1
|
Zola RS, Lenzi EK, da Silva LR, Lenzi MK. Entropy Production in a Fractal System with Diffusive Dynamics. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1578. [PMID: 38136458 PMCID: PMC10742906 DOI: 10.3390/e25121578] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2023] [Revised: 11/09/2023] [Accepted: 11/22/2023] [Indexed: 12/24/2023]
Abstract
We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker-Planck equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered.
Collapse
Affiliation(s)
- Rafael S. Zola
- Departmento de Física, Universidade Tecnológica Federal do Paraná—Campus de Apucarana, Apucarana 86812-460, PR, Brazil
| | - Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
| | - Luciano R. da Silva
- Departamento de Física, Universidade Federal do Rio Grande do Norte, Natal 59078-900, RN, Brazil;
| | - Marcelo K. Lenzi
- Departamento de Engenharia Química, Universidade Federal do Paraná, Curitiba 81531-980, PR, Brazil;
| |
Collapse
|
2
|
Zhang Y, Xu N, Liu Z, Bai Y, Wu C, Guo Z. A Knudsen diffusion model for predicting VOC emissions from porous wood-based panels based on porosimetry tests. ENVIRONMENTAL SCIENCE AND POLLUTION RESEARCH INTERNATIONAL 2023; 30:34598-34611. [PMID: 36513898 DOI: 10.1007/s11356-022-24456-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2022] [Accepted: 11/25/2022] [Indexed: 06/17/2023]
Abstract
Volatile organic compounds (VOCs) emitted from porous wood-based panels with fractal structure severely pollute indoor environment. Different from previous studies which the diffusion type of VOC in building materials is attributed to Fick diffusion, VOC emission from porous wood-based panels belongs to Knudsen diffusion is firstly determined by comparing the pore diameter of internal channel with VOC molecular free path in this paper. Therefore, a time fractional mass transfer model related to the fractal dimension has been proposed to analyze Knudsen diffusion characteristics firstly. This model considers areal porosity has an impact on surface emission. Analytical solution of the present model is obtained for the first time. Furthermore, it is proved that the finite difference scheme is solvable, unconditionally stable, and convergent, and numerical simulation result and experimental data match well. Moreover, the influences of the fractal dimension df, areal porosity ε, and delay time parameter λ on VOC emission are demonstrated and analyzed; results suggest that the higher ε and df, and lower λ promote VOC emission, which can provide guidance for improving indoor air quality.
Collapse
Affiliation(s)
- Yan Zhang
- School of Science Beijing, University of Civil Engineering and Architecture, Beijing, 100044, China.
- Beijing Key Laboratory of Functional Materials for Building Structure and Environment Remediation, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China.
| | - Ning Xu
- School of Science Beijing, University of Civil Engineering and Architecture, Beijing, 100044, China
- Beijing Key Laboratory of Functional Materials for Building Structure and Environment Remediation, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China
| | - Ziyan Liu
- Overseas Chinese College, Capital University of Economics and Business, Beijing, 100070, China
| | - Yu Bai
- School of Science Beijing, University of Civil Engineering and Architecture, Beijing, 100044, China
- Beijing Key Laboratory of Functional Materials for Building Structure and Environment Remediation, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China
| | - Chuandong Wu
- Department of Chemistry, School of Chemistry and Biological Engineering, University of Science and Technology Beijing, Beijing, 100083, China
| | - Zhongbao Guo
- China Building Material Test & Certification Group Co., Ltd, Beijing, 100024, China
| |
Collapse
|
3
|
Universality of temperature behavior of dielectric dispersion characteristic for hopping conductivity in solids in the frame of model of thermally activated effective dipoles. APPLIED NANOSCIENCE 2023. [DOI: 10.1007/s13204-022-02755-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/14/2023]
|
4
|
Yoshida K, Casati G, Watanabe S, Shudo A. Sublinear diffusion in the generalized triangle map. Phys Rev E 2022; 106:014206. [PMID: 35974566 DOI: 10.1103/physreve.106.014206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2022] [Accepted: 06/17/2022] [Indexed: 06/15/2023]
Abstract
Diffusion of the orbits in a nonchaotic area-preserving map called a generalized triangle map (GTM) is numerically and analytically investigated. We provide accurate empirical evidence that the mean-squared displacement of the momentum for generic perturbation parameter settings increases sublinearly in time, and that the distribution of the momentum obeys a time-fractional diffusion equation. We show that the diffusion properties in the GTM do not follow any of the known stochastic processes generating sublinear diffusion since two seemingly incompatible features, non-Markovian yet stationary, coexist in the dynamics.
Collapse
Affiliation(s)
- Kensuke Yoshida
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Giulio Casati
- Center for Nonlinear and Complex Systems, Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy
- International Institute of Physics, Federal University of Rio Grande do Norte, 59708-400 Natal-RN, Brazil
| | - Shingo Watanabe
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| | - Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
| |
Collapse
|
5
|
Zunke C, Bewerunge J, Platten F, Egelhaaf SU, Godec A. First-passage statistics of colloids on fractals: Theory and experimental realization. SCIENCE ADVANCES 2022; 8:eabk0627. [PMID: 35061533 PMCID: PMC8782457 DOI: 10.1126/sciadv.abk0627] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2021] [Accepted: 11/29/2021] [Indexed: 05/30/2023]
Abstract
In nature and technology, particle dynamics frequently occur in complex environments, for example in restricted geometries or crowded media. These dynamics have often been modeled invoking a fractal structure of the medium although the fractal structure was only indirectly inferred through the dynamics. Moreover, systematic studies have not yet been performed. Here, colloidal particles moving in a laser speckle pattern are used as a model system. In this case, the experimental observations can be reliably traced to the fractal structure of the underlying medium with an adjustable fractal dimension. First-passage time statistics reveal that the particles explore the speckle in a self-similar, fractal manner at least over four decades in time and on length scales up to 20 times the particle radius. The requirements for fractal diffusion to be applicable are laid out, and methods to extract the fractal dimension are established.
