Nonextensive thermodynamic relations

The generalized zeroth law of thermodynamics indicates that the physical temperature in nonextensive statistical mechanics is different from the inverse of the Lagrange multiplier, beta. This leads to modifications of some of thermodynamic relations for nonextensive systems. Here, taking the first law of thermodynamics and the Legendre transform structure as the basic premises, it is found that Clausius definition of the thermodynamic entropy has to be appropriately modified, and accordingly the thermodynamic relations proposed by Tsallis, Mendes and Plastino [Physica A 261 (1998) 534] are also to be rectified. It is shown that the definition of specific heat and the equation of state remain form invariant. As an application, the classical gas model is reexamined and, in marked contrast with the previous result obtained by Abe [Phys. Lett. A 263 (1999) 424: Erratum A 267 (2000) 456] using the unphysical temperature and the unphysical pressure, the specific heat and the equation of state are found to be similar to those in ordinary extensive thermodynamics.Comment: 17 pages. The discussion about the Legendre transform structure is modified and some additional comments are mad

Macroscopic thermodynamics of equilibrium characterized by power-law canonical distributions

Macroscopic thermodynamics of equilibrium is constructed for systems obeying power-law canonical distributions. With this, the connection between macroscopic thermodynamics and microscopic statistical thermodynamics is generalized. This is complementary to the Gibbs theorem for the celebrated exponential canonical distributions of systems in contact with a heat bath. Thereby, a thermodynamic basis is provided for power-law phenomena ubiquitous in nature.Comment: 12 page

Stability of Tsallis antropy and instabilities of Renyi and normalized Tsallis entropies: A basis for q-exponential distributions

The q-exponential distributions, which are generalizations of the Zipf-Mandelbrot power-law distribution, are frequently encountered in complex systems at their stationary states. From the viewpoint of the principle of maximum entropy, they can apparently be derived from three different generalized entropies: the Renyi entropy, the Tsallis entropy, and the normalized Tsallis entropy. Accordingly, mere fittings of observed data by the q-exponential distributions do not lead to identification of the correct physical entropy. Here, stabilities of these entropies, i.e., their behaviors under arbitrary small deformation of a distribution, are examined. It is shown that, among the three, the Tsallis entropy is stable and can provide an entropic basis for the q-exponential distributions, whereas the others are unstable and cannot represent any experimentally observable quantities.Comment: 20 pages, no figures, the disappeared "primes" on the distributions are added. Also, Eq. (65) is correcte

Perspectives on Nuclear Structure and Scattering with the Ab Initio No-Core Shell Model

Nuclear structure and reaction theory are undergoing a major renaissance with advances in many-body methods, strong interactions with greatly improved links to Quantum Chromodynamics (QCD), the advent of high performance computing, and improved computational algorithms. Predictive power, with well-quantified uncertainty, is emerging from non-perturbative approaches along with the potential for new discoveries such as predicting nuclear phenomena before they are measured. We present an overview of some recent developments and discuss challenges that lie ahead. Our focus is on explorations of alternative truncation schemes in the harmonic oscillator basis, of which our Japanese--United States collaborative work on the No-Core Monte-Carlo Shell Model is an example. Collaborations with Professor Takaharu Otsuka and his group have been instrumental in these developments.Comment: 8 pages, 5 figures, accepted for publication in Proceedings of Perspectives of the Physics of Nuclear Structure, JPS Conference Proceedings, Japan (to appear

Support Vector Machines in Analysis of Top Quark Production

Multivariate data analysis techniques have the potential to improve physics analyses in many ways. The common classification problem of signal/background discrimination is one example. The Support Vector Machine learning algorithm is a relatively new way to solve pattern recognition problems and has several advantages over methods such as neural networks. The SVM approach is described and compared to a conventional analysis for the case of identifying top quark signal events in the dilepton decay channel amidst a large number of background events.Comment: 8 pages, 8 figures, to be published in the proceedings of the "Advanced Statistical Techniques in Particle Physics" conference in Durham, UK (March, 2002

Information measures based on Tsallis' entropy and geometric considerations for thermodynamic systems

An analysis of the thermodynamic behavior of quantum systems can be performed from a geometrical perspective investigating the structure of the state space. We have developed such an analysis for nonextensive thermostatistical frameworks, making use of the q-divergence derived from Tsallis' entropy. Generalized expressions for operator variance and covariance are considered, in terms of which the fundamental tensor is given.Comment: contribution to 3rd NEXT-SigmaPhi International Conference (August 2005, Kolymbari, Greece

Geometry of escort distributions

Given an original distribution, its statistical and probabilistic attributs may be scanned by the associated escort distribution introduced by Beck and Schlogl and employed in the formulation of nonextensive statistical mechanics. Here, the geometric structure of the one-parameter family of the escort distributions is studied based on the Kullback-Leibler divergence and the relevant Fisher metric. It is shown that the Fisher metric is given in terms of the generalized bit-variance, which measures fluctuations of the crowding index of a multifractal. The Cramer-Rao inequality leads to the fundamental limit for precision of statistical estimate of the order of the escort distribution. It is also quantitatively discussed how inappropriate it is to use the original distribution instead of the escort distribution for calculating the expectation values of physical quantities in nonextensive statistical mechanics.Comment: 12 pages, no figure

Exponents of 2-multiarrangements and multiplicity lattices

We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively.Comment: 14 page