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Gharari F, Arias-Calluari K, Alonso-Marroquin F, Najafi MN. Space-time fractional porous media equation: Application on modeling of S&P500 price return. Phys Rev E 2021; 104:054140. [PMID: 34942814 DOI: 10.1103/physreve.104.054140] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2020] [Accepted: 10/27/2021] [Indexed: 11/07/2022]
Abstract
We present the fractional extensions of the porous media equation (PME) with an emphasis on the applications in stock markets. Three kinds of "fractionalization" are considered: local, where the fractional derivatives for both space and time are local; nonlocal, where both space and time fractional derivatives are nonlocal; and mixed, where one derivative is local, and another is nonlocal. Our study shows that these fractional equations admit solutions in terms of generalized q-Gaussian functions. Each solution of these fractional formulations contains a certain number of free parameters that can be fitted with experimental data. Our focus is to analyze stock market data and determine the model that better describes the time evolution of the probability distribution of the price return. We proposed a generalized PME motivated by recent observations showing that q-Gaussian distributions can model the evolution of the probability distribution. Various phases (weak, strong super diffusion, and normal diffusion) were observed on the time evolution of the probability distribution of the price return separated by different fitting parameters [Phys. Rev. E 99, 062313 (2019)1063-651X10.1103/PhysRevE.99.062313]. After testing the obtained solutions for the S&P500 price return, we found that the local and nonlocal schemes fit the data better than the classic porous media equation.
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Affiliation(s)
- Fatemeh Gharari
- Department of Statistics and Computer Science, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran
| | | | | | - Morteza N Najafi
- Department of Physics, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran
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Barrick TR, Spilling CA, Ingo C, Madigan J, Isaacs JD, Rich P, Jones TL, Magin RL, Hall MG, Howe FA. Quasi-diffusion magnetic resonance imaging (QDI): A fast, high b-value diffusion imaging technique. Neuroimage 2020; 211:116606. [DOI: 10.1016/j.neuroimage.2020.116606] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2019] [Revised: 12/22/2019] [Accepted: 02/02/2020] [Indexed: 12/11/2022] Open
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3
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Nandi S, Pyne A, Ghosh M, Banerjee P, Ghosh B, Sarkar N. Antagonist Effects of l-Phenylalanine and the Enantiomeric Mixture Containing d-Phenylalanine on Phospholipid Vesicle Membrane. LANGMUIR : THE ACS JOURNAL OF SURFACES AND COLLOIDS 2020; 36:2459-2473. [PMID: 32073868 DOI: 10.1021/acs.langmuir.9b03543] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
One of the congenital flaws of metabolism, phenylketonuria (PKU), is known to be related to the self-assembly of toxic fibrillar aggregates of phenylalanine (Phe) in blood at elevated concentrations. Our experimental findings using l-phenylalanine (l-Phe) at millimolar concentration suggest the formation of fibrillar morphologies in the dry phase, which in the solution phase interact strongly with the model membrane composed of 1,2-diacyl-sn-glycero-phosphocholine (LAPC) lipid, thereby decreasing the rigidity (or increasing the fluidity) of the membrane. The hydrophobic interaction, in addition to the electrostatic attraction of Phe with the model membrane, is found to be responsible for such phenomena. On the contrary, various microscopic observations reveal that such fibrillar morphologies of l-Phe are severely ruptured in the presence of its enantiomer d-phenylalanine (d-Phe), thereby converting the fibrillar morphologies into crushed flakes. Various biophysical studies, including the solvation dynamics experiment, suggest that this l-Phe in the presence of d-Phe, when interacting with the same model membrane, now reverts the rigidity of the membrane, i.e., increases the rigidity of the membrane, which was lost due to interaction with l-Phe exclusively. Fluorescence anisotropy measurements also support this reverse rigid character of the membrane in the presence of an enantiomeric mixture of amino acids. A comprehensive understanding of the interaction of Phe with the model membrane is further pursued at the single-molecular fluorescence detection level using fluorescence correlation spectroscopy (FCS) experiments. Therefore, our experimental conclusion interprets a linear correlation between increased permeability and enhanced fluidity of the membrane in the presence of l-Phe and certifies d-Phe as a therapeutic modulator of l-Phe fibrillar morphologies. Further, the study proposes that the rigidity of the membrane lost due to interaction with l-Phe was reinstated-in fact, increased-in the presence of the enantiomeric mixture containing both d- and l-Phe.
