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.
The cross-entropy pendant is the expectation of the negative q-logarithm with respect to a second distribution, . So .
Using , this may be written . For smaller , values all tend towards .
The limit computes the negative of the slope of at and one recovers . So for fixed small , raising this expectation relates to
log-likelihood maximalization.
Properties
Identities
A logarithm can be expressed in terms of a slope through resulting in the following formula for the standard entropy:
Given two independent systems A and B, for which the joint
probability density satisfies
the Tsallis entropy of this system satisfies
From this result, it is evident that the parameter is a measure of the departure from additivity. In the limit when q = 1,
which is what is expected for an additive system. This property is sometimes referred to as "pseudo-additivity".
Exponential families
Many common distributions like the normal distribution belongs to the statistical
exponential families.
Tsallis entropy for an exponential family can be written [3] as
where F is log-normalizer and k the term indicating the carrier measure.
For multivariate normal, term k is zero, and therefore the Tsallis entropy is in closed-form.
In scientific literature, the physical relevance of the Tsallis entropy has been debated.[4][5][6] However, from the years 2000 on, an increasingly wide spectrum of natural, artificial and social
complex systems have been identified which confirm the predictions and consequences that are derived from this nonadditive entropy, such as nonextensive statistical mechanics,[7] which generalizes the Boltzmann–Gibbs theory.
Among the various experimental verifications and applications presently available in the literature, the following ones deserve a special mention:
The distribution characterizing the motion of cold atoms in dissipative
optical lattices predicted in 2003[8] and observed in 2006.[9]
The fluctuations of the magnetic field in the
solar wind enabled the calculation of the q-triplet (or Tsallis triplet).[10]
The velocity distributions in a driven dissipative
dusty plasma.[11]
High energy collisional experiments at LHC/CERN (CMS, ATLAS and ALICE detectors)[14][15] and RHIC/Brookhaven (STAR and PHENIX detectors).[16]
Among the various available theoretical results which clarify the physical conditions under which Tsallis entropy and associated statistics apply, the following ones can be selected:
Several interesting physical systems[28] abide by entropic
functionals that are more general than the standard Tsallis entropy. Therefore, several physically meaningful generalizations have been introduced. The two most general of these are notably:
Superstatistics, introduced by C. Beck and E. G. D. Cohen in 2003[29] and
Spectral Statistics, introduced by G. A. Tsekouras and
Constantino Tsallis in 2005.[30] Both these entropic forms have Tsallis and Boltzmann–Gibbs statistics as special cases; Spectral Statistics has been proven to at least contain Superstatistics and it has been conjectured to also cover some additional cases.[citation needed]
^Tsallis, Constantino (2009). Introduction to nonextensive statistical mechanics : approaching a complex world (Online-Ausg. ed.). New York: Springer.
ISBN978-0-387-85358-1.
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.
The cross-entropy pendant is the expectation of the negative q-logarithm with respect to a second distribution, . So .
Using , this may be written . For smaller , values all tend towards .
The limit computes the negative of the slope of at and one recovers . So for fixed small , raising this expectation relates to
log-likelihood maximalization.
Properties
Identities
A logarithm can be expressed in terms of a slope through resulting in the following formula for the standard entropy:
Given two independent systems A and B, for which the joint
probability density satisfies
the Tsallis entropy of this system satisfies
From this result, it is evident that the parameter is a measure of the departure from additivity. In the limit when q = 1,
which is what is expected for an additive system. This property is sometimes referred to as "pseudo-additivity".
Exponential families
Many common distributions like the normal distribution belongs to the statistical
exponential families.
Tsallis entropy for an exponential family can be written [3] as
where F is log-normalizer and k the term indicating the carrier measure.
For multivariate normal, term k is zero, and therefore the Tsallis entropy is in closed-form.
In scientific literature, the physical relevance of the Tsallis entropy has been debated.[4][5][6] However, from the years 2000 on, an increasingly wide spectrum of natural, artificial and social
complex systems have been identified which confirm the predictions and consequences that are derived from this nonadditive entropy, such as nonextensive statistical mechanics,[7] which generalizes the Boltzmann–Gibbs theory.
Among the various experimental verifications and applications presently available in the literature, the following ones deserve a special mention:
The distribution characterizing the motion of cold atoms in dissipative
optical lattices predicted in 2003[8] and observed in 2006.[9]
The fluctuations of the magnetic field in the
solar wind enabled the calculation of the q-triplet (or Tsallis triplet).[10]
The velocity distributions in a driven dissipative
dusty plasma.[11]
High energy collisional experiments at LHC/CERN (CMS, ATLAS and ALICE detectors)[14][15] and RHIC/Brookhaven (STAR and PHENIX detectors).[16]
Among the various available theoretical results which clarify the physical conditions under which Tsallis entropy and associated statistics apply, the following ones can be selected:
Several interesting physical systems[28] abide by entropic
functionals that are more general than the standard Tsallis entropy. Therefore, several physically meaningful generalizations have been introduced. The two most general of these are notably:
Superstatistics, introduced by C. Beck and E. G. D. Cohen in 2003[29] and
Spectral Statistics, introduced by G. A. Tsekouras and
Constantino Tsallis in 2005.[30] Both these entropic forms have Tsallis and Boltzmann–Gibbs statistics as special cases; Spectral Statistics has been proven to at least contain Superstatistics and it has been conjectured to also cover some additional cases.[citation needed]
^Tsallis, Constantino (2009). Introduction to nonextensive statistical mechanics : approaching a complex world (Online-Ausg. ed.). New York: Springer.
ISBN978-0-387-85358-1.