A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy . More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices q in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation q_stat+q_sens=2 is numerically satisfied, q_stat and q_sens being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.     

Vacuum Radiation,Entropy, and Molecular Chaos

Vacuum radiation causes a particle to make a random walk about its dynamical trajectory. In this random walk the root mean square change in spatial coordinate is proportional to t ^1/2, and the fractional changes in momentum and energy are proportional to t ^−1/2, where t is time. Thus the exchange of energy and momentum between a particle and the vacuum tends to zero over time. At the end of a mean free path the fractional change in momentum of a particle in a gas is very small. However, at the end of the mean free path each particle undergoes an interaction that magnifies the preceding change, and the net result is that the momentum distribution of the particles in a gas is randomized in a few collision times. In this way the random action of vacuum radiation and its subsequent magnification by molecular interaction produces entropy increase. This process justifies the assumption of molecular chaos used in the Boltzmann transport equation.     

Theoretical study of new acceptor and donor molecules based on polycyclic aromatic hydrocarbons

Functionalized polycyclic aromatic hydrocarbons (PAHs) are an interesting class of molecules in which the electronic state of the graphene-like hydrocarbon part is tuned by the functional group. Searching for new types of donor and acceptor molecules, a set of new PAHs has recently been investigated experimentally using ultraviolet photoelectron spectroscopy (UPS). In this work, the electronic structure of the PAHs is studied theoretically with the help of B3LYP hybrid density functionals. Using the ΔSCF method, electron binding energies have been determined which affirm, specify and complement the UPS data. Symmetry properties of molecular orbitals are analyzed for a categorization and an estimate of the related signal strength. While σ-like orbitals are difficult to detect in UPS spectra of condensed film, calculation provides a detailed insight into the hidden parts of the electronic structure of donor and acceptor molecules. In addition, a diffuse basis set (6-311++G**) was used to calculate electron affinity and LUMO eigenvalues. The calculated electron affinity (EA) provides a classification of the donor/acceptor properties of the studied molecules. Coronene-hexaone shows a high EA, comparable to TCNQ, which is a well-known classical acceptor. Calculated HOMO-LUMO gaps using the related eigenvalues have a good agreement with the experimental lowest excitation energies. TD-DFT also accurately predicts the measured optical gap.     

The Renormalization Group and Optimization of Entropy

We illustrate the possible connection that exists between the extremal properties of entropy expressions and the renormalization group (RG) approach when applied to systems with scaling symmetry. We consider three examples: (1) Gaussian fixed-point criticality in a fluid or in the capillary-wave model of an interface; (2) Lévy-like random walks with self-similar cluster formation; and (3) long-ranged bond percolation. In all cases we find a decreasing entropy function that becomes minimum under an appropriate constraint at the fixed point. We use an equivalence between random-walk distributions and order-parameter pair correlations in a simple fluid or magnet to study how the dimensional anomaly at criticality relates to walks with long-tailed distributions.     

Swollen-collapsed transition in random hetero-polymers

A lattice model of a hetero-polymer with random hydrophilic-hydrophobic charges interacting with the solvent is introduced, whose continuum counterpart has been proposed by Garel, Leibler and Orland [#!GLO!#]. The transfer matrix technique is used to study various constrained annealed systems which approximate at various degrees of accuracy the original quenched model. For highly hydrophobic chains an ordinary -point transition is found from a high temperature swollen phase to a low temperature compact phase. Depending on the type of constrained averages, at very low temperatures a swollen phase or a coexistence between compact and swollen phases are found. The results are carefully compared with the corresponding ones obtained in the continuum limit, and various improvements in the original calculations are discussed. Received: 10 April 1998 / Revised: 4 June 1998 / Accepted: 1st July 1998     

Quenched Averages for Self-Avoiding Walks and Polygons on Deterministic Fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and .     

Effective coefficients of quasi-steady Maxwell’s equations with multiscale isotropic random conductivity

The effective coefficients in the quasi-steady Maxwell’s equations are calculated for a multiscale isotropic medium by using a subgrid modeling approach. The conductivity is mathematically represented by a Kolmogorov multiplicative continuous cascade with a lognormal probability distribution. The scale of the solution domain is assumed to be large as compared with the scale of heterogeneities of the medium. The theoretical results obtained in the paper are compared with the results of a direct 3D numerical simulation and the results of the conventional perturbation theory.     

A stochastic model for microtubule length dynamics

In this paper, we use random walk theory to describe the length dynamics of microtubules, one of the principal components of the cytoskeleton. We present a simple two-state model (growing and shrinking) of microtubule length evolution that incorporates a variable rate of switching between the states. Using the generating function technique, we calculate the mean length of microtubule, its variance and diffusion coefficient. We also report analytical and computer simulation results for the mean number of positive monomers in microtubule, and find good qualitative agreement with experimental data.     

John Bell and the Identical Twins

I present a biographical profile of John S. Bell based upon extensive interviews I had with him. I present Bell‘s views on the quantum theory along with a simple explanation of his identity. Jeremy Bernstein is Professor Emeritus of Physics at the Stevens Institute of Technology and a former staff writer for The New Yorker.     

Phase transitions in diluted magnets: Critical behavior,percolation, and random fields

Transition metal halides provide realizations of Ising,XY, and Heisenberg antiferromagnets in one, two, and three dimensions. The interactions, which are of short range, are generally well understood. By dilution with nonmagnetic species such as Zn⁺⁺ or Mg⁺⁺ one is able to prepare site-random alloys which correspond to random systems of particular interest in statistical mechanics. By mixing two magnetic ions such as Fe⁺⁺ and Co⁺⁺ one can produce magnetic crystals with competing interactions-either in the form of competing anisotropies or competing ferromagnetic and antiferromagnetic interactions. In this paper the results of a series of neutron scattering experiments on these systems carried out at Brookhaven over the past several years are briefly reviewed. First the critical behavior in Rb₂Mn_0.5Ni_0.5F₄ and Fe_cZn_1–cF₂ which correspond to two-dimensional and three-dimensional random Ising systems, respectively, are discussed. Percolation phenomena have been studied in Rb₂Mn_cMg_l–cF₄, Rb₂Co_cMg_l–cF₄, KMn_cZ_l-cF₃, and Mn_cZn_l–cF₂ which correspond to two-and three-dimensional Heisenberg and Ising models, respectively. In these casesc is chosen to be in the neighborhood of the nearest-neighbor percolation concentration. Application of a uniform field to the above systems generates a random staggered magnetic field; this has facilitated a systematic study of the random field problem. As we shall discuss in detail, a variety of novel, unexpected phenomena have been observed.