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Skardal PS. Quasiperiodic dynamics and a Neimark-Sacker bifurcation in nonlinear random walks on complex networks. Phys Rev E 2020; 101:012307. [PMID: 32069530 DOI: 10.1103/physreve.101.012307] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2019] [Indexed: 06/10/2023]
Abstract
We study the dynamics of nonlinear random walks on complex networks. In particular, we investigate the role and effect of directed network topologies on long-term dynamics. While a period-doubling bifurcation to alternating patterns occurs at a critical bias parameter value, we find that some directed structures give rise to a different kind of bifurcation that gives rise to quasiperiodic dynamics. This does not occur for all directed network structure, but only when the network structure is sufficiently directed. We find that the onset of quasiperiodic dynamics is the result of a Neimark-Sacker bifurcation, where a pair of complex-conjugate eigenvalues of the system Jacobian pass through the unit circle, destabilizing the stationary distribution with high-dimensional rotations. We investigate the nature of these bifurcations, study the onset of quasiperiodic dynamics as network structure is tuned to be more directed, and present an analytically tractable case of a four-neighbor ring.
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Spectroscopic investigation on alteration of dynamic properties of lipid membrane in presence of Gamma-Aminobutyric Acid (GABA). J Mol Liq 2020. [DOI: 10.1016/j.molliq.2019.111877] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Zhokh A, Strizhak P. Non-Fickian Transport in Porous Media: Always Temporally Anomalous? Transp Porous Media 2018. [DOI: 10.1007/s11242-018-1066-6] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
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Vitali S, Sposini V, Sliusarenko O, Paradisi P, Castellani G, Pagnini G. Langevin equation in complex media and anomalous diffusion. J R Soc Interface 2018; 15:20180282. [PMID: 30158182 PMCID: PMC6127165 DOI: 10.1098/rsif.2018.0282] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2018] [Accepted: 07/30/2018] [Indexed: 11/30/2022] Open
Abstract
The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle's dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.
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Affiliation(s)
- Silvia Vitali
- Department of Physics and Astronomy, Bologna University, Viale Berti Pichat 6/2, 40126 Bologna, Italy
| | - Vittoria Sposini
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
| | - Oleksii Sliusarenko
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
| | - Paolo Paradisi
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
- ISTI-CNR, Institute of Information Science and Technologies 'A. Faedo' (Consiglio Nazionale delle Ricerche), Via Moruzzi 1, 56124 Pisa, Italy
| | - Gastone Castellani
- Department of Physics and Astronomy, Bologna University, Viale Berti Pichat 6/2, 40126 Bologna, Italy
| | - Gianni Pagnini
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
- Ikerbasque-Basque Foundation for Science, Calle de María Díaz de Haro 3, 48013 Bilbao, Basque Country, Spain
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Sánchez R, Newman DE, Mier JA. Modeling transport across the running-sandpile cellular automaton by means of fractional transport equations. Phys Rev E 2018; 97:052123. [PMID: 29906898 DOI: 10.1103/physreve.97.052123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2017] [Indexed: 06/08/2023]
Abstract
Fractional transport equations are used to build an effective model for transport across the running sandpile cellular automaton [Hwa et al., Phys. Rev. A 45, 7002 (1992)PLRAAN1050-294710.1103/PhysRevA.45.7002]. It is shown that both temporal and spatial fractional derivatives must be considered to properly reproduce the sandpile transport features, which are governed by self-organized criticality, at least over sufficiently long or large scales. In contrast to previous applications of fractional transport equations to other systems, the specifics of sand motion require in this case that the spatial fractional derivatives used for the running sandpile must be of the completely asymmetrical Riesz-Feller type. Appropriate values for the fractional exponents that define these derivatives in the case of the running sandpile are obtained numerically.
