In statistics, a Tsallis distribution is a

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

derived from the maximization of the

Tsallis entropy In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. Overview The concept was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics and is identical in ...

under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the Tsallis distribution". The

q-Gaussian The ''q''-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The ''q''-Gaussian is a generalization of the Gaussian in the sam ...

is a generalization of the Gaussian in the same way that

is a generalization of standard Boltzmann–Gibbs entropy or

Shannon entropy Shannon may refer to: People * Shannon (given name) * Shannon (surname) * Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958) * Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum Will ...

. Similarly, if the domain of the variable is constrained to be positive in the maximum entropy procedure, the q-exponential distribution is derived. The Tsallis distributions have been applied to problems in the fields of statistical mechanics,

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economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...

, finance, and

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. The distributions are often used for their heavy tails. Note that Tsallis distributions are obtained as

Box–Cox transformation In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions. It is a data transformation technique used to stabilize variance, make the data more normal distribution-like ...

over usual distributions, with deformation parameter

\lambda=1-q

. This deformation transforms exponentials into q-exponentials.

Procedure

In a similar procedure to how the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...

can be derived using the standard Boltzmann–Gibbs entropy or Shannon entropy, the

can be derived from a maximization of the

subject to the appropriate constraints.

Common Tsallis distributions

q-Gaussian

See

q-exponential distribution

See q-exponential distribution

q-Weibull distribution

See q-Weibull distribution

Notes

External links

{{Tsallis Statistical mechanics Types of probability distributions Probability distributions with non-finite variance