TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Tomovski, Zivorad T1 - Velocity and displacement correlation functions for fractional generalized Langevin equations JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - 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. KW - fractional generalized Langevin equation KW - frictional memory kernel KW - variances KW - mean square displacement KW - anomalous diffusion Y1 - 2012 U6 - https://doi.org/10.2478/s13540-012-0031-2 SN - 1311-0454 VL - 15 IS - 3 SP - 426 EP - 450 PB - Versita CY - Warsaw ER - TY - JOUR A1 - Petreska, Irina A1 - Pejov, Ljupco A1 - Sandev, Trifce A1 - Kocarev, Ljupčo A1 - Metzler, Ralf T1 - Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting JF - Fractal and fractional N2 - The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules. KW - rotational diffusion KW - memory kernel KW - Fokker-Planck equation KW - non-exponential relaxation KW - autocorrelation function KW - complex KW - susceptibility Y1 - 2022 U6 - https://doi.org/10.3390/fractalfract6020088 SN - 2504-3110 VL - 6 IS - 2 PB - MDPI AG, Fractal Fract Editorial Office CY - Basel ER - TY - JOUR A1 - Petreska, Irina A1 - de Castro, Antonio S. M. A1 - Sandev, Trifce A1 - Lenzi, Ervin K. T1 - The time-dependent Schrödinger equation in non-integer dimensions for constrained quantum motion JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - We propose a theoretical model, based on a generalized Schroedinger equation, to study the behavior of a constrained quantum system in non-integer, lower than two-dimensional space. The non-integer dimensional space is formed as a product space X x Y, comprising x-coordinate with a Hausdorff measure of dimension alpha(1) = D -1 (1 < D < 2) and y-coordinate with the Lebesgue measure of dimension of length (alpha(2) = 1). Geometric constraints are set at y = 0. Two different approaches to find the Green's function are employed, both giving the same form in terms of the Fox H-function. For D = 2, the solution for two-dimensional quantum motion on a comb is recovered. (C) 2020 Elsevier B.V. All rights reserved. KW - Schrödinger equation KW - non-integer dimension KW - Green's function KW - Bessel functions KW - Fox H-function Y1 - 2020 U6 - https://doi.org/10.1016/j.physleta.2020.126866 SN - 0375-9601 SN - 1873-2429 VL - 384 IS - 34 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Singh, Rishu Kumar A1 - Metzler, Ralf A1 - Sandev, Trifce T1 - Resetting dynamics in a confining potential JF - Journal of physics : A, Mathematical and theoretical N2 - We study Brownian motion in a confining potential under a constant-rate resetting to a reset position x(0). The relaxation of this system to the steady-state exhibits a dynamic phase transition, and is achieved in a light cone region which grows linearly with time. When an absorbing boundary is introduced, effecting a symmetry breaking of the system, we find that resetting aids the barrier escape only when the particle starts on the same side as the barrier with respect to the origin. We find that the optimal resetting rate exhibits a continuous phase transition with critical exponent of unity. Exact expressions are derived for the mean escape time, the second moment, and the coefficient of variation (CV). KW - diffusion KW - resetting KW - barrier escape KW - first-passage Y1 - 2020 U6 - https://doi.org/10.1088/1751-8121/abc83a SN - 1751-8113 SN - 1751-8121 VL - 53 IS - 50 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Basnarkov, Lasko A1 - Tomovski, Igor A1 - Sandev, Trifce A1 - Kocarev, Ljupčo T1 - Non-Markovian SIR epidemic spreading model of COVID-19 JF - Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science N2 - We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete-and continuous-time versions. The distributions of infection intensity and recovery period may take an arbitrary form. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, Spain and the UK, in the spring, 2020. KW - Epidemic spreading models KW - Non-Markovian processes KW - COVID-19 KW - SIR model Y1 - 2022 U6 - https://doi.org/10.1016/j.chaos.2022.112286 SN - 0960-0779 SN - 1873-2887 VL - 160 PB - Elsevier CY - Oxford [u.a.] ER - TY - JOUR A1 - Xu, Pengbo A1 - Deng, Weihua A1 - Sandev, Trifce T1 - Levy walk with parameter dependent velocity BT - hermite polynomial approach and numerical simulation JF - Journal of physics : A, Mathematical and theoretical N2 - To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Levy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Levy walk. Two approaches are compared for some already known results. We also consider the generalized Levy walk with parameter dependent velocity. Namely, we consider the