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Affiliation(s)
- R Sánchez
- Departamento de Física, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain
| | - D E Newman
- Department of Physics, University of Alaska, Fairbanks, Alaska 99775-5920, USA
| | - J A Mier
- Departamento de Física Aplicada, Universidad de Cantabria, 39005 Santander, Spain
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Sun Z, Zhang J, Yang X, Xu W. Taming stochastic bifurcations in fractional-order systems via noise and delayed feedback. CHAOS (WOODBURY, N.Y.) 2017; 27:083102. [PMID: 28863494 DOI: 10.1063/1.4997050] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The dynamics in fractional-order systems have been widely studied during the past decade due to the potential applications in new materials and anomalous diffusions, but the investigations have been so far restricted to a fractional-order system without time delay(s). In this paper, we report the first study of random responses of fractional-order system coupled with noise and delayed feedback. Stochastic averaging method has been utilized to determine the stationary probability density functions (PDFs) by means of the principle of minimum mean-square error, based on which stochastic bifurcations could be identified through recognizing the shape of the PDFs. It has been found that by changing the fractional order the shape of the PDFs can switch from unimodal distribution to bimodal one, or from bimodal distribution to unimodal one, thus announcing the onset of stochastic bifurcation. Further, we have demonstrated that by merely modulating the time delay, the feedback strengths, or the noise intensity, the shapes of PDFs can transit between a single peak and a double peak. Therefore, it provides an efficient candidate to control, say, induce or suppress, the stochastic bifurcations in fractional-order systems.
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Affiliation(s)
- Zhongkui Sun
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
| | - Jintian Zhang
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
| | - Xiaoli Yang
- College of Mathematics and Information Science, Shaan'xi Normal University, Xi'an 710062, People's Republic of China
| | - Wei Xu
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
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10
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Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation. ENTROPY 2017. [DOI: 10.3390/e19070297] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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11
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12
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Gustafson KB, Bayati BS, Eckhoff PA. Fractional Diffusion Emulates a Human Mobility Network during a Simulated Disease Outbreak. Front Ecol Evol 2017. [DOI: 10.3389/fevo.2017.00035] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022] Open
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13
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14
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Fujioka J, Espinosa A, Rodríguez RF, Malomed BA. Radiating subdispersive fractional optical solitons. CHAOS (WOODBURY, N.Y.) 2014; 24:033121. [PMID: 25273201 DOI: 10.1063/1.4892616] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
It was recently found [Fujioka et al., Phys. Lett. A 374, 1126 (2010)] that the propagation of solitary waves can be described by a fractional extension of the nonlinear Schrödinger (NLS) equation which involves a temporal fractional derivative (TFD) of order α > 2. In the present paper, we show that there is also another fractional extension of the NLS equation which contains a TFD with α < 2, and in this case, the new equation describes the propagation of radiating solitons. We show that the emission of the radiation (when α < 2) is explained by resonances at various frequencies between the pulses and the linear modes of the system. It is found that the new fractional NLS equation can be derived from a suitable Lagrangian density, and a fractional Noether's theorem can be applied to it, thus predicting the conservation of the Hamiltonian, momentum and energy.
