scholarly journals Power-law distributions and fluctuation-dissipation relation in the stochastic dynamics of two-variable Langevin equations

2012 ◽  
Vol 2012 (02) ◽  
pp. P02006 ◽  
Author(s):  
Jiu-Lin Du
Keyword(s):  
Power Law ◽  
Stochastic Dynamics ◽  
Langevin Equations ◽  
Fluctuation Dissipation ◽  
Power Law Distributions

scholarly journals Effective one-component model of binary mixture: molecular arrest induced by the spatially correlated stochastic dynamics

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
M. Majka ◽  
P. F. Góra
Keyword(s):  
Binary Mixtures ◽  
Soft Matter ◽  
Stochastic Dynamics ◽  
Component Model ◽  
Langevin Equations ◽  
Correlated Noise ◽  
Spatially Correlated ◽  
Fluctuation Dissipation ◽  
The Difference ◽  
Spatially Correlated Noise

AbstractSpatially correlated noise (SCN), i.e. the thermal noise that affects neighbouring particles in a similar manner, is ubiquitous in soft matter systems. In this work, we apply the over-damped SCN-driven Langevin equations as an effective, one-component model of the dynamics in dense binary mixtures. We derive the thermodynamically consistent fluctuation-dissipation relation for SCN to show that it predicts the molecular arrest resembling the glass transition, i.e. the critical slow-down of dynamics in the disordered phases. We show that the mechanism of singular dissipation is embedded in the dissipation matrix, accompanying SCN. We are also able to identify the characteristic length of collective dissipation, which diverges at critical packing. This novel physical quantity conveniently describes the difference between the ergodic and non-ergodic dynamics. The model is fully analytically solvable, one-dimensional and admits arbitrary interactions between the particles. It qualitatively reproduces several different modes of arrested disorder encountered in binary mixtures, including e.g. the re-entrant arrest. The model can be effectively compared to the mode coupling theory.

POWER LAWS ARE DISGUISED BOLTZMANN LAWS

2001 ◽  
Vol 12 (03) ◽  
pp. 333-343 ◽  
Author(s):  
PETER RICHMOND ◽  
SORIN SOLOMON
Keyword(s):  
Distribution Function ◽  
Power Law ◽  
Stochastic Dynamics ◽  
Mean Field ◽  
Power Laws ◽  
Direct Consequence ◽  
Boltzmann Distribution ◽  
Langevin Equations ◽  
Global Dynamics ◽  
Distribution Of Wealth

Using a previously introduced model on generalized Lotka–Volterra dynamics together with some recent results for the solution of generalized Langevin equations, we derive analytically the equilibrium mean field solution for the probability distribution of wealth and show that it has two characteristic regimes. For large values of wealth, it takes the form of a Pareto style power law. For small values of wealth, w ≤ wm, the distribution function tends sharply to zero. The origin of this law lies in the random multiplicative process built into the model. Whilst such results have been known since the time of Gibrat, the present framework allows for a stable power law in an arbitrary and irregular global dynamics, so long as the market is "fair", i.e., there is no net advantage to any particular group or individual. We further show that the dynamics of relative wealth is independent of the specific nature of the agent interactions and exhibits a universal character even though the total wealth may follow an arbitrary and complicated dynamics. In developing the theory, we draw parallels with conventional thermodynamics and derive for the system some new relations for the "thermodynamics" associated with the Generalized Lotka–Volterra type of stochastic dynamics. The power law that arises in the distribution function is identified with new additional logarithmic terms in the familiar Boltzmann distribution function for the system. These are a direct consequence of the multiplicative stochastic dynamics and are absent for the usual additive stochastic processes.

scholarly journals Predicting observational signatures of coronal heating by Alfvén waves and nanoflares

2007 ◽  
Vol 3 (S247) ◽  
pp. 279-287
Author(s):  
Patrick Antolin ◽  
Kazunari Shibata ◽  
Takahiro Kudoh ◽  
Daiko Shiota ◽  
David Brooks
Keyword(s):  
Power Law ◽  
Alfven Waves ◽  
Alfvén Waves ◽  
Power Law Distribution ◽  
Longitudinal Modes ◽  
Heating Mechanism ◽  
Set Up ◽  
Power Law Index ◽  
Power Law Distributions ◽  
Release Processes

AbstractAlfvén waves can dissipate their energy by means of nonlinear mechanisms, and constitute good candidates to heat and maintain the solar corona to the observed few million degrees. Another appealing candidate is the nanoflare-reconnection heating, in which energy is released through many small magnetic reconnection events. Distinguishing the observational features of each mechanism is an extremely difficult task. On the other hand, observations have shown that energy release processes in the corona follow a power law distribution in frequency whose index may tell us whether small heating events contribute substantially to the heating or not. In this work we show a link between the power law index and the operating heating mechanism in a loop. We set up two coronal loop models: in the first model Alfvén waves created by footpoint shuffling nonlinearly convert to longitudinal modes which dissipate their energy through shocks; in the second model numerous heating events with nanoflare-like energies are input randomly along the loop, either distributed uniformly or concentrated at the footpoints. Both models are based on a 1.5-D MHD code. The obtained coronae differ in many aspects, for instance, in the simulated intensity profile that Hinode/XRT would observe. The intensity histograms display power law distributions whose indexes differ considerably. This number is found to be related to the distribution of the shocks along the loop. We thus test the observational signatures of the power law index as a diagnostic tool for the above heating mechanisms and the influence of the location of nanoflares.

From the central limit theorem to heavy-tailed distributions

2003 ◽  
Vol 40 (3) ◽  
pp. 803-806 ◽  
Author(s):  
Jinwen Chen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Power Law ◽  
Power Law Distribution ◽  
The Central Limit Theorem ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Random Indices ◽  
Power Law Distributions

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.

scholarly journals Extra investigation of the self-organized critical Manna model at higher critical dimension

2021 ◽  
pp. 1-12
Author(s):  
Andrey Viktorovich Podlazov
Keyword(s):  
Power Law ◽  
Critical Dimension ◽  
The Self ◽  
Two Dimensional ◽  
Upper Critical Dimension ◽  
Self Organized ◽  
Logarithmic Corrections ◽  
Universal Indicator ◽  
Sandpile Models ◽  
Power Law Distributions

I investigate the nature of the upper critical dimension for isotropic conservative sandpile models and calculate the emerging logarithmic corrections to power-law distributions. I check the results experimentally using the case of Manna model with the theoretical solution known for all statement starting from the two-dimensional one. In addition, based on this solution, I construct a non-trivial super-universal indicator for this model. It characterizes the distribution of avalanches by time the border of their region needs to pass its width.

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