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 - 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 - 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 - 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 - Petreska, Irina A1 - Sandev, Trifce A1 - Lenzi, Ervin Kaminski T1 - Comb-like geometric constraints leading to emergence of the time-fractional Schrödinger equation JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - This paper presents an overview over several examples, where the comb-like geometric constraints lead to emergence of the time-fractional Schrodinger 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 Schrodinger equation on a comb, the time-fractional equation for the Green's functions appears, due to these specific geometric constraints. KW - Comb model KW - time-fractional Schrödinger equation KW - Green’ s functions Y1 - 2021 U6 - https://doi.org/10.1142/S0217732321300056 SN - 0217-7323 SN - 1793-6632 VL - 36 IS - 14 PB - World Scientific CY - Singapore ER -