Analytical results for long-time behavior in anomalous diffusion

Ferreira, Rogelma M. S.; Santos, M. V. S.; Donato, C. C.; Andrade, José S.; Oliveira, Fernando A.

doi:10.1103/physreve.86.021121

Cited by 32 publications

(16 citation statements)

References 27 publications

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“…Note that for α = 2 − , where R(t → ∞) → 1/ ln(t) → 0, the MC is not violated. In this way the MC is satisfied for all diffusive processes in the range 0 < α < 2 − [96]. It is interesting to observe that Equation (49) for long time behavior is equivalent to the condition…”

Section: The Kinchin Theorem and Ergodicitymentioning

confidence: 88%

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

et al. 2019

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In this article we review classical and recent results in anomalous diffusion and provide mechanisms useful for the study of the fundamentals of certain processes, mainly in condensed matter physics, chemistry and biology. Emphasis will be given to some methods applied in the analysis and characterization of diffusive regimes through the memory function, the mixing condition (or irreversibility), and ergodicity. Those methods can be used in the study of small-scale systems, ranging in size from single-molecule to particle clusters and including among others polymers, proteins, ion channels and biological cells, whose diffusive properties have received much attention lately.

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Section: The Kinchin Theorem and Ergodicitymentioning

confidence: 88%

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

et al. 2019

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“…We can write x 2 ∼ t α , such that the system is regarded as subdiffusive for 0 < α < 1, normal for α = 1, and superdiffusive for 1 < α ≤ 2; the ballistic diffusion occurs for α = 2. In these cases, one verifies that there is a general relationship between the memory function and the diffusion exponent in Laplace space [20,21]…”

Section: Brownian Dynamicsmentioning

confidence: 85%

“…In non-ergodic systems, we will consider a local equilibrium where lim t→∞ R(t) = κ, such that it must satisfy the second law of thermodynamics [6,7] with 0 ≤ κ 2 < 1. Thus, we have (21) or equivalently…”

Section: Non-equilibrium Fluctuation-dissipation Theoremmentioning

confidence: 99%

Non-equilibrium Fluctuation-Dissipation Theorem for Stationary Anomalous Diffusion

Lapas¹,

Morgado²,

Penna³

et al. 2015

Acta Phys. Pol. B

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We derive a Fluctuation-Dissipation Theorem (FDT) from the generalized Langevin equation for both the equilibrium and far-from-equilibrium states. The equilibrium FDT is obtained under the assumption of energy balance and stationarity condition. We derive a non-equilibrium relation of the FDT, which can be applied in slow relaxation processes and nonergodic systems whenever the second law of thermodynamics is brought to bear. We obtain also a relationship between stationary and non-ergodic behaviour based on the non-equilibrium FDT. Emphasis is placed on ballistic diffusion, which goes to local equilibrium.

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“…Substituting (4) into (2) leads to

where

and

are non-local correlations for growth and competition that weigh the interaction between individuals. Here the nonlocal kernels play a role similar to that of the memory kernel in the Generalized Langevin Equation [51] , [52] , [53] , [54] , [55] , [56] , [57] , [58] . In the case of Langevin dynamics, the memory kernel reflects the fact that events occurred in the past can affect the present.…”

Section: A General Equation For Population Dynamicsmentioning

confidence: 99%

Nonlinear self-organized population dynamics induced by external selective nonlocal processes

Aranda

Penna

Oliveira

2021

Communications in Nonlinear Science and Numerical Simulation

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Highlights It is proposed a model based on non-local operators capable of reproduce several of the equations traditionally used in research on population dynamics, as well as having additional features, useful for describing the phenomenon of pattern formation. The combination of two of the parameters present in the model, namely, α and β , determines the existence of a phase space, characterized by regions with and without patterns. The patterns formed will have M peaks and M valleys, with M ≥ 2. There are ( α, β ) combinations which lead to the formation of degenerate states, where the system seems not to forget its initial conditions. The model allows the study of real systems, such is the case of bacterial populations subjected to inhomogeneous growth conditions. The parameter β represents a macroscopic measure of changes that occur at an individual level in the concentrations of proteins in bacteria, as a product of significant variations in the environment.

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Analytical results for long-time behavior in anomalous diffusion

Cited by 32 publications

References 27 publications

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

Non-equilibrium Fluctuation-Dissipation Theorem for Stationary Anomalous Diffusion

Nonlinear self-organized population dynamics induced by external selective nonlocal processes

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