We introduce a refined way to diffusely explore complex networks with stochastic resetting where ... more We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. ...
The investigation of fluctuations and random processes in complex systems and random environments... more The investigation of fluctuations and random processes in complex systems and random environments has been attracting much attention for years [...]
R. K. Singh, ∗ K. Górska, † and T. Sandev 4, 5, ‡ Department of Physics, Bar-Ilan University, Ram... more R. K. Singh, ∗ K. Górska, † and T. Sandev 4, 5, ‡ Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, Poland Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia (Dated: March 22, 2022)
Journal of Statistical Mechanics: Theory and Experiment
We consider a one-dimensional Brownian search in the presence of trapping. The diffusion equation... more We consider a one-dimensional Brownian search in the presence of trapping. The diffusion equation of the particle is represented by a memory kernel that enters the general waiting time probability density function. We find the general form of the first arrival time density, search reliability and efficiency and analyze several special cases of the memory kernel. We also analyze the Lévy search in the presence of trapping in cases of single and multiple targets, as well as combined Lévy–Brownian search strategies in case of a single target. The presented results are general and could be of interest for further investigation of different optimal search strategies, as well as in the animal foraging or spreading of contamination particles in the environment.
This paper presents an overview over several examples, where the comb-like geometric constraints ... more This paper presents an overview over several examples, where the comb-like geometric constraints lead to emergence of the time-fractional Schrödinger equation. Motion of a quantum object on a comb structure is modeled by a suitable modification of the kinetic energy operator, obtained by insertion of the Dirac delta function in the Laplacian. First, we consider motion of a free particle on two- and three-dimensional comb structures, and then we extend the study to the interacting cases. A general form of a nonlocal term, which describes the interactions of the particle with the medium, is included in the Hamiltonian, and later on, the cases of constant and Dirac delta potentials are analyzed. At the end, we discuss the case of non-integer dimensions, considering separately the case of fractal dimension between one and two, and the case of fractal dimension between two and three. All these examples show that even though we are starting with the standard time-dependent Schrödinger equ...
We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusio... more We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that MSD 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 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.
Communications in Nonlinear Science and Numerical Simulation
In this work we consider the first encounter problems between a fixed and/or mobile target A and ... more 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 probability (SP) of the target A on the both kinds of structures are analyzed 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.
We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a can... more We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for non-stationary and non-ergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains non-ergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the non-ergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.
Pengbo Xu, Weihua Deng, and Trifce Sandev School of Mathematics and Statistics, Gansu Key Laborat... more Pengbo Xu, Weihua Deng, and Trifce Sandev School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, P.O. Box 162, 1001 Skopje, Macedonia and Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
Journal of Statistical Mechanics: Theory and Experiment, 2020
We present an analytical treatment of anomalous diffusion in a three-dimensional comb (xyz-comb) ... more We present an analytical treatment of anomalous diffusion in a three-dimensional comb (xyz-comb) by using the Green's function approach. We derive exact analytical solutions for the propagators for an instantaneous point injection and natural boundary conditions. The marginal distributions for all three directions are obtained and the corresponding mean squared displacements are found. The analytical results are confirmed by numerical simulations in the framework of coupled Langevin equations. We also analyze a random search process on the xyz-comb, and analytical results on the first arrival time distribution, search reliability and efficiency are obtained. Results for multiple targets are presented as well. The developed approach can be useful for further studies 8 Author to whom any correspondence should be addressed.
Recent experimental findings on anomalous diffusion have demanded novel models that combine annea... more Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb-model is a simplified description of diffusion on percolation clusters, where the comb-like structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb-model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comb-like structure by a generalized fractal structure. Our hybrid comb-models thus represent a diffusion where different comb-like structures describe different quenched disorder mechanisms, and the fractional operators account for various annealed disorders mechanisms. We find exact solutions for the diffusion propagator and mean square displacement in terms of different memory kernels used for defining the fractional operators. Among other findings, we show that these models describe crossovers from subdiffusion to Brownian or confined diffusions, situations emerging in empirical results. These results reveal the critical role of interactions between geometrical restrictions and memory effects on modeling anomalous diffusion.
Communications in Nonlinear Science and Numerical Simulation, 2020
Abstract We analyze the diffusion of a system on a backbone structure, by considering the presenc... more Abstract We analyze the diffusion of a system on a backbone structure, by considering the presence of reaction terms. We start our analysis by considering an irreversible reaction process, where the particles are removed from the system. After, we consider the diffusion subjected to a reversible reaction process. The behavior for the system in this scenario depends on the relative rates of diffusion and reaction. For these cases, we obtain exact solutions in terms of the Green function approach and show a rich class of behavior which can be related to anomalous diffusion.
Physica A: Statistical Mechanics and its Applications, 2017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights We study a generalized Langevin equation with truncated Mittag-Leffler memory kernel. The diffusion process is described by help of the Mittag-Leffler functions. Model for description of lateral diffusion of lipids and proteins in cell membranes.
We analyze the generalized time-dependent Schrödinger equation for the force free case, as a gene... more We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.