Collapse
Affiliation(s)
- Christoph Zunke
- Condensed Matter Physics Laboratory, Heinrich Heine University, Universitätsstrasse 1, 40225 Düsseldorf, Germany
| | - Jörg Bewerunge
- Condensed Matter Physics Laboratory, Heinrich Heine University, Universitätsstrasse 1, 40225 Düsseldorf, Germany
| | - Florian Platten
- Condensed Matter Physics Laboratory, Heinrich Heine University, Universitätsstrasse 1, 40225 Düsseldorf, Germany
- Institute of Biological Information Processing, Biomacromolecular Systems and Processes (IBI-4), Forschungszentrum Jülich, 52425 Jülich, Germany
| | - Stefan U. Egelhaaf
- Condensed Matter Physics Laboratory, Heinrich Heine University, Universitätsstrasse 1, 40225 Düsseldorf, Germany
| | - Aljaž Godec
- Mathematical bioPhysics Group, Max Planck Institute for Biophysical Chemistry, 37077 Göttingen, Germany
| |
Collapse
|
6
|
Abstract
We investigate particle diffusion in a heterogeneous medium limited by a surface where sorption–desorption processes are governed by a kinetic equation. We consider that the dynamics of the particles present in the medium are governed by a diffusion equation with a spatial dependence on the diffusion coefficient, i.e., K(x) = D|x|−η, with −1 < η and D = const, respectively. This system is analyzed in a semi-infinity region, i.e., the system is defined in the interval [0,∞) for an arbitrary initial condition. The solutions are obtained and display anomalous spreading, that is, the dynamics may be viewed as anomalous diffusion, which in turn is related, and hence, the model can be directly applied to several complex systems ranging from biological fluids to electrolytic cells.
Collapse
|
7
|
Nikan O, Avazzadeh Z, Machado JT. A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer. J Adv Res 2021; 32:45-60. [PMID: 34484825 PMCID: PMC8408339 DOI: 10.1016/j.jare.2021.03.002] [Citation(s) in RCA: 20] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2020] [Revised: 12/11/2020] [Accepted: 03/04/2021] [Indexed: 11/27/2022] Open
Abstract
INTRODUCTION During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena. OBJECTIVES The purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann-Liouville sense. METHODS The temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant. RESULTS Three examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique. CONCLUSION This paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids.
Collapse
Affiliation(s)
- O. Nikan
- School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
| | - Z. Avazzadeh
- Department of Applied Mathematics, Xi'an Jiaotong-Liverpool University, Suzhou 215123, China
| | - J.A. Tenreiro Machado
- Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal
| |
Collapse
|
8
|
|
9
|
A Fractional Decline Model Accounting for Complete Sequence of Regimes for Production from Fractured Unconventional Reservoirs. Transp Porous Media 2021. [DOI: 10.1007/s11242-020-01516-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
|
10
|
Wang W, Cherstvy AG, Liu X, Metzler R. Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise. Phys Rev E 2020; 102:012146. [PMID: 32794926 DOI: 10.1103/physreve.102.012146] [Citation(s) in RCA: 30] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2020] [Accepted: 06/22/2020] [Indexed: 01/09/2023]
Abstract
Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.
Collapse
Affiliation(s)
- Wei Wang
- College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China.,Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Xianbin Liu
- College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| |
Collapse
|
11
|
Chugunov V, Fomin S. Effect of adsorption, radioactive decay and fractal structure of matrix on solute transport in fracture. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190283. [PMID: 32389092 DOI: 10.1098/rsta.2019.0283] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 02/27/2020] [Indexed: 06/11/2023]
Abstract
Reservoir contamination by various contaminants including radioactive elements is an actual environmental problem for all developed countries. Analysis of mass transport in a complex environment shows that the conventional diffusion equation based on Fick's Law fails to model the anomalous character of the diffusive mass transport observed in the field and laboratory experiments. These complex processes can be modelled by non-local advection-diffusion equations with temporal and spatial fractional derivatives. In the present paper, fractional differential equations are used for modelling the transport of radioactive materials in a fracture surrounded by the porous matrix of fractal structure. A new form of fractional differential equation for modelling migration of the radioactive contaminant in the fracture is derived and justified. Solutions of particular boundary value problems for this equation were found by application of the Laplace transform. Through the use of fractional derivatives, the model accounts for contaminant exchange between fracture and surrounding porous matrix of fractal geometry. For the case of an arbitrary time-dependent source of radioactive contamination located at the inlet of the fracture, the exact solutions for solute concentration in the fracture and surrounding porous medium are obtained. Using the concept of a short memory, an approximate solution of the problem of radioactive contaminant transport along the fracture surrounded by the fractal type porous medium is also obtained and compared with the exact solution. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
Collapse
|
12
|
The Key Factors That Determine the Economically Viable, Horizontal Hydrofractured Gas Wells in Mudrocks. ENERGIES 2020. [DOI: 10.3390/en13092348] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
We assemble a multiscale physical model of gas production in a mudrock (shale). We then tested our model on 45 horizontal gas wells in the Barnett with 12–15 years on production. When properly used, our model may enable shale companies to gain operational insights into how to complete a particular well in a particular shale. Macrofractures, microfractures, and nanopores form a multiscale system that controls gas flow in mudrocks. Near a horizontal well, hydraulic fracturing creates fractures at many scales and increases permeability of the source rock. We model the physical properties of the fracture network embedded in the Stimulated Reservoir Volume (SRV) with a fractal of dimension D < 2 . This fracture network interacts with the poorly connected nanopores in the organic matrix that are the source of almost all produced gas. In the practically impermeable mudrock, the known volumes of fracturing water and proppant must create an equal volume of fractures at all scales. Therefore, the surface area and the number of macrofractures created after hydrofracturing are constrained by the volume of injected water and proppant. The coupling between the fracture network and the organic matrix controls gas production from a horizontal well. The fracture permeability, k f , and the microscale source term, s, affect this coupling, thus controlling the reservoir pressure decline and mass transfer from the nanopore network to the fractures. Particular values of k f and s are determined by numerically fitting well production data with an optimization algorithm. The relationship between k f and s is somewhat hyperbolic and defines the type of fracture system created after hydrofracturing. The extremes of this relationship create two end-members of the fracture systems. A small value of the ratio k f / s causes faster production decline because of the high microscale source term, s. The effective fracture permeability is lower, but gas flow through the matrix to fractures is efficient, thus nullifying the negative effect of the smaller k f . For the high values of k f / s , production decline is slower. In summary, the fracture network permeability at the macroscale and the microscale source term control production rate of shale wells. The best quality wells have good, but not too good, macroscale connectivity.
