Pathway parameter and thermonuclear functions

In the theory of thermonuclear reaction rates, analytical evaluation of thermonuclear functions for non-resonant reactions, including cases with cut-off and depletion of the tail of the Maxwell-Boltzmann distribution function were considered in a series of papers by Mathai and Haubold [A.M. Mathai, H.J. Haubold, Modern Problems in Nuclear and Neutrino Astrophysics, Akademie-Verlag, Berlin, 1988]. In the present paper we study more general classes of thermonuclear functions by introducing a pathway parameter α, so that when α→1 the thermonuclear functions in the Maxwell-Boltzmannian case are recovered. We will also give interpretations for the pathway parameter α in the case of cut-off and in terms of moments. Non-extensive statistical mechanics, as developed by Tsallis [C. Tsallis, What should a statistical mechanics satisfy to reflect nature? Physica D 193 (2004) 3-34], provides the physical basis for the generalized thermonuclear functions considered in this paper.     

A class of asymmetric pathway distributions and an entropy interpretation

Asymmetric distributions are widely used in probability modeling and statistical analysis. Recently, various asymmetric distributions are being developed by many researchers for modeling various data sets in real life contexts. In the present paper, we introduce a new class of q-type asymmetric distributions which include q-analogues of asymmetric Laplace, exponential power, Weibull etc. and corresponding standard distributions as special cases. Also we show that this pathway model can be obtained by optimizing Mathai’s generalized entropy with more general setup, which is a generalization of various entropy measures due to Shannon and others.     

Information‐entropic measures in free and confined hydrogen atom

Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (r_c) for the latter. In a CHA, at small r_c, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate r_c, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.     

基于Tsallis理论的中国股市收益分布研究

应用Tsallis提出的非广延统计力学理论以及与之密切相关的非线性Fokker-Planck方程所描述的动力系统,根据我国上证指数和深证指数2004年1月1日～2008年11月13日的高频数据,分析了在三种不同的时间标度下股指收益的概率分布,发现Tsallis分布可以很好地描述两市收益分布的尖峰厚尾有限方差等特征,同时也给出了市场微观动力学层面的解释。揭示出我国上海和深圳股市的价格过程并不符合随机游走,而是反常扩散过程,两市具有十分接近的非线性动力系统特征。所得结论对于研究我国金融市场的资产配置和定价、风险管理和制度建设都具有重要的意义。     

Generalized ICM for image segmentation based on Tsallis statistics

In this paper, the iterated conditional modes optimization method of a Markov random field technique for image segmentation is generalized based on Tsallis statistics. It is observed that, for some q$q$ entropic index values the new algorithm performs better segmentation than the classical one. The proposed algorithm also does not have a local minimum problem and reaches a global minimum energy point although the number of iterations remains the same as ICM. Based on the findings of the new algorithm, it can be expressed that the new technique can be used for the image segmentation processes in which the objects are Gaussian or nearly Gaussian distributed.     

Landau damping of dust acoustic waves in the plasma with nonextensive distribution

The Landau damping of dust acoustic waves propagating in a dusty plasma composed of nonextensive distributed electrons and ions, and Maxwellian distributed dust grains is investigated based on kinetic theory. The dust acoustic waves are found in the range of k_vd?ω?k_vi?k_ve$k{v}_d?\omega ?k{v}_i?k{v}_e$, where _vα$v_{\alpha }$ is the thermal velocity of species α(=i,e,d)$\alpha (=i,e,d)$. The damping rate is shown to be dependent on the nonextensivity parameter q$q$ as well as the ratio of ion density to electron. In the limit q→1$q\to 1$, the result based on the Maxwellian distribution is recovered. The maximum Landau damping rate is found to be enhanced as the population of the electron density decreases.     

The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

By only requiring the q deformed logarithms (q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm (q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1?q<2, while the interval 0<q?1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation (2−q), which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in (−∞,0]. However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.     

Manipulation with heterogeneity within a species population formulated as an inverse problem

Dependence of the evolutionary dynamics on the population’s heterogeneity has been reliably recognized and studied within the frame of evolutionary optimization theory. As the causal relation between the heterogeneity and dynamics of environment has been revealed, the possibility to influence convergence rate of evolutionary processes by purposeful manipulation with environment emerges.For the above purposes we formulate the task as the inverse problem meaning that desired population heterogeneity, quantified by Tsallis information entropy, represents the model’s input and dynamics of environment leading to desired population heterogeneity is looked for. Here the presented abstract model of evolutionary motion within the inverse model of replicating species is case-independent and it is relevant for the broad range of phenomena observed at cellular, ecological, economic and social scales. We envision relevance of the model for anticancer therapy, in which the effort is to circumvent heterogeneity as it typically correlates with the therapy efficiency.