@article{SandevMetzlerTomovski2012, author = {Sandev, Trifce and Metzler, Ralf and Tomovski, Zivorad}, title = {Velocity and displacement correlation functions for fractional generalized Langevin equations}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {15}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {3}, publisher = {Versita}, address = {Warsaw}, issn = {1311-0454}, doi = {10.2478/s13540-012-0031-2}, pages = {426 -- 450}, year = {2012}, abstract = {We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.}, language = {en} } @article{TomovskiSandevMetzleretal.2012, author = {Tomovski, Zivorad and Sandev, Trifce and Metzler, Ralf and Dubbeldam, Johan}, title = {Generalized space-time fractional diffusion equation with composite fractional time derivative}, series = {Physica : europhysics journal ; A, Statistical mechanics and its applications}, volume = {391}, journal = {Physica : europhysics journal ; A, Statistical mechanics and its applications}, number = {8}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0378-4371}, doi = {10.1016/j.physa.2011.12.035}, pages = {2527 -- 2542}, year = {2012}, abstract = {We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grunwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of the form delta(x). t-beta/Gamma(1-beta) (beta > 0).}, language = {en} } @article{SandevMetzlerTomovski2014, author = {Sandev, Trifce and Metzler, Ralf and Tomovski, Zivorad}, title = {Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise}, series = {Journal of mathematical physics}, volume = {55}, journal = {Journal of mathematical physics}, number = {2}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0022-2488}, doi = {10.1063/1.4863478}, pages = {23}, year = {2014}, abstract = {We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion.}, language = {en} } @article{SandevChechkinKorabeletal.2015, author = {Sandev, Trifce and Chechkin, Aleksei V. and Korabel, Nickolay and Kantz, Holger and Sokolov, Igor M. and Metzler, Ralf}, title = {Distributed-order diffusion equations and multifractality: Models and solutions}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {92}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.92.042117}, pages = {19}, year = {2015}, abstract = {We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.}, language = {en} } @article{SandevChechkinKantzetal.2015, author = {Sandev, Trifce and Chechkin, Aleksei V. and Kantz, Holger and Metzler, Ralf}, title = {Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {18}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {4}, publisher = {De Gruyter}, address = {Berlin}, issn = {1311-0454}, doi = {10.1515/fca-2015-0059}, pages = {1006 -- 1038}, year = {2015}, abstract = {We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.}, language = {en} } @article{SandevIominKantzetal.2016, author = {Sandev, Trifce and Iomin, Alexander and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei V.}, title = {Comb Model with Slow and Ultraslow Diffusion}, series = {Mathematical modelling of natural phenomena}, volume = {11}, journal = {Mathematical modelling of natural phenomena}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/201611302}, pages = {18 -- 33}, year = {2016}, abstract = {We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.}, language = {en} } @article{SandevSokolovMetzleretal.2017, author = {Sandev, Trifce and Sokolov, Igor M. and Metzler, Ralf and Chechkin, Aleksei V.}, title = {Beyond monofractional kinetics}, series = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, volume = {102}, journal = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, publisher = {Elsevier}, address = {Oxford}, issn = {0960-0779}, doi = {10.1016/j.chaos.2017.05.001}, pages = {210 -- 217}, year = {2017}, abstract = {We discuss generalized integro-differential diffusion equations whose integral kernels are not of a simple power law form, and thus these equations themselves do not belong to the family of fractional diffusion equations exhibiting a monoscaling behavior. They instead generate a broad class of anomalous nonscaling patterns, which correspond either to crossovers between different power laws, or to a non-power-law behavior as exemplified by the logarithmic growth of the width of the distribution. We consider normal and modified forms of these generalized diffusion equations and provide a brief discussion of three generic types of integral kernels for each form, namely, distributed order, truncated power law and truncated distributed order kernels. For each of the cases considered we prove the non-negativity of the solution of the corresponding generalized diffusion equation and calculate the mean squared displacement. (C) 2017 Elsevier Ltd. All rights reserved.}, language = {en} } @article{MolinaGarciaSandevSafdarietal.2018, author = {Molina-Garcia, Daniel and Sandev, Trifce and Safdari, Hadiseh and Pagnini, Gianni and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Crossover from anomalous to normal diffusion}, series = {New Journal of Physics}, volume = {20}, journal = {New Journal of Physics}, publisher = {IOP Publishing Ltd}, address = {London und Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/aae4b2}, pages = {28}, year = {2018}, abstract = {Abstract The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive-diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling.}, language = {en} } @article{SandevTomovskiDubbeldametal.2018, author = {Sandev, Trifce and Tomovski, Zivorad and Dubbeldam, Johan L. A. and Chechkin, Aleksei V.}, title = {Generalized diffusion-wave equation with memory kernel}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {1}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aaefa3}, pages = {22}, year = {2018}, abstract = {We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate the mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with a regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling the broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.}, language = {en} } @article{SandevMetzlerChechkin2018, author = {Sandev, Trifce and Metzler, Ralf and Chechkin, Aleksei V.}, title = {From continuous time random walks to the generalized diffusion equation}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {21}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {1}, publisher = {De Gruyter}, address = {Berlin}, issn = {1311-0454}, doi = {10.1515/fca-2018-0002}, pages = {10 -- 28}, year = {2018}, abstract = {We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.}, language = {en} }