Collapse
|
13
|
Kuśmierz Ł, Gudowska-Nowak E. Subdiffusive continuous-time random walks with stochastic resetting. Phys Rev E 2019; 99:052116. [PMID: 31212503 DOI: 10.1103/physreve.99.052116] [Citation(s) in RCA: 21] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2018] [Indexed: 06/09/2023]
Abstract
We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.
Collapse
Affiliation(s)
- Łukasz Kuśmierz
- Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Ewa Gudowska-Nowak
- Marian Smoluchowski Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland and Mark Kac Complex Systems Research Center, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
| |
Collapse
|
14
|
Zhokh A, Trypolskyi A, Strizhak P. Relationship between the anomalous diffusion and the fractal dimension of the environment. Chem Phys 2018. [DOI: 10.1016/j.chemphys.2018.02.015] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
|
15
|
Su Y, Ma XG, Lai PY, Tong P. Colloidal diffusion over a quenched two-dimensional random potential. SOFT MATTER 2017; 13:4773-4785. [PMID: 28653070 DOI: 10.1039/c7sm01056g] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
A two-layer colloidal system is developed for the study of diffusion over a quenched two-dimensional random potential. A mixture of bidisperse silica spheres is used to form a randomly packed colloidal monolayer on the bottom substrate. The corrugated surface of the bottom colloidal monolayer provides a gravitational potential field for the dilute diffusing particles in the top layer. The population probability histogram P(x,y) of the diffusing particles is obtained to fully characterize the random potential landscape U(x,y) via the Boltzmann distribution. The dynamical properties of the top diffusing particles, such as their mean square displacement (MSD), histogram of the escape time, and long-time self-diffusion coefficient, are simultaneously measured from the particle trajectories. A quantitative relationship between the long-time diffusion coefficient and the random potential is obtained, which is in good agreement with the theoretical prediction. The measured MSD reveals a wide region of subdiffusion resulting from the structural disorders. The crossover from subdiffusion to normal diffusion is explained by the Lorentz model for tracer diffusion through a heterogeneous space filled with a set of randomly distributed obstacles.
Collapse
Affiliation(s)
- Yun Su
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
| | - Xiao-Guang Ma
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
| | - Pik-Yin Lai
- Department of Physics and Center for Complex Systems, National Central University, Chungli District, Tao-Yuan City, Taiwan 320, Republic of China.
| | - Penger Tong
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
| |
Collapse
|
16
|
Zhokh A, Strizhak P. Non-Fickian diffusion of methanol in mesoporous media: Geometrical restrictions or adsorption-induced? J Chem Phys 2017; 146:124704. [PMID: 28388159 DOI: 10.1063/1.4978944] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The methanol mass transfer in the mesoporous silica and alumina/zeolite H-ZSM-5 grains has been studied. We demonstrate that the methanol diffusion is characterized as a time-fractional for both solids. Methanol transport occurs in the super-diffusive regime, which is faster comparing to the Fickian diffusion. We show that the fractional exponents defining the regime of transport are different for each porous grain. The difference between the values of the fractional exponents is associated with a difference in the energetic strength of the active sites of the surface of the media of different chemical nature as well as the geometrical restrictions of the porous media. Increasing by six-fold, the pore diameter leads to a 1.1 fold increase of the fractional exponent. Decreasing by three-fold, the methanol desorption energy results into the same increasing the fractional exponent. Our findings support that mainly the adsorption process, which is defined by the energetic disorder of the corresponding surface active sites, is likely to be the driving force of the abnormality of the mass transfer in the porous media. Therefore, the fractional exponent is a fundamental characteristic which is individual for each combination of the porous solid and diffusing species.
