Stationary states in two-dimensional systems driven by bivariate Lévy noises

Role of long jumps in Lévy noise-induced multimodality

Lévy noise is a paradigmatic noise used to describe out-of-equilibrium systems. Typically, properties of Lévy noise driven systems are very different from their Gaussian white noise driven counterparts. In particular, under action of Lévy noise, stationary states in single-well, super-harmonic, potentials are no longer unimodal. Typically, they are bimodal; however, for fine-tuned potentials, the number of modes can be further increased. The multimodality arises as a consequence of the competition between long displacements induced by the non-equilibrium stochastic driving and action of the deterministic force. Here, we explore robustness of bimodality in the quartic potential under action of the Lévy noise. We explore various scenarios of bounding long jumps and assess their ability to weaken and destroy multimodality. In general, we demonstrate that despite its robustness it is possible to destroy the bimodality, however it requires drastic reduction in the length of noise-induced jumps.

Underdamped stochastic harmonic oscillator driven by Lévy noise

We investigate the distribution of potential and kinetic energy in stationary states of the linearly damped stochastic oscillator driven by Lévy noises. In the long time limit distributions of kinetic and potential energies of the oscillator follow the power-law asymptotics and do not fulfill the equipartition theorem. The partition of the mechanical energy is controlled by the damping coefficient. In the limit of vanishing damping a stochastic analog of the equipartition theorem can be proposed, namely, the statistical properties of potential and kinetic energies attain distributions characterized by the same widths. For larger damping coefficient the larger fraction of energy is stored in its potential form. In the limit of very strong damping the contribution of kinetic energy becomes negligible. Finally, we demonstrate that the ratio of instantaneous kinetic and potential energies, which signifies departure from the mechanical energy equipartition, follows universal power-law asymptotics, regardless of the symmetric α-stable noise parameters. Altogether our investigations clearly indicate strongly nonequilibrium character of Lévy-stable fluctuations with the stability index α<2.

Emergence of Lévy Walks from Second-Order Stochastic Optimization

In natural foraging, many organisms seem to perform two different types of motile search: directed search (taxis) and random search. The former is observed when the environment provides cues to guide motion towards a target. The latter involves no apparent memory or information processing and can be mathematically modeled by random walks. We show that both types of search can be generated by a common mechanism in which Lévy flights or Lévy walks emerge from a second-order gradient-based search with noisy observations. No explicit switching mechanism is required-instead, continuous transitions between the directed and random motions emerge depending on the Hessian matrix of the cost function. For a wide range of scenarios, the Lévy tail index is α=1, consistent with previous observations in foraging organisms. These results suggest that adopting a second-order optimization method can be a useful strategy to combine efficient features of directed and random search.

Lévy flights and nonhomogenous memory effects: Relaxation to a stationary state

The non-Markovian stochastic dynamics involving Lévy flights and a potential in the form of a harmonic and nonlinear oscillator is discussed. The subordination technique is applied and the memory effects, which are nonhomogeneous, are taken into account by a position-dependent subordinator. In the nonlinear case, the asymptotic stationary states are found. The relaxation pattern to the stationary state is derived for the quadratic potential: the density decays like a linear combination of the Mittag-Leffler functions. It is demonstrated that in the latter case the density distribution satisfies a fractional Fokker-Planck equation. The densities for the nonlinear oscillator reveal a complex picture, qualitatively dependent on the potential strength, and the relaxation pattern is exponential at large time.

Non-Gaussian polymers described by alpha-stable chain statistics: Model, effective interactions in binary mixtures, and application to on-surface separation

The Gaussian chain model is the classical description of a polymeric chain, which provides analytical results regarding end-to-end distance, the distribution of segments around the mass center of a chain, coarse-grained interactions between two chains and effective interactions in binary mixtures. This hierarchy of results can be calculated thanks to the α stability of the Gaussian distribution. In this paper we show that it is possible to generalize the model of Gaussian chain to the entire class of α-stable distributions, obtaining the analogous hierarchy of results expressed by the analytical closed-form formulas in the Fourier space. This allows us to establish the α-stable chain model. We begin with reviewing the applications of Levy flights in the context of polymer sciences, which include: chains described by the heavy-tailed distributions of persistence length; polymers adsorbed to the surface; and the chains driven by a noise with power-law spatial correlations. Further, we derive the distribution of segments around the mass center of the α-stable chain and construct the coarse-grained interaction potential between two chains. These results are employed to discuss the model of binary mixture consisting of the α-stable chains. In what follows, we establish the spinodal decomposition condition generalized to the mixtures of the α-stable polymers. This condition is further applied to compare the on-surface phase separation of adsorbed polymers (which are known to be described with heavy-tailed statistics) with the phase separation condition in the bulk. Finally, we predict the four different scenarios of simultaneous mixing and demixing in the two- and three-dimensional systems.

Taming Lévy flights in confined crowded geometries

We study two-dimensional diffusive motion of a tracer particle in restricted, crowded anisotropic geometries. The underlying medium is formed from a monolayer of elongated molecules [Cieśla J. Chem. Phys. 140, 044706 (2014)] of known concentration. Within this mesh structure, a tracer molecule is allowed to perform a Cauchy random walk with uncorrelated steps. Our analysis shows that the presence of obstacles significantly influences the motion, which in an obstacle-free space would be of a superdiffusive type. At the same time, the selfdiffusive process reveals different anomalous properties, both at the level of a single trajectory realization and after the ensemble averaging. In particular, due to obstacles, the sample mean squared displacement asymptotically grows sublinearly in time, suggesting a non-Markov character of motion. Closer inspection of survival probabilities indicates, however, that the underlying diffusion is memoryless over long time scales despite a strong inhomogeneity of the motion induced by the orientational ordering.