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Tsallis Statistics
The term Tsallis statistics usually refers to the collection of mathematical functions and associated probability distributions that were originated by Constantino Tsallis. Using that collection, it is possible to derive Tsallis distributions from the optimization of the Tsallis entropic form. A continuous real parameter ''q'' can be used to adjust the distributions, so that distributions which have properties intermediate to that of Gaussian and Lévy distributions can be created. The parameter ''q'' represents the degree of non- extensivity of the distribution. Tsallis statistics are useful for characterising complex, anomalous diffusion. Tsallis functions The ''q''-deformed exponential and logarithmic functions were first introduced in Tsallis statistics in 1994. However, the ''q''-logarithm is the Box–Cox transformation for q=1-\lambda, proposed by George Box and David Cox in 1964. ''q''-exponential The ''q''-exponential is a deformation of the exponential function using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsallis Entropy
In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. It is proportional to the expectation of the q-logarithm of a distribution. History The concept was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics and is identical in form to Havrda–Charvát structural α-entropy, introduced in 1967 within information theory. Definition Given a discrete set of probabilities \ with the condition \sum_i p_i=1, and q any real number, the Tsallis entropy is defined as :S_q() = k \cdot \frac \left( 1 - \sum_i p_i^q \right), where q is a real parameter sometimes called ''entropic-index'' and k a positive constant. In the limit as q \to 1, the usual Boltzmann–Gibbs entropy is recovered, namely :S_\text = S_1(p) = -k \sum_i p_i \ln p_i , where one identifies k with the Boltzmann constant k_B. For continuous probability distributions, we define the entropy as :S_q = \left( 1 - \int (p(x)) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constantino Tsallis
Constantino Tsallis (; ; born 4 November 1943) is a naturalized Brazilian physicist of Greek descent, working in Rio de Janeiro at Centro Brasileiro de Pesquisas Físicas (CBPF), Brazil. Biography Tsallis was born in Greece, and grew up in Argentina, where he studied physics at Instituto Balseiro, in Bariloche. In 1974, he received a ''Doctorat d'État ès Sciences Physiques'' degree from the University of Paris-Sud. He moved to Brazil in 1975 with his wife and daughter. Tsallis is an External Professor of the Santa Fe Institute. Research Tsallis is credited with introducing the notion of what is known as Tsallis entropy and Tsallis statistics in his 1988 paper "Possible generalization of Boltzmann–Gibbs statistics" published in the ''Journal of Statistical Physics''. The generalization is considered to be a good candidate for formulating a theory of non- extensive thermodynamics. The resulting theory is not intended to replace Boltzmann–Gibbs statistics, but rather suppleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-exponential Distribution
The ''q''-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The ''q''-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The exponential distribution is recovered as q \rightarrow 1. Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for q=1-\lambda, a particular case of power transform in statistics. Characterization Probability density function The ''q''-exponential distribution has the probability density function :(2-q) \lambda e_q(-\lambda x) where :e_q(x) = +(1-q)x is the ''q''-exponential if . When , ''e''''q''(x) is just exp(''x''). Derivation In a similar procedure to how the exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