Collapse
Affiliation(s)
- Alexey Zhokh
- L.V. Pisarzhevsky Institute of Physical Chemistry, National Academy of Sciences of Ukraine, Prospect Nauki, 31, Kiev 03028, Ukraine
| | - Peter Strizhak
- L.V. Pisarzhevsky Institute of Physical Chemistry, National Academy of Sciences of Ukraine, Prospect Nauki, 31, Kiev 03028, Ukraine
| |
Collapse
|
17
|
Hernández D, Herrera-Hernández EC, Núñez-López M, Hernández-Coronado H. Self-similar Turing patterns: An anomalous diffusion consequence. Phys Rev E 2017; 95:022210. [PMID: 28297859 DOI: 10.1103/physreve.95.022210] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/14/2016] [Indexed: 11/07/2022]
Abstract
In this work, we show that under specific anomalous diffusion conditions, chemical systems can produce well-ordered self-similar concentration patterns through diffusion-driven instability. We also find spiral patterns and patterns with mixtures of rotational symmetries. The type of anomalous diffusion discussed in this work, either subdiffusion or superdiffusion, is a consequence of the medium heterogeneity, and it is modeled through a space-dependent diffusion coefficient with a power-law functional form.
Collapse
Affiliation(s)
- D Hernández
- Posgrado en Ciencias de la Complejidad, Universidad Autónoma de la Ciudad de México, Ciudad de México, Laboratorio Nacional de Ciencias de la Complejidad, Ciudad de México, Mexico
| | - E C Herrera-Hernández
- CONACYT-Centro de Ingeniería y Desarrollo Industrial, Av. Playa pie de la Cuesta 702, Desarrollo Sn. Pablo, 76125, Querétaro, Qro., Mexico
| | - M Núñez-López
- Departamento de Matemáticas Aplicadas y Sistemas, DMAS Universidad Autónoma Metropolitana, Cuajimalpa, Av. Vasco de Quiroga, 4871, Sta. Fe Cuajimalpa, Cuajimalpa de Morelos, 05300, Mexico
| | - H Hernández-Coronado
- Departamento de Física, Facultad de Ciencias, UNAM, A. P. 50-542, Mexico DF, 04510, Mexico
| |
Collapse
|
18
|
Noetinger B, Roubinet D, Russian A, Le Borgne T, Delay F, Dentz M, de Dreuzy JR, Gouze P. Random Walk Methods for Modeling Hydrodynamic Transport in Porous and Fractured Media from Pore to Reservoir Scale. Transp Porous Media 2016. [DOI: 10.1007/s11242-016-0693-z] [Citation(s) in RCA: 64] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
|
19
|
Balankin AS. Effective degrees of freedom of a random walk on a fractal. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062146. [PMID: 26764671 DOI: 10.1103/physreve.92.062146] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2015] [Indexed: 06/05/2023]
Abstract
We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.
Collapse
Affiliation(s)
- Alexander S Balankin
- Grupo "Mecánica Fractal," ESIME, Instituto Politécnico Nacional, México D.F., 07738, Mexico
| |
Collapse
|
20
|
Kozak JJ, Garza-López RA, Abad E. Lattice statistical theory of random walks on a fractal-like geometry. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032147. [PMID: 24730829 DOI: 10.1103/physreve.89.032147] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2013] [Indexed: 06/03/2023]
Abstract
We have designed a two-dimensional, fractal-like lattice and explored, both numerically and analytically, the differences between random walks on this lattice and a regular, square-planar Euclidean lattice. We study the efficiency of diffusion-controlled processes for flows from external sites to a centrosymmetric reaction center and, conversely, for flows from a centrosymmetric source to boundary sites. In both cases, we find that analytic expressions derived for the mean walk length on the fractal-like lattice have an algebraic dependence on system size, whereas for regular Euclidean lattices the dependence can be transcendental. These expressions are compared with those derived in the continuum limit using classical diffusion theory. Our analysis and the numerical results quantify the extent to which one paradigmatic class of spatial inhomogeneities can compromise the efficiency of adatom diffusion on solid supports and of surface-assisted self-assembly in metal-organic materials.
Collapse
Affiliation(s)
- John J Kozak
- DePaul University, 243 South Wabash, Chicago, Illinois 60604-2301, USA and Beckman Institute, Caltech, Pasadena, California 91125, USA
| | - Roberto A Garza-López
- Department of Chemistry and Seaver Chemistry Laboratory, Pomona College, Claremont, California 60604-2301, USA
| | - Enrique Abad
- Departamento de Física Aplicada, Centro Universitario de Mérida, Universidad de Extremadura, E-06800 Mérida, Spain
| |
Collapse
|
21
|
Herrera-Hernández EC, Coronado M, Hernández-Coronado H. Fractal continuum model for tracer transport in a porous medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:063004. [PMID: 24483554 DOI: 10.1103/physreve.88.063004] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2013] [Revised: 08/21/2013] [Indexed: 06/03/2023]
Abstract
A model based on the fractal continuum approach is proposed to describe tracer transport in fractal porous media. The original approach has been extended to treat tracer transport and to include systems with radial and uniform flow, which are cases of interest in geoscience. The models involve advection due to the fluid motion in the fractal continuum and dispersion whose mathematical expression is taken from percolation theory. The resulting advective-dispersive equations are numerically solved for continuous and for pulse tracer injection. The tracer profile and the tracer breakthrough curve are evaluated and analyzed in terms of the fractal parameters. It has been found in this work that anomalous transport frequently appears, and a condition on the fractal parameter values to predict when sub- or superdiffusion might be expected has been obtained. The fingerprints of fractality on the tracer breakthrough curve in the explored parameter window consist of an early tracer breakthrough and long tail curves for the spherical and uniform flow cases, and symmetric short tailed curves for the radial flow case.
