Jan Naudts

Deformed exponentials and logarithms in generalized thermostatistics

Criteria are given that κ-deformed logarithmic and exponential functions should satisfy. With a pair of such functions one can associate another function, called the deduced logarithmic function. It is shown that generalized thermostatistics can be formulated in terms of κ-deformed exponential functions together with the associated deduced logarithmic functions.

Generalised Exponential Families and Associated Entropy Functions

A generalised notion of exponential families is introduced. It is based on the variational principle, borrowed from statistical physics. It is shown that inequivalent generalised entropy functions lead to distinct generalised exponential families. The well-known result that the inequality of Cram´er and Rao becomes an equality in the case of an exponential family can be generalised. However, this requires the introduction of escort probabilities.

Generalized thermostatistics based on deformed exponential and logarithmic functions

The equipartition theorem states that inverse temperature equals the log-derivative of the density of states. This relation can be generalized by introducing a proportionality factor involving an increasing positive function φ(x). It is shown that this assumption leads to an equilibrium distribution of the Boltzmann–Gibbs form with the exponential function replaced by a deformed exponential function. In this way one obtains a formalism of generalized thermostatistics introduced previously by the author. It is shown that Tsallis’ thermostatistics, with a slight modification, is the most obvious example of this formalism and corresponds with the choice φ(x)=xq.

The Effect of Spin-Flip Symmetry on the Performance of the Simple GA

We use the one-dimensional nearest neighbor interaction functions (NNIs) to show how the presence of symmetry in a fitness function greatly influences the convergence behavior of the simple genetic algorithm (SGA). The effect of symmetry on the SGA supports the statement that it is not the amount of interaction present in a fitness function, measured e.g. by Davidors epistasis variance and the experimental design techniques introduced by Reeves and Wright, which is important, but the kind of interaction. The NNI functions exhibit a minimal amount of second order interaction, are trivial to optimize deterministically and yet show a wide range of SGA behavior. They have been extensively studied in statistical physics; results from this field explain the negative effect of symmetry on the convergence behavior of the SGA. This note intends to introduce them to the GA-community.

Linear Response Theory and the KMS Condition

AbstractThe response, relaxation and correlation functions are defined for any vector state ε of a von Neumann algebra

Generalized thermostatistics and mean-field theory

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Dual description of nonextensive ensembles

, acting on a Hilbert space ℋ, satisfying the KMS-condition. An operator representation of these functions is given on a particular Hilbert space.With this technique we prove the existence of the static admittance and the relaxation function. Finally we generalize the fluctuation-dissipation theorem and other relations between the above mentionned functions to infinite systems.

A generalized quantum microcanonical ensemble

The notion of a generalized exponential family is considered in the restricted context of non-extensive statistical physics. Examples are given of models belonging to this family. In particular, the q-Gaussians are discussed and it is shown that the configurational probability distributions of the micro-canonical ensemble belong to the q-exponential family.