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Affiliation(s)
- J Fujioka
- Departamento de Física Química, Instituto de Física, Universidad Nacional Autónoma de México, Mexico, DF 04510, Mexico
| | - A Espinosa
- Departamento de Física Química, Instituto de Física, Universidad Nacional Autónoma de México, Mexico, DF 04510, Mexico
| | - R F Rodríguez
- Departamento de Física Química, Instituto de Física, Universidad Nacional Autónoma de México, Mexico, DF 04510, Mexico
| | - B A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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15
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Basit H, Lopez SG, Keyes TE. Fluorescence correlation and lifetime correlation spectroscopy applied to the study of supported lipid bilayer models of the cell membrane. Methods 2014; 68:286-99. [DOI: 10.1016/j.ymeth.2014.02.005] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2013] [Revised: 01/21/2014] [Accepted: 02/06/2014] [Indexed: 10/25/2022] Open
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16
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Abstract
A novel method is presented for the simulation of a discrete state space, continuous time Markov process subject to fractional diffusion. The method is based on Lie-Trotter operator splitting of the diffusion and reaction terms in the master equation. The diffusion term follows a multinomial distribution governed by a kernel that is the discretized solution of the fractional diffusion equation. The algorithm is validated and simulations are provided for the Fisher-KPP wavefront. It is shown that the wave speed is dictated by the order of the fractional derivative, where lower values result in a faster wave than in the case of classical diffusion. Since many physical processes deviate from classical diffusion, fractional diffusion methods are necessary for accurate simulations.
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Affiliation(s)
- Basil S Bayati
- Intellectual Ventures Laboratory, 1600 132nd Ave. NE, Bellevue, Washington 98004, USA.
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17
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Yu Q, Liu F, Turner I, Burrage K. Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20120150. [PMID: 23547227 DOI: 10.1098/rsta.2012.0150] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here, the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments, where processes are often anisotropic. Anomalous diffusion in the human brain using fractional-order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional-order calculus with respect to time and space. However, effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) in both fractional Laplacian and Riesz derivative form is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated. We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.
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Affiliation(s)
- Qiang Yu
- School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Queensland 4001, Australia
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18
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Dybiec B, Gudowska-Nowak E. Discriminating between normal and anomalous random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:061122. [PMID: 20365133 DOI: 10.1103/physreve.80.061122] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2009] [Revised: 10/16/2009] [Indexed: 05/26/2023]
Abstract
Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, {x2(t) proportional t, while anomalous behavior is expected to show a different time dependence, x2(t) proportional t{delta} with delta<1 for subdiffusive and delta>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2(t) proportional t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).
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Affiliation(s)
- Bartłomiej Dybiec
- Marian Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagellonian University, ul Reymonta 4, 30-059 Kraków, Poland.
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Roman HE, Porto M. Fractional derivatives of random walks: time series with long-time memory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:031127. [PMID: 18851013 DOI: 10.1103/physreve.78.031127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2008] [Revised: 05/23/2008] [Indexed: 05/26/2023]
Abstract
We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly decaying autocorrelation functions. The relation between these correlated walks and the well-known fractionally integrated autoregressive models with conditional heteroskedasticity (FIGARCH), commonly used in econometric studies, is discussed. The application of correlated random walks to simulate empirical financial times series is considered and compared with the predictions from FIGARCH and the simpler FIARCH processes. A comparison with empirical data is performed.
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Affiliation(s)
- H Eduardo Roman
- Dipartimento di Fisica, Università di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
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20
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Dybiec B, Gudowska-Nowak E, Sokolov IM. Transport in a Lévy ratchet: group velocity and distribution spread. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:011117. [PMID: 18763929 DOI: 10.1103/physreve.78.011117] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2008] [Indexed: 05/26/2023]
Abstract
We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric, white, Lévy noise, being a minimal setup for a "Lévy ratchet." Due to the nonthermal character of the Lévy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the Lévy ratchet has to be based on the characteristics of directionality which are different from typically used measures such as mean current and the dispersion of particle positions, since these become inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport such as the position of the median of the particle displacement distribution characterizing the group velocity and the interquantile distance giving the measure of the distribution width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length, unveiling qualitative differences between the noises with Lévy indices below and above unity.
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Affiliation(s)
- B Dybiec
- M Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagellonian University, ul. Reymonta 4, Kraków, Poland.