Levy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Levy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement. KW - Hermite polynomial expansion KW - Levy walk KW - anomalous diffusion KW - parameter KW - dependent velocity Y1 - 2020 U6 - https://doi.org/10.1088/1751-8121/ab7420 SN - 1751-8113 SN - 1751-8121 VL - 53 IS - 11 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Stojkoski, Viktor A1 - Jolakoski, Petar A1 - Pal, Arnab A1 - Sandev, Trifce A1 - Kocarev, Ljupco A1 - Metzler, Ralf T1 - Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity JF - Philosophical transactions of the Royal Society A: Mathematical, physical and engineering sciences N2 - We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and income mobility, the feasibility of an individual to change their position in the income rankings. For this purpose, we use the properties of an established model for income growth that includes 'resetting' as a stabilizing force to ensure stationary dynamics. We find that the dynamics of inequality is regime-dependent: it may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequality is steady and the system efficiently mimics ergodicity. Mobility measures, conversely, are always stable over time, but suggest that economies become less mobile in non-ergodic regimes. By fitting the model to empirical data for the income share of the top earners in the USA, we provide evidence that the income dynamics in this country is consistently in a regime in which non-ergodicity characterizes inequality and immobility. Our results can serve as a simple rationale for the observed real-world income dynamics and as such aid in addressing non-ergodicity in various empirical settings across the globe.This article is part of the theme issue 'Kinetic exchange models of societies and economies'. KW - income inequality KW - income mobility KW - geometric Brownian motion KW - non-ergodicity KW - stochastic resetting Y1 - 2022 U6 - https://doi.org/10.1098/rsta.2021.0157 SN - 1364-503X SN - 1471-2962 VL - 380 IS - 2224 PB - Royal Society CY - London ER - TY - JOUR A1 - Sandev, Trifce A1 - Iomin, Alexander A1 - Kocarev, Ljupco T1 - Hitting times in turbulent diffusion due to multiplicative noise JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study a distribution of times of the first arrivals to absorbing targets in turbulent diffusion, which is due to a multiplicative noise. Two examples of dynamical systems with a multiplicative noise are studied. The first one is a random process according to inhomogeneous diffusion, which is also known as a geometric Brownian motion in the Black-Scholes model. The second model is due to a random processes on a two-dimensional comb, where inhomogeneous advection is possible only along the backbone, while Brownian diffusion takes place inside the branches. It is shown that in both cases turbulent diffusion takes place as the one-dimensional random process with the log-normal distribution in the presence of absorbing targets, which are characterized by the Levy-Smirnov distribution for the first hitting times. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.042109 SN - 2470-0045 SN - 2470-0053 VL - 102 IS - 4 PB - American Institute of Physics CY - Woodbury, NY ER - TY - JOUR A1 - Sandev, Trifce A1 - Domazetoski, Viktor A1 - Kocarev, Ljupco A1 - Metzler, Ralf A1 - Chechkin, Aleksei T1 - Heterogeneous diffusion with stochastic resetting JF - Journal of physics : A, Mathematical and theoretical N2 - We study a heterogeneous diffusion process (HDP) with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyse the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like t (1/2) while the length scale xi(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the HDP with diffusion length increasing like t ( p/2) the length scale xi(t) grows like t ( p ). The obtained results are verified by numerical solutions of the corresponding Langevin equation. KW - heterogeneous diffusion KW - Fokker-Planck equation KW - Langevin equation KW - stochastic resetting KW - nonequilibrium stationary state KW - large deviation function Y1 - 2022 U6 - https://doi.org/10.1088/1751-8121/ac491c SN - 1751-8113 SN - 1751-8121 VL - 55 IS - 7 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Tomovski, Zivorad A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Dubbeldam, Johan T1 - Generalized space-time fractional diffusion equation with composite fractional time derivative JF - Physica : europhysics journal ; A, Statistical mechanics and its applications N2 - 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). KW - Fractional diffusion equation KW - Composite fractional derivative KW - Riesz-Feller fractional derivative KW - Mittag-Leffler functions KW - Fox H-function KW - Fractional moments