Collapse
Affiliation(s)
- E C Herrera-Hernández
- Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, 07730, México D.F., Mexico
| | - M Coronado
- Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, 07730, México D.F., Mexico
| | - H Hernández-Coronado
- Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, 07730, México D.F., Mexico, and Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, C. U., 04510, México D.F., Mexico
| |
Collapse
|
22
|
Balankin AS, Elizarraraz BE. Reply to "Comment on 'Hydrodynamics of fractal continuum flow' and 'Map of fluid flow in fractal porous medium into fractal continuum flow'". PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:057002. [PMID: 24329395 DOI: 10.1103/physreve.88.057002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2013] [Indexed: 06/03/2023]
Abstract
The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. E 85, 025302(R) (2012); 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds' transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged.
Collapse
Affiliation(s)
- Alexander S Balankin
- Grupo "Mecánica Fractal," ESIME-Zacatenco, Instituto Politécnico Nacional, México Distrito Federal 07738, Mexico
| | | |
Collapse
|
23
|
Balakrishnan V, Kozak JJ. Analytic expression for the mean time to absorption for a random walker on the Sierpinski fractal. III. The effect of non-nearest-neighbor jumps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052139. [PMID: 24329246 DOI: 10.1103/physreve.88.052139] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2013] [Indexed: 06/03/2023]
Abstract
We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d≥2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearest-neighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.
Collapse
Affiliation(s)
- V Balakrishnan
- Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India
| | - John J Kozak
- DePaul University, 243 South Wabash Avenue, Chicago, Illinois 60604-2301, USA
| |
Collapse
|
24
|
Moreles MA, Peña J, Botello S, Iturriaga R. On Modeling Flow in Fractal Media form Fractional Continuum Mechanics and Fractal Geometry. Transp Porous Media 2013. [DOI: 10.1007/s11242-013-0179-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
|
25
|
Balankin AS, Mena B, Martínez-González CL, Matamoros DM. Random walk in chemical space of Cantor dust as a paradigm of superdiffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:052101. [PMID: 23214828 DOI: 10.1103/physreve.86.052101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2012] [Revised: 07/31/2012] [Indexed: 06/01/2023]
Abstract
We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.
Collapse
Affiliation(s)
- Alexander S Balankin
- Grupo Mecánica Fractal, Instituto Politécnico Nacional, México Distrito Federal 07738, Mexico
| | | | | | | |
Collapse
|
26
|
Hernandez-Coronado H, Coronado M, Herrera-Hernandez EC. Transport in fractal media: an effective scale-invariant approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:066316. [PMID: 23005215 DOI: 10.1103/physreve.85.066316] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2012] [Indexed: 06/01/2023]
Abstract
In this paper an advective-dispersion equation with scale-dependent coefficients is proposed for describing transport through fractals. This equation is obtained by imposing scale invariance and assuming that the porosity, the dispersion coefficient, and the velocity follow fractional power laws on the scale. The model incorporates the empirically found trends in highly heterogeneous media, regarding the dependence of the dispersivity on the scale and the dispersion coefficient on the velocity. We conclude that the presence of nontrivial fractal parameters produces anomalous dispersion, as expected, and that the presence of convective processes induces a reescalation in the concentration and shifts the tracer velocity to different values with respect to the nonfractal case.
Collapse
Affiliation(s)
- H Hernandez-Coronado
- Instituto Mexicano del Petróleo, Eje central Lázaro Cárdenas 152, 07730, México D.F., Mexico
| | | | | |
Collapse
|
27
|
Balankin AS, Elizarraraz BE. Map of fluid flow in fractal porous medium into fractal continuum flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:056314. [PMID: 23004869 DOI: 10.1103/physreve.85.056314] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2012] [Indexed: 06/01/2023]
Abstract
This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided.
Collapse
Affiliation(s)
- Alexander S Balankin
- Grupo "Mecánica Fractal," Instituto Politécnico Nacional, México, Distrito Federal, Mexico
| | | |
Collapse
|
28
|
Frisch U, Pomyalov A, Procaccia I, Ray SS. Turbulence in noninteger dimensions by fractal Fourier decimation. PHYSICAL REVIEW LETTERS 2012; 108:074501. [PMID: 22401207 DOI: 10.1103/physrevlett.108.074501] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/05/2011] [Indexed: 05/20/2023]
Abstract
Fractal decimation reduces the effective dimensionality D of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius k is proportional to k(D) for large k. At the critical dimension D(c)=4/3 there is an equilibrium Gibbs state with a k(-5/3) spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89, 064501 (2002). Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D=2 with a rapidly rising Kolmogorov constant, likely to diverge as (D-4/3)(-2/3).
Collapse
Affiliation(s)
- Uriel Frisch
- UNS, CNRS, OCA, Laboratoire Lagrange, Nice, France
| | | | | | | |
Collapse
|
29
|
Meyer B, Chevalier C, Voituriez R, Bénichou O. Universality classes of first-passage-time distribution in confined media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051116. [PMID: 21728499 DOI: 10.1103/physreve.83.051116] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2010] [Indexed: 05/31/2023]
Abstract
We study the first-passage time (FPT) distribution to a target site for a random walker evolving in a bounded domain. We show that in the limit of large volume of the confining domain, this distribution falls into universality classes indexed by the walk dimension d(w) and the fractal dimension d(f) of the medium, which have been recently identified previously [Bénichou et al., Nat. Chem. 2, 472 (2010)]. We present in this paper a complete derivation of these universal distributions, discuss extensively the range of applicability of the results, and extend the method to continuous-time random walks. This analysis puts forward the importance of the geometry, and in particular the position of the starting point, in first-passage statistics. Analytical results are validated by numerical simulations, applied to various models of transport in disordered media, which illustrate the universality classes of the FPT distribution.
