965 research outputs found
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
phi(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 phi(x)=x^q.Comment: Invited talk at Next2003, uses Elsevier LaTeX macro
Generalized thermostatistics and mean-field theory
The present paper studies a large class of temperature dependent probability
distributions and shows that entropy and energy can be defined in such a way
that these probability distributions are the equilibrium states of a
generalized thermostatistics. This generalized thermostatistics is obtained
from the standard formalism by deformation of exponential and logarithmic
functions. Since this procedure is non-unique, specific choices are motivated
by showing that the resulting theory is well-behaved. In particular, the
equilibrium state of any system with a finite number of degrees of freedom is,
automatically, thermodynamically stable and satisfies the variational
principle. The equilibrium probability distribution of open systems deviates
generically from the Boltzmann-Gibbs distribution. If the interaction with the
environment is not too strong then one can expect that a slight deformation of
the exponential function, appearing in the Boltzmann-Gibbs distribution, can
reproduce the observed temperature dependence. An example of a system, where
this statement holds, is a single spin of the Ising chain. The connection
between the present formalism and Tsallis' thermostatistics is discussed. In
particular, the present generalization sheds some light onto the historical
development of the latter formalism.Comment: version accepted for publication in Physica
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 Cramer and
Rao becomes an equality in the case of an exponential family can be
generalised. However, this requires the introduction of escort probabilities.Comment: 20 page
Parameter estimation in nonextensive thermostatistics
Equilibrium statistical physics is considered from the point of view of
statistical estimation theory. This involves the notions of statistical model,
of estimators, and of exponential family. A useful property of the latter is
the existence of identities, obtained by taking derivatives of the logarithm of
the partition sum. It is shown that these identities still exist for models
belonging to generalised exponential families, in which case they involve
escort probability distributions. The percolation model serves as an example. A
previously known identity is derived. It relates the average number of sites
belonging to the finite cluster at the origin, the average number of perimeter
sites, and the derivative of the order parameter.Comment: 7 pages in revtex4, part of the talk given at the conference NEXT200
Quantum statistical manifolds: the linear growth case
A class of vector states on a von Neumann algebra is constructed. These
states belong to a deformed exponential family. One specific deformation is
considered. It makes the exponential function asymptotically linear.
Difficulties arising due to non-commutativity are highlighted.Comment: 24 pages, 12pt, A4; improved version, now making use of the commutant
algebr
Escort density operators and generalized quantum information measures
Parametrized families of density operators are studied. A generalization of
the lower bound of Cramer and Rao is formulated. It involves escort density
operators. The notion of phi-exponential family is introduced. This family,
together with its escort, optimizes the generalized lower bound. It also
satisfies a maximum entropy principle and exhibits a thermodynamic structure in
which entropy and free energy are related by Legendre transform.Comment: 10 page
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