KW - Asymptotic expansions KW - Grunwald-Letnikov approximation Y1 - 2012 U6 - https://doi.org/10.1016/j.physa.2011.12.035 SN - 0378-4371 SN - 1873-2119 VL - 391 IS - 8 SP - 2527 EP - 2542 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Sandev, Trifce A1 - Tomovski, Zivorad A1 - Dubbeldam, Johan L. A. A1 - Chechkin, Aleksei V. T1 - Generalized diffusion-wave equation with memory kernel JF - Journal of physics : A, Mathematical and theoretical N2 - 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. KW - diffusion-wave equation KW - Mittag-Leffler function KW - anomalous diffusion Y1 - 2018 U6 - https://doi.org/10.1088/1751-8121/aaefa3 SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 1 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Stojkoski, Viktor A1 - Sandev, Trifce A1 - Basnarkov, Lasko A1 - Kocarev, Ljupco A1 - Metzler, Ralf T1 - Generalised geometric Brownian motion BT - theory and applications to option pricing JF - Entropy N2 - Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness. KW - geometric Brownian motion KW - Fokker– Planck equation KW - Black– Scholes model KW - option pricing Y1 - 2020 U6 - https://doi.org/10.3390/e22121432 SN - 1099-4300 VL - 22 IS - 12 PB - MDPI CY - Basel ER - TY - JOUR A1 - Singh, Rishu Kumar A1 - Górska, Katarzyna A1 - Sandev, Trifce T1 - General approach to stochastic resetting JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that mean square displacement relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the probability density function (PDF) of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevE.105.064133 SN - 2470-0045 SN - 2470-0053 VL - 105 IS - 6 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - From continuous time random walks to the generalized diffusion equation JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - 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. KW - continuous time random walk (CTRW) KW - generalized diffusion equation KW - Mittag-Leffler functions KW - anomalous diffusion Y1 - 2018 U6 - https://doi.org/10.1515/fca-2018-0002 SN - 1311-0454 SN - 1314-2224 VL - 21 IS - 1 SP - 10 EP - 28 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Peng, Junhao A1 - Sandev, Trifce A1 - Kocarev, Ljupco T1 - First encounters on Bethe lattices and Cayley trees JF - Communications in nonlinear science & numerical simulation N2 - In this work we consider the first encounter problems between a fixed and/or mobile target A and a moving trap B on Bethe lattices and Cayley trees. The survival probabilities (SPs) of the target A on the both kinds of structures are considered analytically and compared. On Bethe lattices, the results show that the fixed target will still prolong its survival time, whereas, on Cayley trees, there are some initial positions where the target should move to prolong its survival time. The mean first encounter time (MFET) for mobile target A is evaluated numerically and compared with the mean first passage time (MFPT) for the fixed target A. Different initial settings are addressed and clear boundaries are obtained. These findings are helpful for optimizing the strategy to prolong the survival time of the target or to speed up the search process on Cayley trees, in relation to the target's movement and the initial position configuration of the two walkers. We also present a new method, which uses a small amount of memory, for simulating random walks on Cayley trees. (C) 2020 Elsevier B.V. All rights reserved. KW - Random walks KW - Survival probability KW - Mean first encounter time KW - Bethe KW - lattices KW - Cayley trees Y1 - 2021 U6 - https://doi.org/10.1016/j.cnsns.2020.105594 SN - 1007-5704 SN - 1878-7274 VL - 95 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Sandev, Trifce A1 - Chechkin, Aleksei V. A1 - Korabel, Nickolay A1 - Kantz, Holger A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Distributed-order diffusion equations and multifractality: Models and solutions JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1103/PhysRevE.92.042117 SN - 1539-3755 SN - 1550-2376 VL - 92 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Sandev, Trifce A1 - Chechkin, Aleksei V. A1 - Kantz, Holger A1 - Metzler, Ralf T1 - Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - 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. KW - continuous time random walk (CTRW) KW - Fokker-Planck-Smoluchowski equation KW - Mittag-Leffler functions KW - anomalous diffusion KW - multi-scaling Y1 - 2015 U6 - https://doi.org/10.1515/fca-2015-0059 SN - 1311-0454 SN - 1314-2224 VL - 18 IS - 4 SP - 1006 EP - 1038 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Molina-Garcia, Daniel A1 - Sandev, Trifce A1 - Safdari, Hadiseh A1 - Pagnini, Gianni