Collapse
Affiliation(s)
- B Meyer
- Laboratoire de Physique Théorique de la matière Condensée (UMR 7600), Université Paris 6, Paris, France
| | | | | | | |
Collapse
|
30
|
Abstract
It has long been appreciated that the transport properties of molecules can control reaction kinetics. This effect can be characterized by the time it takes a diffusing molecule to reach a target-the first-passage time (FPT). Determining the FPT distribution in realistic confined geometries has until now, however, seemed intractable. Here, we calculate this FPT distribution analytically and show that transport processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes. Beyond the theoretical aspect, this result changes our views on standard reaction kinetics and we introduce the concept of 'geometry-controlled kinetics'. More precisely, we argue that geometry-and in particular the initial distance between reactants in 'compact' systems-can become a key parameter. These findings could help explain the crucial role that the spatial organization of genes has in transcription kinetics, and more generally the impact of geometry on diffusion-limited reactions.
Collapse
|
31
|
Haynes CP, Roberts AP. Continuum diffusion on networks: trees with hyperbranched trunks and fractal branches. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:031111. [PMID: 19391906 DOI: 10.1103/physreve.79.031111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2008] [Indexed: 05/27/2023]
Abstract
The probability that a random walker returns to its origin for large times scales as t;{-d[over ]2} , where d[over ] is the spectral dimension. We calculate d[over ] for a class of tree structures using a renormalization technique on an infinite continued fraction. We consider a wide range of homogeneous networks based on replacing the branches of a self-similar tree with arbitrary fractals and composite fractals. We also consider a new class of inhomogeneous hyperbranched trees.
Collapse
Affiliation(s)
- C P Haynes
- School of Mathematics and Physics, University of Queensland, Queensland 4072, Australia
| | | |
Collapse
|
32
|
Brault P, Josserand C, Bauchire JM, Caillard A, Charles C, Boswell RW. Anomalous diffusion mediated by atom deposition into a porous substrate. PHYSICAL REVIEW LETTERS 2009; 102:045901. [PMID: 19257443 DOI: 10.1103/physrevlett.102.045901] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2008] [Indexed: 05/27/2023]
Abstract
Constant flux atom deposition into a porous medium is shown to generate a dense overlayer and a diffusion profile. Scaling analysis shows that the overlayer acts as a dynamic control for atomic diffusion in the porous substrate. This is modeled by generalizing the porous diffusion equation with a time-dependent diffusion coefficient equivalent to a nonlinear rescaling of time.
Collapse
Affiliation(s)
- Pascal Brault
- Groupe de Recherches sur l'Energétique des Milieux Ionisés, UMR6606, Université d'Orléans, France
| | | | | | | | | | | |
Collapse
|
33
|
Leorato S, Orsingher E. Branching on a Sierpinski graph. Stat Probab Lett 2009. [DOI: 10.1016/j.spl.2008.07.032] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
|
34
|
Arkhincheev VE. Random walks on the Comb model and its generalizations. CHAOS (WOODBURY, N.Y.) 2007; 17:043102. [PMID: 18163766 DOI: 10.1063/1.2772179] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
Microscopic models with anomalous diffusion, which include the Comb model and its generalization for the finite width of the backbone, have been considered in this paper. The physical mechanisms of the subdiffusion random walks have been established. The first comes from the permanent return of the diffusing particle to the initial point of the diffusion due to "effective reducing" of the dimensionality of the considered system to the quasi-one-dimensional system. This physical mechanism has been obtained in the Comb model and in the model with a strip. The second mechanism of the subdiffusion is connected with random capture on the traps of diffusing particles and their ensuing random release from the traps. It has been shown that these different mechanisms of subdiffusion have been described by the different generalized diffusion equations of fractional order. The solutions of these different equations have been obtained, and the physical sense of the fractional order generalized equations has been discussed.
Collapse
Affiliation(s)
- V E Arkhincheev
- Buryat Science Center, Siberian Branch of Russian Academy of Sciences, 670047, str. Sakhyanovoi 6, Ulan-Ude, Russia
| |
Collapse
|
35
|
|
36
|
Sellers S, Barker JA. Generalized diffusion equation for anisotropic anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:061103. [PMID: 17280034 DOI: 10.1103/physreve.74.061103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2006] [Indexed: 05/13/2023]
Abstract
Motivated by studies of comblike structures, we present a generalization of the classical diffusion equation to model anisotropic, anomalous diffusion. We assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density, where each component of the diffusion tensor can have its own scaling law. We also assume scaling laws that have an explicit power-law dependence on space and time. Solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures.