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21
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Leiderman K, Steinberg S. High-Resolution Models of Motion of Macromolecules in Cell Membranes. MATHEMATICS AND COMPUTERS IN SIMULATION 2008; 77:383-399. [PMID: 19347036 PMCID: PMC2597861 DOI: 10.1016/j.matcom.2007.03.008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
The path of a macromolecule on a cell membrane is modeled by a sum of independent identically distributed random variables. Random variables with simple discrete distribution functions capture some important aspects of the jump or hop diffusion reported from single particle tracking experiments that measure the motion of single molecules on a cell membrane. The detail provided by the distribution function for the random variables is critical for accurate simulations of the motion and interactions of macromolecules on the cell membrane. Additionally, the probability distribution for the random variables is easily estimated from single-particle tracking data. The diffusion constant is given by the second moment of the probability distribution, which agrees with the diffusion constant estimated from the mean-square displacement, and thus represents far less information than the distribution function.
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Affiliation(s)
- Karin Leiderman
- Department of Mathematics and Statistics, University of New Mexico, Albuquerque NM 87131-1141 USA
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Magin RL, Abdullah O, Baleanu D, Zhou XJ. Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. JOURNAL OF MAGNETIC RESONANCE (SAN DIEGO, CALIF. : 1997) 2008; 190:255-70. [PMID: 18065249 DOI: 10.1016/j.jmr.2007.11.007] [Citation(s) in RCA: 114] [Impact Index Per Article: 7.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2007] [Revised: 11/08/2007] [Accepted: 11/08/2007] [Indexed: 05/25/2023]
Abstract
Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm(2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.
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Affiliation(s)
- Richard L Magin
- Department of Bioengineering, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7052, USA.
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Guo L, Chowdhury P, Fang J, Gai F. Heterogeneous and anomalous diffusion inside lipid tubules. J Phys Chem B 2007; 111:14244-9. [PMID: 18052149 DOI: 10.1021/jp076562n] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Self-assembled lipid tubules with crystalline bilayer walls are promising candidates for controlled drug delivery vehicles on the basis of their ability to release preloaded biological molecules in a sustained manner. While a previous study has shown that the release rate of protein molecules from lipid tubules depends on the associated molecular mass, suggesting that the pertinent diffusion follows the well-known Stokes-Einstein relationship, only a few attempts have been made toward investigating the details of molecular diffusion in the tubule interior. Herein, we have characterized the diffusion rates of several molecules encapsulated in lipid tubules formed by 1,2-bis(10,12-tricosadiynoyl)-sn-glycero-3-phosphocholine (DC8,9PC) using the techniques of fluorescence recovery after photobleaching (FRAP) and fluorescence correlation spectroscopy (FCS). Our results show that the mobility of these molecules depends not only on their positions in the DC8,9PC tubules but also on their respective concentrations. While the former indicates that the interior of the DC8,9PC tubules is heterogeneous in terms of diffusion, the latter further highlights the possibility of engineering specific conditions for achieving sustained release of a "drug molecule" over a targeted period of time. In addition, our FCS results indicate that the molecular diffusions inside the crystalline bilayer walls of the DC8,9PC tubules strongly deviate from the normal, stochastic processes, with features characterizing not only anomalous subdiffusions but also motions that are superdiffusive in nature.
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Affiliation(s)
- Lin Guo
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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Dybiec B, Gudowska-Nowak E, Sokolov IM. Stationary states in Langevin dynamics under asymmetric Lévy noises. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041122. [PMID: 17994951 DOI: 10.1103/physreve.76.041122] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2007] [Indexed: 05/25/2023]
Abstract
Properties of systems driven by white non-Gaussian noises can be very different from these of systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by alpha-stable Lévy-type noises, which provide natural extension to the Gaussian noise having, however, a new property, namely a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic, and in generic double well potential models subjected to the action of alpha-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.
