Roberto Osório

Nonextensive statistical mechanics and economics

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann–Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accommodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy SBG=−k∑ipilnpi, and typically provides exponential laws for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form Sq=k(1−∑ipiq)/(q−1) (with S1=SBG), and typically provides, for the same type of description, (asymptotic) power laws.

Superlattice energetics and alloy thermodynamics of GaAs/Ge

We re-examine the validity of the three-dimensional bulk thermodynamic description for the zinc-blende-to-diamond transition observed in (GaAs)1−xGe2x alloys. We use interaction parameters extracted from first-principles local-density total-energy calculations for (GaAs)p/(Ge2)p superlattices and a cluster-variation solution to the corresponding Ising-like hamiltonian. The resulting phase diagram obtained with such a realistic energy model fails to account for the observed critical composition in the relevant temperature range, suggesting that, contrary to a previous view, the transition is not driven by bulk thermodynamics.

Correlation effects in metastable (Ga As)1−x Ge2x alloys

An improved model is presented for the zinc-blende -- diamond order-disorder transition proposed originally by Newman and Dow (ND) to occur in (Ga As)1−x Ge2x metastable alloys. A better statistical description which incorporates fully nearest-neighbor correlations yields a considerable reduction in the antisite disorder in comparison to the ND mean field calculation. The direct band-gap energy is calculated within the ND modified virtual-crystal approximation, but environmental disorder effects are also considered at Ge sites. Results for the band gap dependence on concentration show a better qualitative and quantitative agreement with experiment.

Lennard-Jones triangular lattice gas in the Kikuchi approximation

The triangular lattice-gas problem for a model with infinitely repulsive nearest-neighbor and attractive second-, J2, and third-neighbor, J3, interactions is studied in the Kikuchi approximation with a triangle of nearest-neighbor sites as the basic cluster. Values of the ratio α ≡ J3J2 0.9481.

Thermodynamics of surface‐induced ordering in the Ga1−xInxP alloy

The occurrence of spontaneous CuPt‐like ordering in epitaxial Ga1−xInxP alloys is examined by means of a thermodynamic theory. We treat both cation‐ and anion‐terminated surfaces. We find that the order parameter of the observed CuPtB variant is significant for reconstructed cation‐terminated surfaces even at preparation temperatures. A possible mechanism for the formation of the three‐dimensional CuPtB structure involves partial equilibration of the cation‐terminated reconstructed surface followed by freezing of the cations as the surface is buried by newly deposited layers.

Cluster approach to dilute magnetism

A cluster algebra is developed for the definition of independent correlation functions in the cluster-variation method (CVM) for the spin-1 Ising model. A scheme is then introduced for the study of site-dilute spin-12 Ising models by means of the CVM. The procedure regards the site-dilute spin-12 model as the spin-1 model with additional constraints due to dilution. The Desjardins-Steinsvoll algortihm is used for the transformation of the CVM equations into a set of differential equations for the independent correlation functions with the inverse temperature as parameter. The evolution of the correlation functions with temperature and the behavior of response functions such as the specific heat and the susceptability are then obtained for any degree of dilution. As an introduction to this scheme, its detailed application is presented here for the simple case of the pair approximation.

Dilute bcc antiferromagnet with first- and second-neighbor interactions

Abstract The thermodynamics of the site-dilute bcc Ising model with antiferromagnetic first-neighbor and ferro or antiferromagnetic second-neighbor interactions is discussed within the tetrahedron approximation of the cluster-variation method. The joint effect of dilution and frustration is shown to cause an anomaly in the temperature dependence of the uniform susceptibility in the ordered phase.