A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Crossover from anomalous to normal diffusion BT - truncated power-law noise correlations and applications to dynamics in lipid bilayers JF - New Journal of Physics N2 - 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. KW - anomalous diffusion KW - truncated power-law correlated noise KW - lipid bilayer membrane dynamics Y1 - 2018 U6 - https://doi.org/10.1088/1367-2630/aae4b2 SN - 1367-2630 VL - 20 PB - IOP Publishing Ltd CY - London und Bad Honnef ER - TY - GEN A1 - Molina-Garcia, Daniel A1 - Sandev, Trifce A1 - Safdari, Hadiseh A1 - Pagnini, Gianni A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Crossover from anomalous to normal diffusion BT - truncated power-law noise correlations and applications to dynamics in lipid bilayers T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 507 KW - anomalous diffusion KW - truncated power-law correlated noise KW - lipid bilayer membrane dynamics Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-422590 SN - 1866-8372 IS - 507 ER - TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Tomovski, Zivorad T1 - Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise JF - Journal of mathematical physics N2 - 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. Y1 - 2014 U6 - https://doi.org/10.1063/1.4863478 SN - 0022-2488 SN - 1089-7658 VL - 55 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Mendez, Vicenc A1 - Maso-Puigdellosas, Axel A1 - Sandev, Trifce A1 - Campos, Daniel T1 - Continuous time random walks under Markovian resetting JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power-law probability density functions. We prove the existence of a nonequilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target, and the characteristic transport exponents. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.103.022103 SN - 2470-0045 SN - 2470-0053 VL - 103 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Sandev, Trifce A1 - Iomin, Alexander A1 - Kantz, Holger A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - Comb Model with Slow and Ultraslow Diffusion JF - Mathematical modelling of natural phenomena N2 - 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. KW - comb-like model KW - anomalous diffusion KW - mean squared displacement KW - probability density function Y1 - 2016 U6 - https://doi.org/10.1051/mmnp/201611302 SN - 0973-5348 SN - 1760-6101 VL - 11 SP - 18 EP - 33 PB - EDP Sciences CY - Les Ulis ER - TY - JOUR A1 - Sandev, Trifce A1 - Sokolov, Igor M. A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - Beyond monofractional kinetics JF - Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science N2 - 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. KW - Distributed order diffusion-wave equations KW - Complete Bernstein function KW - Completely monotone function Y1 - 2017 U6 - https://doi.org/10.1016/j.chaos.2017.05.001 SN - 0960-0779 SN - 1873-2887 VL - 102 SP - 210 EP - 217 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Stojkoski, Viktor A1 - Sandev, Trifce A1 - Kocarev, Ljupco A1 - Pal, Arnab T1 - Autocorrelation functions and ergodicity in diffusion with stochastic resetting JF - Journal of physics : A, Mathematical and theoretical N2 - Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift-diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting. KW - autocorrelations KW - ergodicity KW - diffusion KW - stochastic resetting Y1 - 2022 U6 - https://doi.org/10.1088/1751-8121/ac4ce9 SN - 1751-8113 SN - 1751-8121 VL - 55 IS - 10 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Padash, Amin A1 - Sandev, Trifce A1 - Kantz, Holger A1 - Metzler, Ralf A1 - Chechkin, Aleksei T1 - Asymmetric Levy flights are more efficient in random search JF - Fractal and fractional N2 - We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Levy flights by leveraging the Fokker-Planck equation with a delta-sink and an asymmetric space-fractional derivative operator with stable index alpha and asymmetry (skewness) parameter beta. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Levy flights with beta <= 0 (with a rightward bias) for short initial distances, while for beta>0 (with a leftward bias) Levy flights with alpha -> 1 are more efficient. When increasing the initial distance of the searcher to the target, Levy flight search (except for alpha=1 with beta=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Levy search compared to symmetric Levy flights at both short and long distances, and the effect is more pronounced for stable indices alpha close to unity. KW - asymmetric Levy flights KW - first-arrival density KW - search efficiency Y1 - 2022 U6 - https://doi.org/10.3390/fractalfract6050260 SN - 2504-3110 VL - 6 IS - 5 PB - MDPI CY - Basel ER -