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 - GEN A1 - Xu, Pengbo A1 - Zhou, Tian A1 - Metzler, Ralf A1 - Deng, Weihua T1 - Stochastic harmonic trapping of a Lévy walk: transport and first-passage dynamics under soft resetting strategies T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1262 KW - diffusion KW - anomalous diffusion KW - stochastic resetting KW - Levy walks Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-560402 SN - 1866-8372 SP - 1 EP - 28 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Xu, Pengbo A1 - Zhou, Tian A1 - Metzler, Ralf A1 - Deng, Weihua T1 - Stochastic harmonic trapping of a Lévy walk BT - transport and first-passage dynamics under soft resetting strategies JF - New journal of physics : the open-access journal for physics / Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics N2 - We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic. KW - diffusion KW - anomalous diffusion KW - stochastic resetting KW - Levy walks Y1 - 2022 U6 - https://doi.org/10.1088/1367-2630/ac5282 SN - 1367-2630 VL - 24 IS - 3 SP - 1 EP - 28 PB - Deutsche Physikalische Gesellschaft CY - Bad Honnef ER -