scholarly journals A phenomenological approach to anomalous transport in complex or disordered media

2021 ◽  
Author(s):  
Ugur Saglam ◽  
Deniz Deger
Keyword(s):  
Master Equation ◽  
Initial Conditions ◽  
Stochastic Theory ◽  
Planck Equation ◽  
Phenomenological Approach ◽  
Transport Processes ◽  
Amorphous Semiconductors ◽  
Anomalous Transport ◽  
Disordered Media ◽  
Conductivity Behavior

We aim to derive a phenomenological approach to link the theories of anomalous transport governed by fractional calculus and stochastic theory with the conductivity behavior governed by the semi-empirical conductivity formalism involving Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami type conductivity equations. We want to determine the anomalous transport processes in the amorphous semiconductors and insulators by developing a theoretical approach over some mathematical instruments and methods. In this paper, we obtain an analytical expression for the average behavior of conductivity in complex or disordered media via using the fractional-stochastic differential equation, the Fourier-Laplace transform, some natural boundary-initial conditions, and familiar physical relations. We start with the stochastic equation of motion called the Langevin equation, develop its equivalent master equation called Klein-Kramers or Fokker-Planck equation, and consider the time-fractional generalization of the master equation. Once we derive the fractional master equation, then determine the expressions for the mean value of the variables or observables through some calculations and conditions. Finally, we use these expressions in the current density relation to obtain the average conductivity behavior.

scholarly journals Fokker-Planck Models for Rotating Stellar Systems

1996 ◽  
Vol 174 ◽  
pp. 363-364 ◽  
Author(s):  
Christian Einsel ◽  
Rainer Spurzem
Keyword(s):  
Initial Conditions ◽  
Planck Equation ◽  
Rotational Energy ◽  
Momentum Transport ◽  
Core Collapse ◽  
Initial Model ◽  
Fundamental Parameters ◽  
Angular Momentum Transport ◽  
Collapse Phase ◽  
Fokker Planck

Observations of Globular Cluster ellipticity distributions related to some fundamental parameters give strong evidence for a decay of rotational energy in these systems with time. In order to study the effectiveness of angular momentum transport (or loss, resp.) a code has been written which solves the Fokker-Planck equation in (E, Jz)-space and follows the evolution from some initial conditions through core collapse (and possibly gravothermal oscillations) up to the post-collapse phase. For the purpose of comparability with N-body simulations rotating initial model configurations according to the prescriptions of Lupton & Gunn (1987) have been constructed. These models are intended to continue previous work by Goodman (1983, Fokker-Planck) and Akiyama & Sugimoto (1989, N-Body). In this contribution the derivation of the flux coefficients is given.

scholarly journals Selected Topics in Homogenization of Transport Processes in Historical Masonry Structures

2012 ◽  
Vol 6 (1) ◽  
pp. 148-159
Author(s):  
Jan Sykora ◽  
Jan Zeman ◽  
Michal Ŝejnoha
Keyword(s):  
Initial Conditions ◽  
Heat Conduction Problem ◽  
Transport Processes ◽  
Random Pattern ◽  
Binary Sample ◽  
Homogenization Approach ◽  
Heat And Moisture Transport ◽  
Historical Masonry ◽  
Steady State Heat ◽  
Heat And Moisture

The paper reviews several topics associated with the homogenization of transport processed in historical ma-sonry structures. Since these often experience an irregular or random pattern, we open the subject by summarizing essen-tial steps in the formulation of a suitable computational model in the form of Statistically Equivalent Periodic Unit Cell (SEPUC). Accepting SEPUC as a reliable representative volume element is supported by application of the Fast Fourier Transform to both the SEPUC and large binary sample of real masonry in search for effective thermal conductivities lim-ited here to a steady state heat conduction problem. Fully coupled non-stationary heat and moisture transport is addressed next in the framework of two-scale first-order homogenization approach with emphases on the application of boundary and initial conditions on the meso-scale.

scholarly journals Information Geometry of Spatially Periodic Stochastic Systems

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 681
Author(s):  
Rainer Hollerbach ◽  
Eun-jin Kim
Keyword(s):  
Equilibrium Point ◽  
Stochastic Systems ◽  
Initial Conditions ◽  
Planck Equation ◽  
Information Geometry ◽  
Initial Position ◽  
Equilibrium Solutions ◽  
Gaussian Peak ◽  
Unstable Fixed Point ◽  
Spatially Periodic

We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , with f - chosen to be particularly flat (locally cubic) at the equilibrium point x = 0 , and f + particularly flat at the unstable fixed point x = 1 . We numerically solve the Fokker–Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x = μ , with μ in the range [ 0 , 1 ] . The strength D of the stochastic noise is in the range 10 - 4 – 10 - 6 . We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point x = 0 , the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x = 1 , there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length L ∞ , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that L ∞ as a function of initial position μ is qualitatively similar to the force, including the differences between f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , illustrating the value of information length as a useful diagnostic of the underlying force in the system.

scholarly journals Stochastic scattering model of anomalous diffusion in arrays of steady vortices

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200183
Author(s):  
Salvatore Buonocore ◽  
Mihir Sen ◽  
Fabio Semperlotti
Keyword(s):  
Transport Mechanism ◽  
Transport Processes ◽  
Anomalous Transport ◽  
Continuum Models ◽  
Particle Dynamic ◽  
Tracer Particles ◽  
Non Local ◽  
Heavy Tailed ◽  
Steady Vortices ◽  
Local Nature

We investigate the occurrence of anomalous transport phenomena associated with tracer particles propagating through arrays of steady vortices. The mechanism responsible for the occurrence of anomalous transport is identified in the particle dynamic, which is characterized by long collision-less trajectories (Lévy flights) interrupted by chaotic interactions with vortices. The process is studied via stochastic molecular models that are able to capture the underlying non-local nature of the transport mechanism. These models, however, are not well suited for problems where computational efficiency is an enabling factor. We show that fractional-order continuum models provide an excellent alternative that is able to capture the non-local nature of anomalous transport processes in turbulent environments. The equivalence between stochastic molecular and fractional continuum models is demonstrated both theoretically and numerically. In particular, the onset and the temporal evolution of heavy-tailed diffused fields are shown to be accurately captured, from a macroscopic perspective, by a fractional diffusion equation. The resulting anomalous transport mechanism, for the selected ranges of density of the vortices, shows a superdiffusive nature.

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