A Shannon–Tsallis transformation

Via a first-order linear differential equation, we determine a general link between two different solutions of the MaxEnt variational problem, namely, the ones that correspond to using either Shannon’s or Tsallis’ entropies in the concomitant variational problem. It is shown that the two variations lead to equivalent solutions that have different appearances but contain the same information. These solutions are linked by our transformation. However, the so-called collision entropy (Tsallis’ one with q=2$q=2$) does not have a Shannon counterpart.     

Study of the q-Gaussian distribution with the scale index and calculating entropy by normalized inner scalogram

In this paper, we discuss a method based on wavelet analysis for the study of the q-index of the Gaussian distribution. We derive q-index from the scale index, $i_{\mathit{scale}}$, using the expression; $q\equiv 1+2i_{\mathit{scale}}$ where $i_{\mathit{scale}}$ is a wavelet based tool for measuring the degree of aperiodicity of a dynamical system in the range of $0\le i_{\mathit{scale}}\le 1$. We show that this expression gives consistent results with the numerical approach of q-Gaussian distribution which determines the degree of non-extensivity of a dynamical system in the range of $1<q<3$. We also suggest a new entropy calculation method based on the normalized inner scalogram for studying the chaotic characteristics of nonlinear dynamical systems.     

Tsallis Mapping in Growing Complex Networks with Fitness

We introduce Tsallis mapping in Bianconi-Barab'asi (B-B) fitness model of growing networks.This mapping addresses the dynamical behavior of the fitness model within the framework of nonextensive statistics mechanics,which is characterized by a dimensionless nonextensivity parameter q.It is found that this new phenomenological parameter plays an important role in the evolution of networks:the underlying evolving networks may undergo a different phases depending on the q exponents,comparing to the original B-B fitness model,and the corresponding critical transition temperature T C is also identified.     

Benchmark calculations of Rényi,Tsallis entropies,and Onicescu information energy for ground state helium using correlated Hylleraas wave functions

Accurate values of physical quantities serve as the stepping stone for further researches. Consequently, we provide benchmark values of Shannon, Rényi, Tsallis entropies, and Onicescu information energy for ground state helium. With the highly correlated Hylleraas wave functions, our calculations fully considered the effect of electron correlation. Presented numerical results converge with increasing size of basis set, fulfill analytic relations between the quantities, and satisfactorily agree with those in the literature. In particular, we present these information-theoretic quantities with high accuracy, and it is believed that the reported data would be a valuable reference for further research on information-theoretic quantities of atomic and molecular systems.     

Heat and entropy in nonextensive thermodynamics

The concepts of the quantity of heat and work are deduced in the context of non-extensive statistical mechanics, following steps parallel to those employed in extensive statistical mechanics.Received: 31 January 2003, Accepted: 6 May 2003, Published online: 21 November 2003PACS: 05.70.-a, 05.30.-d, 05.90. + m     

Generalized Informational Entropy and Noncanonical Distribution in Equilibrium Statistical Mechanics

Based on the Jaynes principle of maximum for informational entropy, we find a generalized probability distribution and construct a generalized equilibrium statistical mechanics (ESM) for a wide class of objects to which the usual (canonical) ESM cannot be applied. We consistently consider the case of a continuous, not discrete, random variable characterizing the state of the object. For large values of the argument, the resulting distribution is characterized by a power-law, not exponential, asymptotic behavior, and the corresponding power asymptotic expression agrees with the empirical laws established for these objects. The -deformed Boltzmann–Gibbs–Shannon functional satisfying the requirements of the entropy axiomatics and leading to the canonical ESM for =0 is used as the original entropy functional. We also consider nonlinear transformations of this functional. We show that depending on how the averages of the dynamical characteristics of the object are defined, the different (Tsallis, Renyi, and Hardy–Littlewood–Pólya) versions of the generalized ESM can be used, and we give their comparative analysis. We find conditions under which the Gibbs–Helmholtz thermodynamic relations hold and the Legendre transformation can be applied to the generalized entropy and the Massieu–Planck function. We consider the Tsallis and Renyi ESM versions in detail for the case of a one-dimensional probabilistic object with a single dynamical characteristic whose role is played by a generalized positive energy with a monotonic power growth. We obtain constraints on the Renyi index under which the equilibrium distribution relates to a definite class of stable Gaussian or Levy–Khinchin distributions.     

Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions

The distance and divergence of the probability measures play a central role in statistics, machine learning, and many other related fields. The Wasserstein distance has received much attention in recent years because of its distinctions from other distances or divergences. Although computing the Wasserstein distance is costly, entropy-regularized optimal transport was proposed to computationally efficiently approximate the Wasserstein distance. The purpose of this study is to understand the theoretical aspect of entropy-regularized optimal transport. In this paper, we focus on entropy-regularized optimal transport on multivariate normal distributions and q-normal distributions. We obtain the explicit form of the entropy-regularized optimal transport cost on multivariate normal and q-normal distributions; this provides a perspective to understand the effect of entropy regularization, which was previously known only experimentally. Furthermore, we obtain the entropy-regularized Kantorovich estimator for the probability measure that satisfies certain conditions. We also demonstrate how the Wasserstein distance, optimal coupling, geometric structure, and statistical efficiency are affected by entropy regularization in some experiments. In particular, our results about the explicit form of the optimal coupling of the Tsallis entropy-regularized optimal transport on multivariate q-normal distributions and the entropy-regularized Kantorovich estimator are novel and will become the first step towards the understanding of a more general setting.     

From Rényi Entropy Power to Information Scan of Quantum States

In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called “cat states”, which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory, are also briefly discussed.     

Stochastic molecular optimization using generalized simulated annealing

We propose a stochastic optimization technique based on a generalized simulated annealing (GSA) method for mapping minima points of molecular conformational energy surfaces. The energy maps are obtained by coupling a classical molecular force field (THOR package) with a GSA procedure. Unlike the usual molecular dynamics (MD) method, the method proposed in this study is force independent; that is, we obtain the optimized conformation without calculating the force, and only potential energy is involved. Therefore, we do not need to know the conformational energy gradient to arrive at equilibrium conformations. Its utility in molecular mechanics is illustrated by applying it to examples of simple molecules (H₂O and H₂O₃) and to polypeptides. The results obtained for H₂O and H₂O₃ using Tsallis thermostatistics suggest that the GSA approach is faster than the other two conventional methods (Boltzmann and Cauchy machines). The results for polypeptides show that pentalanine does not form a stable α-helix structure, probably because the number of hydrogen bonds is insufficient to maintain the helical array. On the contrary, the icoalanine molecule forms an α-helix structure. We obtain this structure simulating all Φ, Ψ pairs using only a few steps, as compared with conventional methods. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 647–657, 1998     

Kinetic approach on the DIA wave propagation in a non-extensive distributed dusty plasma

The Landau damping of the dust ion-acoustic wave (DIAW) in a dusty plasma with non-extensive distributed components is analysed relying on the kinetic approach. The electron, ion, and dust particles are effectively modelled by the non-extensive distribution function of the Tsallis statistics. For a collisionless plasma with different values of plasma components indices, the general dispersion relation is achieved, and the non-extensivity effects on the frequency, as well as the Landau damping of the DIAW, are studied. We show that for , the preliminary results of the Maxwellian plasma are obtained. The decrease of wave damping is achieved by increasing the coefficient q index and the ion-to-electron density ratio. The damping rate also increases with an increasing ion-to-electron temperature ratio.