We present a physical example, where a fractional (both in space and time) Schrödinger equation a... more We present a physical example, where a fractional (both in space and time) Schrödinger equation appears only as a formal effective description of diffusive wave transport in complex inhomogeneous media. This description is a result of the parabolic equation approximation that corresponds to the paraxial small angle approximation of the fractional Helmholtz equation. The obtained effective quantum dynamics is fractional in both space and time. As an example, Lévy flights in an infinite potential well are considered numerically. An analytical expression for the effective wave function of the quantum dynamics is obtained as well.
An analytical treatment of a generalized Langevin equation for a harmonic oscillator driven by ge... more An analytical treatment of a generalized Langevin equation for a harmonic oscillator driven by generalized noises of Mittag-Leffler type is presented. The overdamped limit (cases of high viscous damping) will be considered as a model of conformational dynamics of proteins. The behavior of the oscillator will be analyzed by calculation of the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. The results will be expressed through the Mittag-Leffler type functions. The case of a free particle will be recovered. From the asymptotic behavior of the relaxation functions an appearance of anomalous diffusion will be shown.
FGLEs are generalizations of the GLE where the integer order derivatives is substituted by fracti... more FGLEs are generalizations of the GLE where the integer order derivatives is substituted by fractional derivatives. Recently, some GLE models for a particle driven by single or multiple fractional Gaussian noise have been investigated in order to describe generalized diffusion processes, such as accelerating and retarding diffusion.
We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms... more We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms of the heterogeneous telegrapher's equation. In the diffusion limit of infinite-velocity propagation we recover the results for the heterogeneous diffusion process. The heterogeneous telegrapher's process exhibits a rich variety of diffusion regimes including hyperdiffusion, ballistic motion, superdiffusion, normal diffusion and subdiffusion, and different crossover dynamics characteristic for complex systems in which anomalous diffusion is observed. The anomalous diffusion exponent in the short time limit is twice the exponent in the long time limit, in accordance to the crossover dynamics from ballistic diffusion to normal diffusion in the standard telegrapher's process. We also analyze the finite-velocity heterogeneous diffusion process in presence of stochastic Poissonian resetting. We show that the system reaches a non-equilibrium stationary state. The transition to this non-equilibrium steady state is analysed in terms of the large deviation function.
We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random se... more We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a δ-sink and an asymmetric space-fractional derivative operator with stable index α and asymmetry (skewness) parameter β. 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 Lévy flights with β≤0 (with a rightward bias) for short initial distances, while for β>0 (with a leftward bias) Lévy flights with α→1 are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for α=1 with β=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essen...
We introduce a refined way to diffusely explore complex networks with stochastic resetting where ... more We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. ...
The investigation of fluctuations and random processes in complex systems and random environments... more The investigation of fluctuations and random processes in complex systems and random environments has been attracting much attention for years [...]
R. K. Singh, ∗ K. Górska, † and T. Sandev 4, 5, ‡ Department of Physics, Bar-Ilan University, Ram... more R. K. Singh, ∗ K. Górska, † and T. Sandev 4, 5, ‡ Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, Poland Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia (Dated: March 22, 2022)
Journal of Statistical Mechanics: Theory and Experiment
We consider a one-dimensional Brownian search in the presence of trapping. The diffusion equation... more We consider a one-dimensional Brownian search in the presence of trapping. The diffusion equation of the particle is represented by a memory kernel that enters the general waiting time probability density function. We find the general form of the first arrival time density, search reliability and efficiency and analyze several special cases of the memory kernel. We also analyze the Lévy search in the presence of trapping in cases of single and multiple targets, as well as combined Lévy–Brownian search strategies in case of a single target. The presented results are general and could be of interest for further investigation of different optimal search strategies, as well as in the animal foraging or spreading of contamination particles in the environment.
This paper presents an overview over several examples, where the comb-like geometric constraints ... more This paper presents an overview over several examples, where the comb-like geometric constraints lead to emergence of the time-fractional Schrödinger equation. Motion of a quantum object on a comb structure is modeled by a suitable modification of the kinetic energy operator, obtained by insertion of the Dirac delta function in the Laplacian. First, we consider motion of a free particle on two- and three-dimensional comb structures, and then we extend the study to the interacting cases. A general form of a nonlocal term, which describes the interactions of the particle with the medium, is included in the Hamiltonian, and later on, the cases of constant and Dirac delta potentials are analyzed. At the end, we discuss the case of non-integer dimensions, considering separately the case of fractal dimension between one and two, and the case of fractal dimension between two and three. All these examples show that even though we are starting with the standard time-dependent Schrödinger equ...
We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusio... more We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that MSD 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 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.
Communications in Nonlinear Science and Numerical Simulation
In this work we consider the first encounter problems between a fixed and/or mobile target A and ... more 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 probability (SP) of the target A on the both kinds of structures are analyzed 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.
We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a can... more We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for non-stationary and non-ergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains non-ergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the non-ergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.
