Generalized statistical mechanics for superstatistical systems

Beck, Christian

doi:10.1098/rsta.2010.0280

Cited by 65 publications

(53 citation statements)

References 60 publications

(155 reference statements)

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“…This approach has been applied to many areas of complex systems research, including turbulence, high energy scattering processes, heterogenous nonequilibrium systems, and econophysics (see e.g. [11] for a short review). In finance early applications of the superstatistics concept were worked out by Duarte Queiros et al [6,7] and Ausloos et al [5].…”

Section: Introductionmentioning

confidence: 99%

Transition from lognormal toχ2-superstatistics for financial time series

Xu¹,

Beck²

2016

Physica A: Statistical Mechanics and its Applications

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Share price returns on different time scales can be well modeled by a superstatistical dynamics. Here we provide an investigation which type of superstatistics is most suitable to properly describe share price dynamics on various time scales. It is shown that while χ 2 -superstatistics works well on a time scale of days, on a much smaller time scale of minutes the price changes are better described by lognormal superstatistics. The system dynamics thus exhibits a transition from lognormal to χ 2 superstatistics as a function of time scale. We discuss a more general model interpolating between both statistics which fits the observed data very well. We also present results on correlation functions of the extracted superstatistical volatility parameter, which exhibits exponential decay for returns on large time scales, whereas for returns on small time scales there are long-range correlations and power-law decay.

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Section: Introductionmentioning

confidence: 99%

Transition from lognormal toχ2-superstatistics for financial time series

Xu¹,

Beck²

2016

Physica A: Statistical Mechanics and its Applications

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“…The Mittag-Leffler functions and their generalizations to vector and matrix variable cases, discussed in this paper will be useful for investigators in disciplines of physical sciences, particularly the superstatistics (Beck [1], [2], [3], [4]) and the nonextensive statistical mechanics (Tsallis [42], [43], [44], [45]). The importance of Mittag-Leffler function in physics is steadily increasing.…”

Section: Discussionmentioning

confidence: 99%

“…It is easy to note that the Mittag-Leffler functions of (1)- (4) are special cases of the Wright function (Wright [46]) which is defined as …”

Section: Mittag-leffler Functionmentioning

confidence: 99%

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Matrix-variate statistical distributions and fractional calculus

Mathai

Haubold

2011

fcaa

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Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayA connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional differential equations, Mittag-Leffler functions and Fox Hfunction appear naturally. Some results connected with generalized MittagLeffler density and their asymptotic behavior will be considered. Reference is made to applications in physics, particularly superstatistics and nonextensive statistical mechanics.MSC 2010 : 15A15, 15A52, 33C60, 33E12, 44A20, 62E15

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“…In these systems, local transport is normal and given by standard advection and diffusion, while the global observed behaviour is a sum over all the heterogeneous regions. It is worth mentioning that a similar approach, i.e., the random slow modulation of a parameter due to a non-homogeneous, disordered environment, is the basis of the well-known superstatistics [10,11], where the global statistics is derived by a linear average computed over the parameter distribution. If complexity should be identified with the emergence of selforganized structures, it could be questionable if heterogeneous systems should be considered complex.…”

Section: Complexity In Heterogeneous Systemsmentioning

confidence: 99%