Collapse
Affiliation(s)
- S Sellers
- Mechanical and Aerospace Engineering, Washington University, St Louis, Missouri 63119, USA
| | | |
Collapse
|
37
|
Kiessling V, Crane JM, Tamm LK. Transbilayer effects of raft-like lipid domains in asymmetric planar bilayers measured by single molecule tracking. Biophys J 2006; 91:3313-26. [PMID: 16905614 PMCID: PMC1614489 DOI: 10.1529/biophysj.106.091421] [Citation(s) in RCA: 180] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023] Open
Abstract
Cell membranes have complex lipid compositions, including an asymmetric distribution of phospholipids between the opposing leaflets of the bilayer. Although it has been demonstrated that the lipid composition of the outer leaflet of the plasma membrane is sufficient for the formation of raft-like liquid-ordered (l(o)) phase domains, the influence that such domains may have on the lipids and proteins of the inner leaflet remains unknown. We used tethered polymer supports and a combined Langmuir-Blodgett/vesicle fusion (LB/VF) technique to build asymmetric planar bilayers that mimic plasma membrane asymmetry in many ways. We show that directly supported LB monolayers containing cholesterol-rich l(o) phases are inherently unstable when exposed to water or vesicle suspensions. However, tethering the LB monolayer to the solid support with the lipid-anchored polymer 1,2-dimyristoyl phophatidylethanolamine-N-[poly(ethylene glycol)-triethoxysilane] significantly improves stability and allows for the formation of complex planar-supported bilayers that retain >90% asymmetry for 1-2 h. We developed a single molecule tracking (SPT) system for the study of lipid diffusion in asymmetric bilayers with coexisting liquid phases. SPT allowed us to study in detail the diffusion of individual lipids inside, outside, or directly opposed to l(o) phase domains. We show here that l(o) phase domains in one monolayer of an asymmetric bilayer do not induce the formation of domains in the opposite leaflet when this leaflet is composed of palmitoyl-oleoyl phosphatidylcholine and cholesterol but do induce domains when this leaflet is composed of porcine brain phosphatidylcholine, phosphatidylethanolamine, phosphatidylserine, and cholesterol. The diffusion of lipids is similar in l(o) and liquid-disordered phase domains and is not affected by transbilayer coupling, indicating that lateral and transverse lipid interactions that give rise to the domain structure are weak in the biological lipid mixtures that were employed in this work.
Collapse
Affiliation(s)
- Volker Kiessling
- Department of Molecular Physiology and Biological Physics, University of Virginia, Charlottesville, 22908-0736, USA
| | | | | |
Collapse
|
38
|
Ren FY, Liang JR, Qiu WY, Xiao JB. Answer to an open problem proposed by R Metzler and J Klafter. ACTA ACUST UNITED AC 2006. [DOI: 10.1088/0305-4470/39/18/009] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
|
39
|
Fractional Diffusion Equation Approach to the Anomalous Diffusion on Fractal Lattices. B KOREAN CHEM SOC 2005. [DOI: 10.5012/bkcs.2005.26.11.1723] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
|
40
|
Pedron IT, Mendes RS, Buratta TJ, Malacarne LC, Lenzi EK. Logarithmic diffusion and porous media equations: a unified description. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:031106. [PMID: 16241410 DOI: 10.1103/physreve.72.031106] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2005] [Indexed: 05/04/2023]
Abstract
In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a Lorentzian form, consequently this equation characterizes a superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.
Collapse
Affiliation(s)
- I T Pedron
- Universidade Estadual do Oeste do Paraná, Rua Pernambuco, 1777, 85960-000, Marechal Cândido Rondon, Paraná, Brazil
| | | | | | | | | |
Collapse
|
41
|
Lenzi EK, Mendes RS, Andrade JS, da Silva LR, Lucena LS. N-dimensional fractional diffusion equation and Green function approach: spatially dependent diffusion coefficient and external force. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:052101. [PMID: 16089577 DOI: 10.1103/physreve.71.052101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2004] [Revised: 12/08/2004] [Indexed: 05/03/2023]
Abstract
We investigate an N-dimensional fractional diffusion equation with radial symmetry by using the Green function approach. We consider, in our analysis, the spatial dependence on the diffusion coefficient and the presence of an external force. In particular, we employ boundary conditions in a finite interval and after we extend it to a semi-infinite interval. We also show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.
Collapse
Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá PR, Brazil
| | | | | | | | | |
Collapse
|
42
|
Gadomski A, Rubí J, Łuczka J, Ausloos M. On temperature- and space-dimension dependent matter agglomerations in a mature growing stage. Chem Phys 2005. [DOI: 10.1016/j.chemphys.2004.10.024] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
|
43
|
Deverall MA, Gindl E, Sinner EK, Besir H, Ruehe J, Saxton MJ, Naumann CA. Membrane lateral mobility obstructed by polymer-tethered lipids studied at the single molecule level. Biophys J 2004; 88:1875-86. [PMID: 15613633 PMCID: PMC1305241 DOI: 10.1529/biophysj.104.050559] [Citation(s) in RCA: 142] [Impact Index Per Article: 7.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Obstructed long-range lateral diffusion of phospholipids (TRITC-DHPE) and membrane proteins (bacteriorhodopsin) in a planar polymer-tethered 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine bilayer is studied using wide-field single molecule fluorescence microscopy. The obstacles are well-controlled concentrations of hydrophobic lipid-mimicking dioctadecylamine moieties in the polymer-exposed monolayer of the model membrane. Diffusion of both types of tracer molecules is well described by a percolating system with different percolation thresholds for lipids and proteins. Data analysis using a free area model of obstructed lipid diffusion indicates that phospholipids and tethered lipids interact via hard-core repulsion. A comparison to Monte Carlo lattice calculations reveals that tethered lipids act as immobile obstacles, are randomly distributed, and do not self-assemble into large-scale aggregates for low to moderate tethering concentrations. A procedure is presented to identify anomalous subdiffusion from tracking data at a single time lag. From the analysis of the cumulative distribution function of the square displacements, it was found that TRITC-DHPE and W80i show normal diffusion at lower concentrations of tethered lipids and anomalous diffusion at higher ones. This study may help improve our understanding of how lipids and proteins in biomembranes may be obstructed by very small obstacles comprising only one or very few molecules.
