Superstatistics: Theoretical Concepts and Physical Applications

Beck, Christian

doi:10.1002/9783527622979.ch15

Cited by 19 publications

(14 citation statements)

References 54 publications

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“…Figure 4 shows the age of the entity with maximum size against the total size X of the system for the independence case with σ = 0.8557 (panel b-1), σ = 1 (panel b-2) and σ = 1.2533 (panel b-3): there is a broad distribution of ages of the maximum size entities for large values of X and not a unique age. This observation suggests that the power-law behavior in this case results from the superposition of the statistics of single multiplicative processes at different ages with appropriate statistical weights, as in the superstatistics mechanism mentioned in the introduction [14][15][16]. It is notable that studies following similar ideas with exponential weights established the so-called double Pareto distribution, with power-law tail [41,42].…”

Section: Study Of the Entities Contributing To The Sum Of Sizesmentioning

confidence: 72%

“…Another iconic conceptual framework is self-organized criticality, captured by the metaphor of avalanches in a sandpile, in which power-laws are spontaneous emergent properties, with no apparent need to adjust control parameters [10,11]. Among many other mechanisms, let us mention the superposition of probability distributions [12,13], formalized in the concept of superstatistics [14][15][16]; the key idea is that the basic non-power-law probability distribution of a system has parameters that are random variables themselves and, when including this second source of stochasticity, a power-law distribution may arise. A particular case of superstatistics that found wide applicability is the Tsallis statistics, where the extremization of its generalized entropy produces power-laws [17].…”

Section: Introductionmentioning

confidence: 99%

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Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

Sousa

Takayasu

Sornette

et al. 2017

Entropy

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Abstract:We introduce a simple growth model in which the sizes of entities evolve as multiplicative random processes that start at different times. A novel aspect we examine is the dependence among entities. For this, we consider three classes of dependence between growth factors governing the evolution of sizes: independence, Kesten dependence and mixed dependence. We take the sum X of the sizes of the entities as the representative quantity of the system, which has the structure of a sum of product terms (Sigma-Pi), whose asymptotic distribution function has a power-law tail behavior. We present evidence that the dependence type does not alter the asymptotic power-law tail behavior, nor the value of the tail exponent. However, the structure of the large values of the sum X is found to vary with the dependence between the growth factors (and thus the entities). In particular, for the independence case, we find that the large values of X are contributed by a single maximum size entity: the asymptotic power-law tail is the result of such single contribution to the sum, with this maximum contributing entity changing stochastically with time and with realizations.

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Section: Study Of the Entities Contributing To The Sum Of Sizesmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

Sousa

Takayasu

Sornette

et al. 2017

Entropy

View full text Add to dashboard Cite

show abstract

“…Finally, one can also try to extend the applicability of the present model to experiments that are performed under non-equilibrium conditions. To this purpose one can assume, along the lines of superstatistics [25], that the model parameters are stochastic variables which have some probability distribution. The major problem is then to determine this time-dependent probability distribution.…”

Section: Discussionmentioning

confidence: 99%

The 3-dimensional random walk with applications to overstretched DNA and the protein titin

Straeten

Naudts

2008

Physica A: Statistical Mechanics and its Applications

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a b s t r a c tWe study the three-dimensional persistent random walk with drift. Then we develop a thermodynamic model that is based on this random walk without assuming the Boltzmann-Gibbs form for the equilibrium distribution. The simplicity of the model allows us to perform all calculations in closed form. We show that, despite its simplicity, the model can be used to describe different polymer stretching experiments. We study the reversible overstretching transition of DNA and the static force-extension relation of the protein titin.

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“…[24][25][26][27] Quantities with fat-tailed probability density functions (p.d.f.s) are characteristic of a class of complex dynamic systems that exhibit what is termed "superstatistics." [28][29][30] Acceleration-based SLM formulations that yield fat-tailed non-Gaussian probability density functions for acceleration, similar to the experimental results, have been developed by and Lamorgese et al 32 A review of recent SLM approaches for turbulence, including acceleration-based approaches, is given by Pope. 33 It was observed by Ayyalasomayajula et al 24 that while stochastic Lagrangian methods have been very successful for predicting particle dispersion, they nonetheless fail to correctly predict the strongly heterogeneous particle clustering phenomenon.…”

Section: Introductionmentioning

confidence: 99%

Stochastic vortex structure method for modeling particle clustering and collisions in homogeneous turbulence

Sala

Marshall

2013

Physics of Fluids

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Current particle dispersion models do not accurately predict the particle clustering that occurs in turbulent flow due to interaction of the particles with turbulent eddies. This clustering arises due to the effects of centrifugal forces which act to throw heavy particles out of the turbulent eddies, causing the particles to collect in high-concentration sheets lying between the eddies. The current paper proposes a stochastic vortex structure (SVS) model for simulation of particle clustering and collisions in turbulent flows. A new measure for particle drift relative to the fluid velocity is proposed that is related to the cross product of the fluid acceleration and velocity fields. Tests were conducted comparing the predictions of the SVS model with direct numerical simulation (DNS) and with three different stochastic Lagrangian methods in statistically stationary homogeneous isotropic turbulence with particles having Stokes numbers based on integral length scaling of order unity, assuming one-way fluid-particle coupling. The tests examined different turbulent flow features that are important for particle dispersion and clustering, as well as for prediction of the particle collision rate and collision distribution. The results indicate that the SVS model performs reasonably well for predicting particle concentration heterogeneity and collision rate, and that differences between the SVS and DNS results can be attributed to the fact that the SVS model neglects the small-scale velocity fluctuations within the turbulent flow.

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Superstatistics: Theoretical Concepts and Physical Applications

Abstract: A review of the superstatistics concept is provided, including various recent applications to complex systems.

Cited by 19 publications

References 54 publications

Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

The 3-dimensional random walk with applications to overstretched DNA and the protein titin

Stochastic vortex structure method for modeling particle clustering and collisions in homogeneous turbulence

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