@article{PalyulinBlackburnLomholtetal.2019, author = {Palyulin, Vladimir V. and Blackburn, George and Lomholt, Michael A. and Watkins, Nicholas W. and Metzler, Ralf and Klages, Rainer and Chechkin, Aleksei V.}, title = {First passage and first hitting times of Levy flights and Levy walks}, series = {New journal of physics : the open-access journal for physics}, volume = {21}, journal = {New journal of physics : the open-access journal for physics}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab41bb}, pages = {23}, year = {2019}, abstract = {For both L{\´e}vy flight and L{\´e}vy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For L{\´e}vy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the L{\´e}vy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.}, language = {en} } @article{PadashChechkinDybiecetal.2019, author = {Padash, Amin and Chechkin, Aleksei V. and Dybiec, Bartlomiej and Pavlyukevich, Ilya and Shokri, Babak and Metzler, Ralf}, title = {First-passage properties of asymmetric Levy flights}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {45}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ab493e}, pages = {48}, year = {2019}, abstract = {L{\´e}vy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of L{\´e}vy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering L{\´e}vy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of L{\´e}vy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.}, language = {en} } @article{PalyulinChechkinKlagesetal.2016, author = {Palyulin, Vladimir V. and Chechkin, Aleksei V. and Klages, Rainer and Metzler, Ralf}, title = {Search reliability and search efficiency of combined Levy-Brownian motion: long relocations mingled with thorough local exploration}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {49}, journal = {Journal of physics : A, Mathematical and theoretical}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/49/39/394002}, pages = {2189 -- 2193}, year = {2016}, abstract = {A combined dynamics consisting of Brownian motion and Levy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and Levy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and Levy flights with stable exponent alpha < 1, by itself implying zero probability of hitting a point on a line, leads to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent a of the Levy flight component.}, language = {en} } @article{PavlyukevichLiXuetal.2015, author = {Pavlyukevich, Ilya and Li, Yongge and Xu, Yong and Chechkin, Aleksei V.}, title = {Directed transport induced by spatially modulated Levy flights}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {48}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {49}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/48/49/495004}, pages = {21}, year = {2015}, abstract = {In this paper we study the dynamics of a particle in a ratchet potential subject to multiplicative alpha-stable Levy noise, alpha is an element of(0, 2), in the limit of a noise amplitude epsilon -> 0. We compare the dynamics for Ito and Marcus multiplicative noises and obtain the explicit asymptotics of the escape time in the wells and transition probabilities between the wells. A detailed analysis of the noise-induced current is performed for the Seebeck ratchet with a weak multiplicative noise for alpha is an element of(0, 2].}, language = {en} } @article{ChechkinZaidLomholtetal.2013, author = {Chechkin, Aleksei V. and Zaid, I. M. and Lomholt, M. A. and Sokolov, Igor M. and Metzler, Ralf}, title = {Bulk-mediated surface diffusion on a cylinder in the fast exchange limit}, series = {Mathematical modelling of natural phenomena}, volume = {8}, journal = {Mathematical modelling of natural phenomena}, number = {2}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/20138208}, pages = {114 -- 126}, year = {2013}, abstract = {In various biological systems and small scale technological applications particles transiently bind to a cylindrical surface. Upon unbinding the particles diffuse in the vicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise to an effective surface translation, for which we here derive and discuss the dynamic equations, including additional surface diffusion. We discuss the time evolution of the number of surface-bound particles, the effective surface mean squared displacement, and the surface propagator. In particular, we observe sub- and superdiffusive regimes. A plateau of the surface mean-squared displacement reflects a stalling of the surface diffusion at longer times. Finally, the corresponding first passage problem for the cylindrical geometry is analysed.}, language = {en} }