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Affiliation(s)
- B Dybiec
- M. Smoluchowski Institute of Physics and Mark Kac Center for Complex Systems Research, Jagellonian University, ul. Reymonta 4, 30-059 Kraków, Poland.
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Zoia A, Rosso A, Kardar M. Fractional Laplacian in bounded domains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:021116. [PMID: 17930015 DOI: 10.1103/physreve.76.021116] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2007] [Indexed: 05/25/2023]
Abstract
The fractional Laplacian operator -(-delta)(alpha/2) appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.
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Affiliation(s)
- A Zoia
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
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Marseguerra M, Zoia A. Normal and anomalous transport across an interface: Monte Carlo and analytical approach. ANN NUCL ENERGY 2006. [DOI: 10.1016/j.anucene.2006.09.012] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Demontis P, Suffritti GB. Fractional diffusion interpretation of simulated single-file systems in microporous materials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:051112. [PMID: 17279882 DOI: 10.1103/physreve.74.051112] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2006] [Indexed: 05/13/2023]
Abstract
The single-file diffusion of water in the straight channels of two different crystalline microporous aluminosilicates (zeolites bikitaite and Li-ABW) was studied by comparing the results of molecular dynamics computer simulations with the predictions of anomalous diffusion theory modeled by using fractional diffusion equations. At high coverage, the agreement is reasonably good, in particular for sufficiently large displacements and sufficiently long times. At low coverage, interesting phenomena appear in the simulation results, such as multimodal propagators, which could be interpreted on the basis of fractional Fokker-Planck equations. The results are discussed also in view of different theories that have been proposed to model the single-file diffusion process.
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Affiliation(s)
- Pierfranco Demontis
- Dipartimento di Chimica, Università di Sassari and Consorzio Interuniversitario Nazionale per la Scienza e Tecnologia dei Materiali (INSTM), Unità di ricerca di Sassari, Via Vienna, 2, I-07100 Sassari, Italy.
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Chen W. A speculative study of 23-order fractional Laplacian modeling of turbulence: some thoughts and conjectures. CHAOS (WOODBURY, N.Y.) 2006; 16:023126. [PMID: 16822029 DOI: 10.1063/1.2208452] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/10/2023]
Abstract
This study makes the first attempt to use the 23-order fractional Laplacian modeling of Kolmogorov -53 scaling of fully developed turbulence and enhanced diffusing movements of random turbulent particles. Nonlinear inertial interactions and molecular Brownian diffusivity are considered to be the bifractal mechanism behind multifractal scaling of moderate Reynolds number turbulence. Accordingly, a stochastic equation is proposed to describe turbulence intermittency. The 23-order fractional Laplacian representation is also used to model nonlinear interactions of fluctuating velocity components, and then we conjecture a fractional Reynolds equation, underlying fractal spacetime structures of Levy 23 stable distribution and the Kolmogorov scaling at inertial scales. The new perspective of this study is that the fractional calculus is an effective approach to modeling the chaotic fractal phenomena induced by nonlinear interactions.
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Affiliation(s)
- Wen Chen
- Department of Engineering Mechanics, Hohai University, No. 1 Xikang Road, Nanjing City, Jiangsu Province 210098, China.
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Krepysheva N, Di Pietro L, Néel MC. Space-fractional advection-diffusion and reflective boundary condition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:021104. [PMID: 16605326 DOI: 10.1103/physreve.73.021104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2005] [Revised: 11/29/2005] [Indexed: 05/08/2023]
Abstract
Anomalous diffusive transport arises in a large diversity of disordered media. Stochastic formulations in terms of continuous time random walks (CTRWs) with transition probability densities showing space- and/or time-diverging moments were developed to account for anomalous behaviors. A broad class of CTRWs was shown to correspond, on the macroscopic scale, to advection-diffusion equations involving derivatives of noninteger order. In particular, CTRWs with Lévy distribution of jumps and finite mean waiting time lead to a space-fractional equation that accounts for superdiffusion and involves a nonlocal integral-differential operator. Within this framework, we analyze the evolution of particles performing symmetric Lévy flights with respect to a fluid moving at uniform speed . The particles are restricted to a semi-infinite domain limited by a reflective barrier. We show that the introduction of the boundary condition induces a modification in the kernel of the nonlocal operator. Thus, the macroscopic space-fractional advection-diffusion equation obtained is different from that in an infinite medium.
