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Register a new userShould I stay or should I go? Zero-size jumps in random walks for Levy flights
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We study Markovian continuous-time random walk models for Levy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Levy-like motion for wild animals.
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Mass and energy injection throughout the lifetime of a star cluster contributes to the gas reservoir available for subsequent episodes of star formation and the feedback energy budget responsible for ejecting material from the cluster. In addition, mass processed in stellar interiors and ejected as winds has the potential to augment the abundance ratios of currently forming stars, or stars which form at a later time from a retained gas reservoir. Here we present hydrodynamical simulations that explore a wide range of cluster masses, compactnesses, metallicities and stellar population age combinations in order to determine the range of parameter space conducive to stellar wind retention or wind powered gas expulsion in star clusters. We discuss the effects of the stellar wind prescription on retention and expulsion effectiveness, using MESA stellar evolutionary models as a test bed for exploring how the amounts of wind retention/expulsion depend upon the amount of mixing between the winds from stars of different masses and ages. We conclude by summarizing some implications for gas retention and expulsion in a variety of compact ($sigma_v gtrsim 20 , {rm km s^{-1}}$) star clusters including young massive star clusters ($10^5 lesssim M/M_odot lesssim 10^7$, $age lesssim 500$~Myrs), intermediate age clusters ($10^5 lesssim M/M_odot lesssim 10^7$, $age approx 1-4$~Gyrs), and globular clusters ($10^5 lesssim M/M_odot lesssim 10^7$, $age gtrsim 10$~Gyrs).
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 and 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.
Among Markovian processes, the hallmark of Levy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Levy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very slow diffusion, logarithmic in time. These processes are path-dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the Central Limit Theorem is modified in this context, keeping the usual distinction between analytic and non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.
Let L(t) be a Levy flights process with a stability index alphain(0,2), and U be an external multi-well potential. A jump-diffusion Z satisfying a stochastic differential equation dZ(t)=-U(Z(t-))dt+sigma(t)dL(t) describes an evolution of a Levy particle of an `instant temperature sigma(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. sigma(t)approx t^{-theta} for some theta>0. We discover two different cooling regimes. If theta<1/alpha (slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as tto infty, which is concentrated at the potentials local minima. If theta>1/alpha (fast cooling) the Levy particle gets trapped in one of the potential wells.
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.
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