Collapse
Affiliation(s)
- M A Deverall
- Department of Chemistry, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA
| | | | | | | | | | | | | |
Collapse
|
44
|
|
45
|
de Dreuzy JR, Davy P, Erhel J, de Brémond d'Ars J. Anomalous diffusion exponents in continuous two-dimensional multifractal media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:016306. [PMID: 15324168 DOI: 10.1103/physreve.70.016306] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2004] [Indexed: 05/24/2023]
Abstract
We study diffusion in heterogeneous multifractal continuous media that are characterized by the second-order dimension of the multifractal spectrum D2, while the fractal dimension of order 0, D0, is equal to the embedding Euclidean dimension 2. We find that the mean anomalous and fracton dimensions, d(w) and d(s), are equal to those of homogeneous media showing that, on average, the key parameter is the fractal dimension of order 0 D0, equal to the Euclidean dimension and not to the correlation dimension D2. Beyond their average, the anomalous diffusion and fracton exponents, d(w) and d(s), are highly variable and consistently range in the interval [1,4]. d(w) can be consistently either larger or lower than 2, indicating possible subdiffusive and superdiffusive regimes. On a realization basis, we show that the exponent variability is related to the local conductivity at the medium inlet through the conductivity scaling.
Collapse
Affiliation(s)
- Jean-Raynald de Dreuzy
- Géosciences Rennes, UMR CNRS 6118, Université de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France
| | | | | | | |
Collapse
|
46
|
Méndez V, Fort J, Rotstein HG, Fedotov S. Speed of reaction-diffusion fronts in spatially heterogeneous media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:041105. [PMID: 14682921 DOI: 10.1103/physreve.68.041105] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2002] [Revised: 05/15/2003] [Indexed: 05/24/2023]
Abstract
The front speed problem for nonuniform reaction rate and diffusion coefficient is studied by using singular perturbation analysis, the geometric approach of Hamilton-Jacobi dynamics, and the local speed approach. Exact and perturbed expressions for the front speed are obtained in the limit of large times. For linear and fractal heterogeneities, the analytic results have been compared with numerical results exhibiting a good agreement. Finally we reach a general expression for the speed of the front in the case of smooth and weak heterogeneities.
Collapse
Affiliation(s)
- Vicenç Méndez
- Departament de Medicina, Facultat de Ciències de la Salut, Universitat Internacional de Catalunya, c/ Gomera s/n, 08190-Sant Cugat del Vallès (Barcelona), Spain
| | | | | | | |
Collapse
|
47
|
Isliker H, Vlahos L. Random walk through fractal environments. PHYSICAL REVIEW E 2003; 67:026413. [PMID: 12636828 DOI: 10.1103/physreve.67.026413] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2002] [Indexed: 11/07/2022]
Abstract
We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e., of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D(F) of the fractal is less than 2, there is though, always a finite rate of unaffected escape. Random walks through fractal sets with D(F)< or =2 can thus be considered as defective Levy walks. The distribution of jump increments for D(F)>2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk increments. It is shown that the particles undergo anomalous, enhanced diffusion for D(F)<2, the diffusion is dominated by the finite escape rate. Diffusion for D(F)>2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality models give rise to enhanced diffusion. The analytical results are illustrated by Monte Carlo simulations.
Collapse
Affiliation(s)
- H Isliker
- Association Euratom-Hellenic Republic, Section of Astrophysics, Astronomy and Mechanics Department of Physics, University of Thessaloniki, GR 54006 Thessaloniki, Greece.
| | | |
Collapse
|
48
|
Malacarne LC, Mendes RS, Pedron IT, Lenzi EK. N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:052101. [PMID: 12059613 DOI: 10.1103/physreve.65.052101] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2001] [Indexed: 05/23/2023]
Abstract
An N-dimensional nonlinear Fokker-Planck equation is investigated here by considering the time dependence of the coefficients, where drift-controlled and source terms are present. We exhibit the exact solution based on the generalized Gaussian function related to the Tsallis statistics. Furthermore, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained by changing the time dependence of the coefficients.
Collapse
Affiliation(s)
- L C Malacarne
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, PR, Brazil.
| | | | | | | |
Collapse
|
49
|
Pedron IT, Mendes RS, Malacarne LC, Lenzi EK. Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:041108. [PMID: 12005807 DOI: 10.1103/physreve.65.041108] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2001] [Indexed: 05/23/2023]
Abstract
In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the N-dimensional nonlinear diffusion equation partial differential rho/ partial differential t=nabla.(Knablarho(nu))-nabla.(muFrho)-alpharho, where K=Dr(-theta), nu, theta, mu, and D are real parameters, F is the external force, and alpha is a time-dependent source. This equation unifies the O'Shaughnessy-Procaccia anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.
Collapse
Affiliation(s)
- I T Pedron
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
| | | | | | | |
Collapse
|
50
|
Kozak JJ, Balakrishnan V. Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:021105. [PMID: 11863501 DOI: 10.1103/physreve.65.021105] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2001] [Indexed: 05/23/2023]
Abstract
The exact analytic expression for the mean time to absorption (or mean walk length) for a particle performing a random walk on a finite Sierpinski gasket with a trap at one vertex is found to be T((n))=[3(n)5(n+1)+4(5(n))-3(n)]/(3(n+1)+1) where n denotes the generation index of the gasket, and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket. In terms of the number N(n) of sites on the gasket and the spectral dimension d of the gasket, the precise asymptotic behavior for large N(n) is T((n))-->1/3(2N(n))(2/d)-N1.464. This serves as a partial check on our result, as it is (a) intermediate between the known results T-N2 (d=1) and T-N ln N (d=2) for random walks on d-dimensional Euclidean lattices and (b) consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice.
Collapse
Affiliation(s)
- John J Kozak
- Department of Chemistry, Iowa State University, Ames, Iowa 50011-3111, USA
| | | |
Collapse
|