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Affiliation(s)
- Natalia Krepysheva
- INRA d'Avignon, UMRA Climat Sol Environnement, Domaine Saint Paul--Site Agroparc, F-84914 Avignon Cedex 9, France.
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Banks DS, Fradin C. Anomalous diffusion of proteins due to molecular crowding. Biophys J 2005; 89:2960-71. [PMID: 16113107 PMCID: PMC1366794 DOI: 10.1529/biophysj.104.051078] [Citation(s) in RCA: 487] [Impact Index Per Article: 25.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2004] [Accepted: 08/03/2005] [Indexed: 11/18/2022] Open
Abstract
We have studied the diffusion of tracer proteins in highly concentrated random-coil polymer and globular protein solutions imitating the crowded conditions encountered in cellular environments. Using fluorescence correlation spectroscopy, we measured the anomalous diffusion exponent alpha characterizing the dependence of the mean-square displacement of the tracer proteins on time, r(2)(t) approximately t(alpha). We observed that the diffusion of proteins in dextran solutions with concentrations up to 400 g/l is subdiffusive (alpha < 1) even at low obstacle concentration. The anomalous diffusion exponent alpha decreases continuously with increasing obstacle concentration and molecular weight, but does not depend on buffer ionic strength, and neither does it depend strongly on solution temperature. At very high random-coil polymer concentrations, alpha reaches a limit value of alpha(l) approximately 3/4, which we take to be the signature of a coupling between the motions of the tracer proteins and the segments of the dextran chains. A similar, although less pronounced, subdiffusive behavior is observed for the diffusion of streptavidin in concentrated globular protein solutions. These observations indicate that protein diffusion in the cell cytoplasm and nucleus should be anomalous as well, with consequences for measurements of solute diffusion coefficients in cells and for the modeling of cellular processes relying on diffusion.
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Affiliation(s)
- Daniel S Banks
- Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada
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Gorenflo R, Mainardi F. Simply and multiply scaled diffusion limits for continuous time random walks. ACTA ACUST UNITED AC 2005. [DOI: 10.1088/1742-6596/7/1/001] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Logvinova K, Néel MC. A fractional equation for anomalous diffusion in a randomly heterogeneous porous medium. CHAOS (WOODBURY, N.Y.) 2004; 14:982-987. [PMID: 15568911 DOI: 10.1063/1.1796211] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
A fractional partial differential equation is derived for the spreading of matter in a saturated porous medium starting from precise hypotheses concerning the medium itself, which is a collection of intertwisted tubes with randomly distributed slopes, filled with quiescent fluid. Examining the fundamental solution of the fractional equation indicates that the second moment is not proportional to time, which is the signature of anomalous diffusion. The equation derived preserves non-negativity and also the total mass of matter.
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Affiliation(s)
- Kira Logvinova
- UMRA Climat, Sol, Environnement, Université d'Avignon-INRA d'Avignon, Faculté des Sciences, 33 rue Louis Pasteur, F-84000 Avignon, France
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Chechkin AV, Metzler R, Gonchar VY, Klafter J, Tanatarov LV. First passage and arrival time densities for Lévy flights and the failure of the method of images. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/41/l01] [Citation(s) in RCA: 122] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Fractional diffusion Processes: Probability Distributions and Continuous Time Random Walk. PROCESSES WITH LONG-RANGE CORRELATIONS 2003. [DOI: 10.1007/3-540-44832-2_8] [Citation(s) in RCA: 60] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
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