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Papers by T Sandev

Physical Review E

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 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.

Fractal and Fractional

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...

EPL (Europhysics Letters)

A Cattaneo equation for a comb structure is considered. We present a rigorous analysis of the obt... more A Cattaneo equation for a comb structure is considered. We present a rigorous analysis of the obtained fractional diffusion equation, and corresponding solutions for the probability distribution function are obtained in the form of the Fox H-function and its infinite series. The mean square displacement along the backbone is obtained as well in terms of the infinite series of the Fox H-function. The obtained solutions describe the transition from normal diffusion to subdiffusion, which results from the comb geometry.

Fractal and Fractional

The application of the fractional calculus in the mathematical modelling of relaxation processes ... more 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 susceptibili...

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 is presented. The overdamped limit (cases of high viscous damping) as a model of conformational dynamics of proteins is considered. The behavior of the oscillator is analyzed by calculation of the mean square displacement and normalized displacement correlation function. The results are expressed in terms of Mittag-Leffler type functions. Standard Brownian motion is a special case of the considered model. It is shown a good agreement with some experimental results. PACS: 02.50.−r, 05.40.−a, 05.10.Gg, 05.40.Ca

T. Sandev Research Center for Computer Science and Information Technologies, Macedonian Academy o... more T. Sandev 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 and Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia (Dated: June 11, 2020)

Journal of Physics A: Mathematical and Theoretical

We study the diffusive motion of a test particle in a two-dimensional comb structure consisting o... more We study the diffusive motion of a test particle in a two-dimensional comb structure consisting of a main backbone channel with continuously distributed side branches, in the presence of stochastic Markovian resetting to the initial position of the particle. We assume that the motion along the infinitely long branches is biased by a confining potential. The crossover to the steady state is quantified in terms of a large deviation function, which is derived for the first time for comb structures in present paper. We show that the relaxation region is demarcated by a nonlinear "light-cone" beyond which the system is evolving in time. We also investigate the first-passage times along the backbone and calculate the mean first-passage time and optimal resetting rate. Backbone diffusion and first-passage dynamics in a comb structure with confining branches.. . 2 [12, 13, 14, 15]. Sometimes for α > 2 the term hyperdiffusion is used [16, 17, 18]. We note that the case α = 1 in heterogeneous media does not necessarily imply that the process has a Gaussian probability density function (PDF), instead, for instance, exponential or stretched Gaussian forms may be observed [19, 20]. We also note that the MSD may also grow exponentially, for a multiplicative noise such as geometric Brownian motion or heterogeneous diffusion processes [21, 22, 23, 24], or logarithmically in strongly disordered environments [25, 26]. A by-now classical model for heterogeneous systems, popularised by Mandelbrot, are fractals, such as the Sierpiński gasket [27, 28, 29]. However, such ideal mathematical fractals are often insufficient to adequately describe real fractals such as networks of rivers, blood vessels, or nerve fibres, for which random fractals such as percolation clusters are more appropriate [27, 28, 29]. In many cases such structures have a characteristic backbone from which various branches emerge [28, 29]. A highly effective model addressing transport on such random loopless structures is a comb, in which infinite branches branch off the central backbone. The comb model was introduced to understand anomalous transport in percolation clusters [30, 31, 32, 33, 34]. Now, comb-like models are widely employed to describe various experimental applications. Comb-like structures are particularly important from a biophysical point of view as they provide a way to address transport along spiny dendrites [35, 36, 37], in which the transport properties crucially depend on the underlying geometry [38]. Similar approaches are being used in the modelling of river basins with their often very ramified geometry [39, 40]. In fact, long time retention data of tracers in water catchments reveal scaling exponents consistent with comb dynamics [41, 42]. Depending on the specific setting the geometry of comb structures effects both subdiffusion [43, 44, 45], including ultraslow diffusion [46], and superdiffusion [17, 47, 48]. The nontrivial nature of transport along a comb is discernible from the fact that motion along the branches results in a long-range memory for motion along the backbone which is generically responsible for the anomalous behaviour of transport [49]. In fact, the comb model can be regarded as the discrete version of a continuous time random walk, in which the return time distribution from a side branch to the main backbone effects power-law waiting times with diverging mean [33], and thus weak ergodicity breaking and ageing effects [50, 51]. Given the wide relevance and interesting properties of comb-like, loopless structures, these represent very powerful mathematical constructs to address motion in heterogeneous media. We here combine the analysis of diffusion on a comb with the idea of stochastic resetting. The concept of stochastic resetting (SR) has attracted considerable attention in non-equilibrium statistical physics [52]. In SR a moving particle is reset, i.e., returned to its initial location at regular or stochastic intervals. This results in a non-equilibrium steady-state even in cases in which the system under consideration does not relax to a steady-state in absence of any resets, see, e.g., free Brownian motion in d dimensions [53, 54]. The effect of resetting is particularly relevant for the first-passage properties of the motion of interest [55, 56, 57]. Indeed, even in generic cases in finite domains

$Research paper thumbnail of From continuous time random walks to the fractional Fokker-Planck equation$

Physical Review E, 2000

We generalize the continuous time random walk ͑CTRW͒ to include the effect of space dependent jum... more We generalize the continuous time random walk ͑CTRW͒ to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation ͑FFPE͒. This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.

$Research paper thumbnail of Generalized space–time fractional diffusion equation with composite fractional time derivative$

ABSTRACT We investigate the solution of space–time fractional diffusion equations with a generali... more 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 Fox’ H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grünwald-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 form \delta(x)t^{-\beta}/\Gamma(1-\beta) (\beta&gt;0).

Journal of Statistical Mechanics: Theory and Experiment

Journal of Physics A: Mathematical and Theoretical

We study Brownian motion in a confining potential under a constant-rate resetting to a reset posi... more 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).

Entropy

Classical option pricing schemes assume that the value of a financial asset follows a geometric B... more 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 ...

New Journal of Physics

The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour... more 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.

Fractional Calculus and Applied Analysis

We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous ... more 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.

$Research paper thumbnail of Generalized distributed order diffusion equations with composite time fractional derivative$

Computers & Mathematics with Applications, 2016

In this paper we investigate the solution of generalized distributed order diffusion equations wi... more In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier-Laplace transform method. We represent solutions in terms of infinite series in Fox H-functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obtained results are special cases of those presented in this paper.