en>ChrisGualtieri: /* Examples */Typo fixing, typos fixed: , → , using AWB

2012-07-29T03:49:57Z

Examples: Typo fixing, typos fixed: , → , using AWB

New page

{{Refimprove|date=December 2007}}

'''Superstatistics'''<ref name=beck>{{cite journal|last1=Beck|first1=C.|last2=Cohen|first2=E.G.D.|title=Superstatistics|journal=[[Physica A]]|volume=322|pages=267–275|year=2003|doi=10.1016/S0378-4371(03)00019-0}}</ref><ref name=cohen>{{cite journal|last1=Cohen|first1=E.G.D.|title=Superstatistics|journal=[[Physica D]]|volume=139|number=1|pages=35–52|year=2004|doi=10.1016/j.physd.2004.01.007}}</ref> is a branch of [[statistical mechanics]] or [[statistical physics]] devoted to the study of [[non-linear]] and non-[[Stability theory|equilibrium]] [[system]]s. It is characterized by using the [[Superposition principle|superposition]] of multiple differing [[statistical model]]s to achieve the desired non-linearity. In terms of ordinary statistical ideas, this is equivalent to compounding the distributions of random variables and it may be considered a simple case of a [[doubly stochastic model]].

Consider<ref name=hanel>{{cite journal|last1=Hanel|first1=R.|last2=Thurner|first2=S.|last3=Gell-Mann|first3=M.|authorlink3=Murray Gell-Mann|title=Generalized entropies and the transformation group of superstatistics|journal=[[Proceedings of the National Academy of Sciences]]| volume=108|issue=16|year=2011|pages=6390–6394| doi=10.1073/pnas.1103539108|arxiv = 1103.0580 |bibcode = 2011PNAS..108.6390H }}</ref> an extended thermodynamical system which is locally in equilibrium and has a [[Boltzmann distribution]], that is the probability of finding the system in a state with energy <math>E</math> is proportional to <math>\exp(-\beta E)</math>. Here <math>\beta</math> is the local inverse temperature. A non-equilibrium thermodynamical system is modeled by considering macroscopic fluctuations of the local inverse temperature. These fluctuations happen on time scales which are much larger than the microscopic relaxation times to the Boltzmann distribution. If the fluctuations of <math>\beta</math> are characterized by a distribution <math>f(\beta)</math>, the ''superstatistical Boltzmann factor'' of the system is given by

:<math>
B(E)=\int_0^\infty d\beta f(\beta)\exp(-\beta E).
</math>

This defines the superstatistical partition function

:<math>
Z = \sum_{i=1}^W B(E_i),
</math>

where the system can assume discrete energy states <math>\{E_i\}_{i=1}^W</math>. The probability of finding the system in state <math>E_i</math> is then given by

:<math>
p_i=\frac{1}{Z}B(E_i).
</math>

Modeling the fluctuations of <math>\beta</math> leads to a description in terms of statistics of Boltzmann statistics, or "superstatistics". One needs to note here that the word super here is short for superposition of the statistics.

==See also==
* [[Maxwell-Boltzmann statistics]]

==References==
{{reflist}}

[[Category:Statistical mechanics]]
[[Category:Nonlinear systems]]

{{physics-stub}}

Multiplicative partition - Revision history

en>ChrisGualtieri: /* Examples */Typo fixing, typos fixed: , → , using AWB