Levy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs-their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some longstanding puzzles around LWs.
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.
Pollen influx (number of pollen grains cm−2 year−1) can objectively reflect the dispersal and deposition features of pollen within a certain time and space, and is often used as a basis for the quantitative reconstruction of palaeovegetation; however, little is known about the features and mechanisms of vertical dispersal of pollen. Here we present the results from a 5 year (2006–2010) monitoring program using pollen traps placed at different heights from ground level up to 60 m and surface soil samples in a mixed coniferous and deciduous broad-leaved woodland in the Changbai mountains, northeastern China. The pollen percentages and pollen influx from the traps have very similar characteristics to the highest values for Betula, Fraxinus, Quercus and Pinus, among the tree taxa and Artemisia, Chenopodiaceae and Asteraceae among the herb taxa. Pollen influx values vary significantly with height and show major differences between three distinct layers, above-canopy (≥32 m), within the trunk layer (8 ≤ 32 m) and on the ground (0 m). These differences in pollen influx are explained by differences in (i) the air flows in each of these layers and (ii) the fall speed of pollen of the various taxa. We found that the pollen recorded on the ground surface is a good representation of the major part of the pollen transported in the trunk space of the woodland. Comparison of the pollen influx values with the theoretical, calculated “characteristic pollen source area” (CPSA) of 12 selected taxa indicates that the pollen deposited on the ground surface of the woodland is a fair representation with 85–90 % of the total pollen deposited at a wind speed of 2.4 m s−1 coming from within ca. 1–5 km for Pinus and Quercus, ca. 5–10 km for Ulmus, Tilia, Oleaceae and Betula, ca. 20–40 km for Fraxinus, Poaceae, Chenopodiaceae, Populus and Salix, and ca. 30–60 km for Artemisia; it is also a good representation with 90–98 % of the total pollen deposited coming from within 60 km at a wind speed of 2.4 m s−1, or 100 km at a wind speed: 6 m s−1, for the 12 selected taxa used in the CPSA calculation. Furthermore, comparison with the vegetation map of the area around the sampling site shows that the pollen deposited on the ground represents all plant communities which grow in the study area within 70 km radius of the sampling site. In this study, the pollen percentages obtained from the soil surface samples are significantly biased towards pollen taxa with good preservation due to thick and robust pollen walls. Therefore, if mosses are available instead, soil samples should be avoided for pollen studies, in particular for the study of pollen-vegetation relationships, the estimation of pollen productivities and quantitative reconstruction of past vegetation. The results also indicate that the existing model of pollen dispersal and deposition, Prentice’s model, provides a fair description of the actual pollen dispersal and deposition in this kind of woodland, which suggests that the application of the landscape reconstruction algorithm would be relevant for reconstruction of this type of woodland in the past.
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.