Super statistic

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The Super statistics is a branch of statistical mechanics and statistical physics , which deals with non-linear systems and non-equilibrium systems concerned. It is characterized by the superimposition ( superposition of) a plurality of different statistical models to achieve the desired non-linearity. In terms of ordinary statistical ideas, this is equivalent to composing the distributions of random variables and can be viewed as a simple case of a doubly stochastic model. It was introduced in 2003 by EGD Cohen and Christian Beck with the motivation not to assume a priori Maxwell-Boltzmann distribution in physical systems .

Consider an extended thermodynamic system , which is locally in equilibrium and has a Boltzmann distribution . Then the probability of finding the system in a state with energy is proportional to . Here is the local inverse temperature. A thermodynamic non-equilibrium system is modeled taking into account macroscopic fluctuations in the local inverse temperature. These fluctuations occur on time scales that are much larger than the microscopic relaxation times for the Boltzmann distribution. If the fluctuations of are characterized by a distribution , the superstatistic Boltzmann factor of the system is given by ${\displaystyle E}$ ${\displaystyle \exp(-\beta E)}$ ${\displaystyle \beta }$ ${\displaystyle \beta }$ ${\displaystyle f(\beta )}$

${\displaystyle B(E)=\int _{0}^{\infty }d\beta f(\beta )\exp(-\beta E)\ .}$

For a system that can assume discrete energy states , the above equation defines the superstatistic partition function ${\displaystyle \{E_{i}\}_{i=1}^{W}}$

${\displaystyle Z=\sum _{i=1}^{W}B(E_{i})\ .}$

The likelihood that the system is in the state is through ${\displaystyle E_{i}}$

${\displaystyle p_{i}={\frac {1}{Z}}B(E_{i})}$ given.

The modeling of the fluctuations in leads to a statistical description of the Boltzmann statistics or "superstatistics". For example, if a gamma distribution is followed, the resulting superstatistic will be the Tsallis statistic. Superstats can also lead to other statistics like power law distributions or stretched exponentials. ${\displaystyle \beta }$ ${\displaystyle \beta }$

Individual evidence

1. ↑ C. Beck, EGD Cohen: Super Statistics . In: Physica A . 322, 2003, pp. 267-275. arxiv : cond-mat / 0205097 . bibcode : 2003PhyA..322..267B . doi : 10.1016 / S0378-4371 (03) 00019-0 .
2. ↑ EGD Cohen: Super Statistics . In: Physica D . 139, 2004, pp. 35-52. bibcode : 2004PhyD..193 ... 35C . doi : 10.1016 / j.physd.2004.01.007 .
3. ^ R. Hanel, S. Thurner, M. Gell-Mann : Generalized entropies and the transformation group of superstatistics . In: Proceedings of the National Academy of Sciences . 108, No. 16, 2011, pp. 6390-6394. arxiv : 1103.0580 . bibcode : 2011PNAS..108.6390H . doi : 10.1073 / pnas.1103539108 .
4. ↑ http://tsallis.cat.cbpf.br/biblio.htm
5. ^ Christian Beck: Stretched exponentials . In: Physica A . 365, pp. 96-101. arxiv : cond-mat / 0510841 . doi : 10.1016 / j.physa.2006.01.030 .
6. ↑ K Ourabah, LA Gougam, M Tribeche: Nonthermal and suprathermal distributions as a consequence of superstatistics . In: Physical Review E . 91, No. 1, 2015, p. 012133. bibcode : 2015PhRvE..91a2133O . doi : 10.1103 / PhysRevE.91.012133 . PMID 25679596 .