th site of the pre-existing graph, and ri its distance to the new site). Then we numerically determine that the probability distribution for a site to have k links is asymptotically given, for all values of αG, by P(k) eq−k/κ, where eqx ≡ [1 + (1 − q)x]1/(1 − q) is the function naturally emerging within nonextensive statistical mechanics. The entropic index is numerically given (at least for αA not too large) by q = 1 + (1/3)e−0.526 αA, and the characteristic number of links by κ 0.1 + 0.08 αA. The αA = 0 particular case belongs to the same universality class to which the Barabasi-Albert model belongs. In addition to this, we have numerically studied the rate at which the average number of links ki increases with the scaled time t/i; asymptotically, ki (t/i)β, the exponent being close to β = ½(1 − αA) for 0 ≤ αA ≤ 1, and zero otherwise. The present results reinforce the conjecture that the microscopic dynamics of nonextensive systems typically build (for instance, in Gibbs Γ-space for Hamiltonian systems) a scale-free network.

Precise radial velocity measurements of G and K giants Multiple systems and variability trend along the Red Giant Branch

We present the results of our radial velocity (RV) measurements of G and K giants, concentrating on the presence of multiple systems in our sample. Eighty-three giants have been observed for 2.5 years with the fiber-fed echelle spectrograph FEROS at the 1.52 m ESO telescope in La Silla, Chile. Seventy-seven stars (93%) of the targets have been analyzed for RV variability using simultaneous Th-Ar calibration and a cross-correlation technique. We estimate the long-term precision of our measurement as better than 25 m s −1 . Projected rotational velocities have been measured for most stars of the sample. Within our time-base only 21 stars (or 27%) show variability below 2σ, while the others show RV variability with amplitudes up to several km s −1 . The large amplitude (several km s −1 ) and shape (high eccentricity) of the RV variations for 11 of the program stars are consistent with stellar companions, and possibly brown dwarf companions for two of the program stars. In those systems for which a full orbit could be derived, the companions have minimum masses from ∼0.6 Mdown to 0.1 M� .T o these multiple systems we add the two candidates of giant planets already discovered in the sample. This analysis shows that multiple systems contribute substantially to the long-term RV variability of giant stars, with about 20% of the sample being composed of multiple systems despite screening our sample for known binary stars. After removing binaries, the range of RV variability in the whole sample clearly decreases, but the remaining stars retain a statistical trend of RV variability with luminosity: luminous cool giants with B − V ≥ 1.2 show RV variations with σRV > 60 m s −1 , while giants with B − V < 1.2 including those in the clump region exhibit less variability or they are constant within our accuracy. The same trend is observed with respect to absolute visual magnitudes: brighter stars show a larger degree of variability and, when plotted in the RV variability vs. magnitude diagram a trend of increasing RV scatter with luminosity is seen. The amplitude of RV variability does not increase dramatically, as predicted, for instance, by simple scaling laws. At least two luminous and cooler stars of the sample show a correlation between RV and chromospheric activity and bisector asymmetry, indicating that in these two objects RV variability is likely induced by the presence of (chromospheric) surface structures.

Search dynamics at the edge of extinction : Anomalous diffusion as a critical survival state

We investigate the general problem of autonomous random walkers whose sole source of energy are search targets that are themselves diffusing random walkers. We study how the energy accumulated by the searcher varies with the target density via numerical simulations and compare the results with an analytical model for fixed targets. We report that superdiffusion of either searcher or target confers substantial energetic advantages to the former. While superdiffusion may not play a crucial role for high target densities, in contrast it confers a vital advantage in the limit of low densities at the edge of extinction: diffusive searchers rapidly die but superdiffusive searchers can survive for long periods without entering into the extinction state. The validity and relevance of our findings in broader contexts are also discussed.

Anomalous diffusion: Fractional Fokker–Planck equation and its solutions

We analyze a linear fractional Fokker–Planck equation for the case of an external force F(x)∝x|x|α−1 and diffusion coefficient D(x)∝|x|−θ (α,θ∈R). We also discuss the connection of the solutions found here with the Fox functions and the nonextensive statistics based on the Tsallis entropy.

Damage spreading in a gradient

We propose a new method of analyzing the frozen-chaotic transition in a cellular automaton by propagating damage in a gradient. We obtain estimations forpc and for the critical exponents for the Kauffman model and the mixture of OR and XOR rules.

Ising-correlated clusters in the Cont-Bouchaud stock market model

Clusters of parallel spins in the square-lattice Ising model are defined as groups of traders acting together on the Cont-Bouchaud stock market model. As in the random percolation case, we see a crossover from a power-law behaviour to a more Gaussian distribution for the market fluctuations.

Nonlinear diffusion equation, Tsallis formalism and exact solutions

We address this work to analyze a nonlinear diffusion equation in the presence of an absorption term taking external forces and spatial time-dependent diffusion coefficient into account. The nonlinear terms present in this equation are due to a nonlinear generalization of the Darcy law and the presence of an absorbent (source) term. We obtain new exact solutions and investigate nonlinear effects produced on the solutions by these terms. We also connect the results found here within the Tsallis formalism.

Anomalous diffusion and anisotropic nonlinear Fokker–Planck equation

We analyse a N-dimensional anisotropic nonlinear Fokker–Planck equation by considering stationary and time-dependent solutions. The stationary solutions are obtained for very general situations, including those when the diffusion coefficients are spatial dependents. Time-dependent solutions are found in the absence of external force and with constant diffusion coefficients. When restricted to the bi-dimensional case, our investigation about time-dependent solutions focuses on situations where the diffusion coefficient are Dx∝|x|−θx and Dy∝|y|−θy with θx,θy∈R. In general, we verify an anomalous behavior induced in a given direction due to the other directions.