By Jan Naudts

The area of non-extensive thermostatistics has been topic to extensive study over the last 20 years and has matured considerably. Generalised Thermostatistics cuts throughout the traditionalism of many statistical physics texts through supplying a clean viewpoint and trying to get rid of parts of doubt and confusion surrounding the area.

The publication is split into elements - the 1st masking themes from traditional statistical physics, when adopting the viewpoint that statistical physics is facts utilized to physics. the second one constructing the formalism of non-extensive thermostatistics, of which the principal function is performed through the thought of a deformed exponential family members of likelihood distributions.

Presented in a transparent, constant, and deductive demeanour, the e-book makes a speciality of conception, a part of that's built through the writer himself, but additionally offers a few references in the direction of application-based texts.

Written by means of a number one contributor within the box, this e-book will supply a great tool for studying approximately contemporary advancements in generalized types of statistical mechanics and thermodynamics, in particular with recognize to self-study. Written for researchers in theoretical physics, arithmetic and statistical mechanics, in addition to graduates of physics, arithmetic or engineering. A prerequisite wisdom of easy notions of statistical physics and a considerable mathematical history are required.

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**Sample text**

17) This function θ(U ) solves the estimation problem what parameter values θk correspond with the thermodynamic values Uk . 15) is reached. Then these functions Uk (θ) are given by Uk = Hk θ =− ∂Φ . 18) In any case, given a matching pair (U, θ) one has the thermodynamic relation S(U ) − Φ(θ) − θk Uk = 0. 18) together form a pair of dual identities. The Massieu function Φ(θ), although of respectable age (1869), is not very well known. The notion used instead is that of free energy, which is a contact transform of the energy, not of the entropy.

45) with θk ≡ θk (r) and Z(r) = Tr e−θ k σk = 2 cosh |θ|. 46) In particular, this model belongs to the curved quantum exponential family. 46), chose a basis in which ρr = 12 I − r k σk is diagonal. This is equivalent with assuming r1 = r2 = 0. In that case it is clear 1 that ρr = Z exp(−θ3 σ3 ) with tanh θ3 = r3 and Z = 2 cosh θ3 . By going back to the original basis θ3 σ3 transforms into a matrix of the form θk σk . The trace of a matrix does not depend on the choice of basis. Hence, Z = 2 cosh θ3 = 2/ 1 − tanh2 θ3 = 2/ 1 − r32 .

4 Applying the Method of Lagrange 41 Fig. 4 Applying the Method of Lagrange The direct application of the maximum entropy principle is not very easy. What one needs is a method to ﬁnd the probability distribution that maximises the entropy functional within a set of distributions all having the right expectations for the thermodynamic variables. Of course, in the case of the BGS-entropy we know (part of) the answer because the probability distributions belonging to the exponential family maximise this entropy.