اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدTruncated Levy Random Walks and Generalized Cauchy Processes
74
0
0.0
(
0
)
تأليف
Ihor Lubashevsky
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
A continuous Markovian model for truncated Levy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Levy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method and the obtained numerical data were employed to calculate the geometric mean of the particle displacement during a certain time interval and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Levy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Levy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather then their characteristics individually.
قيم البحث
اقرأ أيضاً
Bernoulli random walks, a simple avalanche model, and a special branching process are essesntially identical. The identity gives alternative insights into the properties of these basic model sytems.
We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions $f(eta)$, including the case of Levy flights. We study the expected maximum ${mathbb E}[M_n]$ of bridge RWs, i.e., RWs starting an
d ending at the origin after $n$ steps. We obtain an exact analytical expression for ${mathbb E}[M_n]$ valid for any $n$ and jump distribution $f(eta)$, which we then analyze in the large $n$ limit up to second leading order term. For jump distributions whose Fourier transform behaves, for small $k$, as $hat f(k) sim 1 - |a, k|^mu$ with a Levy index $0<mu leq 2$ and an arbitrary length scale $a>0$, we find that, at leading order for large $n$, ${mathbb E}[M_n]sim a, h_1(mu), n^{1/mu}$. We obtain an explicit expression for the amplitude $h_1(mu)$ and find that it carries the signature of the bridge condition, being different from its counterpart for the free random walk. For $mu=2$, we find that the second leading order term is a constant, which, quite remarkably, is the same as its counterpart for the free RW. For generic $0< mu < 2$, this second leading order term is a growing function of $n$, which depends non-trivially on further details of $hat f (k)$, beyond the Levy index $mu$. Finally, we apply our results to compute the mean perimeter of the convex hull of the $2d$ Rouse polymer chain and of the $2d$ run-and-tumble particle, as well as to the computation of the survival probability in a bridge version of the well-known lamb-lion capture problem.
Comb geometry, constituted of a backbone and fingers, is one of the most simple paradigm of a two dimensional structure, where anomalous diffusion can be realized in the framework of Markov processes. However, the intrinsic properties of the structur
e can destroy this Markovian transport. These effects can be described by the memory and spatial kernels. In particular, the fractal structure of the fingers, which is controlled by the spatial kernel in both the real and the Fourier spaces, leads to the Levy processes (Levy flights) and superdiffusion. This generalization of the fractional diffusion is described by the Riesz space fractional derivative. In the framework of this generalized fractal comb model, Levy processes are considered, and exact solutions for the probability distribution functions are obtained in terms of the Fox $H$-function for a variety of the memory kernels, and the rate of the superdiffusive spreading is studied by calculating the fractional moments. For a special form of the memory kernels, we also observed a competition between long rests and long jumps. Finally, we considered the fractional structure of the fingers controlled by a Weierstrass function, which leads to the power-law kernel in the Fourier space. It is a special case, when the second moment exists for superdiffusion in this competition between long rests and long jumps.
The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. Accordin
g to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Levy walks, a class of stochastic processes with power law distributed step durations, which model complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.
The various types of generalized Cattaneo, called also telegraphers equation, are studied. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, textit{i.e.} are normalizable and n
on-negative on their domains. Analysis of the relevant mean squared displacements enables us to classify diffusion processes described by such obtained solutions and to identify them with either ordinary or anomalous super- or subdiffusion. To complete our study we analyse derivations of just considered examples the generalized Cattaneo equations using the continuous time random walk and the persistent random walk approaches.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