Pengbo Xu, Weihua Deng, and Trifce Sandev School of Mathematics and Statistics, Gansu Key Laborat... more Pengbo Xu, Weihua Deng, and Trifce Sandev School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, P.O. Box 162, 1001 Skopje, Macedonia and Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
Journal of Statistical Mechanics: Theory and Experiment, 2020
We present an analytical treatment of anomalous diffusion in a three-dimensional comb (xyz-comb) ... more We present an analytical treatment of anomalous diffusion in a three-dimensional comb (xyz-comb) by using the Green's function approach. We derive exact analytical solutions for the propagators for an instantaneous point injection and natural boundary conditions. The marginal distributions for all three directions are obtained and the corresponding mean squared displacements are found. The analytical results are confirmed by numerical simulations in the framework of coupled Langevin equations. We also analyze a random search process on the xyz-comb, and analytical results on the first arrival time distribution, search reliability and efficiency are obtained. Results for multiple targets are presented as well. The developed approach can be useful for further studies 8 Author to whom any correspondence should be addressed.
Recent experimental findings on anomalous diffusion have demanded novel models that combine annea... more Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb-model is a simplified description of diffusion on percolation clusters, where the comb-like structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb-model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comb-like structure by a generalized fractal structure. Our hybrid comb-models thus represent a diffusion where different comb-like structures describe different quenched disorder mechanisms, and the fractional operators account for various annealed disorders mechanisms. We find exact solutions for the diffusion propagator and mean square displacement in terms of different memory kernels used for defining the fractional operators. Among other findings, we show that these models describe crossovers from subdiffusion to Brownian or confined diffusions, situations emerging in empirical results. These results reveal the critical role of interactions between geometrical restrictions and memory effects on modeling anomalous diffusion.
Communications in Nonlinear Science and Numerical Simulation, 2020
Abstract We analyze the diffusion of a system on a backbone structure, by considering the presenc... more Abstract We analyze the diffusion of a system on a backbone structure, by considering the presence of reaction terms. We start our analysis by considering an irreversible reaction process, where the particles are removed from the system. After, we consider the diffusion subjected to a reversible reaction process. The behavior for the system in this scenario depends on the relative rates of diffusion and reaction. For these cases, we obtain exact solutions in terms of the Green function approach and show a rich class of behavior which can be related to anomalous diffusion.
Physica A: Statistical Mechanics and its Applications, 2017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights We study a generalized Langevin equation with truncated Mittag-Leffler memory kernel. The diffusion process is described by help of the Mittag-Leffler functions. Model for description of lateral diffusion of lipids and proteins in cell membranes.
We analyze the generalized time-dependent Schrödinger equation for the force free case, as a gene... more We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.
We present a physical example, where a fractional (both in space and time) Schrödinger equation a... more We present a physical example, where a fractional (both in space and time) Schrödinger equation appears only as a formal effective description of diffusive wave transport in complex inhomogeneous media. This description is a result of the parabolic equation approximation that corresponds to the paraxial small angle approximation of the fractional Helmholtz equation. The obtained effective quantum dynamics is fractional in both space and time. As an example, Lévy flights in an infinite potential well are considered numerically. An analytical expression for the effective wave function of the quantum dynamics is obtained as well.
An analytical treatment of a generalized Langevin equation for a harmonic oscillator driven by ge... more An analytical treatment of a generalized Langevin equation for a harmonic oscillator driven by generalized noises of Mittag-Leffler type is presented. The overdamped limit (cases of high viscous damping) will be considered as a model of conformational dynamics of proteins. The behavior of the oscillator will be analyzed by calculation of the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. The results will be expressed through the Mittag-Leffler type functions. The case of a free particle will be recovered. From the asymptotic behavior of the relaxation functions an appearance of anomalous diffusion will be shown.
FGLEs are generalizations of the GLE where the integer order derivatives is substituted by fracti... more FGLEs are generalizations of the GLE where the integer order derivatives is substituted by fractional derivatives. Recently, some GLE models for a particle driven by single or multiple fractional Gaussian noise have been investigated in order to describe generalized diffusion processes, such as accelerating and retarding diffusion.
We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms... more We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms of the heterogeneous telegrapher's equation. In the diffusion limit of infinite-velocity propagation we recover the results for the heterogeneous diffusion process. The heterogeneous telegrapher's process exhibits a rich variety of diffusion regimes including hyperdiffusion, ballistic motion, superdiffusion, normal diffusion and subdiffusion, and different crossover dynamics characteristic for complex systems in which anomalous diffusion is observed. The anomalous diffusion exponent in the short time limit is twice the exponent in the long time limit, in accordance to the crossover dynamics from ballistic diffusion to normal diffusion in the standard telegrapher's process. We also analyze the finite-velocity heterogeneous diffusion process in presence of stochastic Poissonian resetting. We show that the system reaches a non-equilibrium stationary state. The transition to this non-equilibrium steady state is analysed in terms of the large deviation function.
We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random se... more We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a δ-sink and an asymmetric space-fractional derivative operator with stable index α and asymmetry (skewness) parameter β. 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 Lévy flights with β≤0 (with a rightward bias) for short initial distances, while for β>0 (with a leftward bias) Lévy flights with α→1 are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for α=1 with β=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essen...
Uploads
Papers by Trifce Sandev