Space-time velocity correlation function for random walks

Space-time correlation functions constitute a useful instrument from the research toolkit of continuous-media and many-body physics. We adopt here this concept for single-particle random walks and demonstrate that the corresponding space-time velocity auto-correlation functions reveal correlations which extend in time much longer than estimated with the commonly employed temporal correlation functions. A generic feature of considered random-walk processes is an effect of velocity echo identified by the existence of time-dependent regions where most of the walkers are moving in the direction opposite to their initial motion. We discuss the relevance of the space-time velocity correlation functions for the experimental studies of cold atom dynamics in an optical potential and charge transport on micro- and nano-scales.Comment: Phys. Rev. Lett., in pres

L\'evy walks

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The L\'{e}vy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to L\'{e}vy walks, surveys their existing applications, including latest advances, and outlines further perspectives.Comment: 50 page

Levy walks with velocity fluctuations

The standard Levy walk is performed by a particle that moves ballistically between randomly occurring collisions, when the intercollision time is a random variable governed by a power-law distribution. During instantaneous collision events the particle randomly changes the direction of motion but maintains the same constant speed. We generalize the standard model to incorporate velocity fluctuations into the process. Two types of models are considered, namely, (i) with a walker changing the direction and absolute value of its velocity during collisions only, and (ii) with a walker whose velocity continuously fluctuates. We present full analytic evaluation of both models and emphasize the importance of initial conditions. We show that the type of the underlying Levy walk process can be identified by looking at the ballistic regions of the diffusion profiles. Our analytical results are corroborated by numerical simulations

Nuclear reactions in hot stellar matter and nuclear surface deformation

Cross-sections for capture reactions of charged particles in hot stellar matter turn out be increased by the quadrupole surface oscillations, if the corresponding phonon energies are of the order of the star temperature. The increase is studied in a model that combines barrier distribution induced by surface oscillations and tunneling. The capture of charged particles by nuclei with well-deformed ground-state is enhanced in stellar matter. It is found that the influence of quadrupole surface deformation on the nuclear reactions in stars grows, when mass and proton numbers in colliding nuclei increase.Comment: 12 pages, 10 figure