Random walk in a quasicontinuum

Anomalous spatial diffusion and multifractality in optical lattices

The transport of cold atoms in shallow optical lattices is characterized by slow, nonstationary momentum relaxation. We develop a projector operator method able to derive, in this case, a generalized Smoluchowski equation for the position variable. We show that this explicitly non-markovian equation can be written as a systematic expansion involving higher-order derivatives. We use the latter to compute arbitrary moments of the spatial distribution and analyze their multifractal properties.

Quasicontinuum Fokker-Planck equation

Building on the work [C. R. Doering, P. S. Hagan, and P. Rosenau, Phys. Rev. A 36, 985 (1987)] we present a regularized Fokker-Planck equation for discrete-state systems with more accurate short-time behavior than its standard, Kramers-Moyal counterpart. This regularization leads to a quasicontinuum Fokker-Planck equation with several key features: it preserves crucial aspects of state-space discreteness ordinarily lost in the standard Kramers-Moyal expansion; it is well posed, and it is more amenable to analytical and numerical tools currently available for continuum systems. In order to expose the basic idea underlying the regularization, it suffices for us to focus on two simple problems--the chemical reaction kinetics of a one-component system and a two-dimensional symmetric random walk on a square lattice. We then describe the path to applying this approach to more complex, discrete-state stochastic systems.

Non-Markovian Lévy diffusion in nonhomogeneous media

We study the diffusion equation with a position-dependent, power-law diffusion coefficient. The equation possesses the Riesz-Weyl fractional operator and includes a memory kernel. It is solved in the diffusion limit of small wave numbers. Two kernels are considered in detail: the exponential kernel, for which the problem resolves itself to the telegrapher's equation, and the power-law one. The resulting distributions have the form of the Lévy process for any kernel. The renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the system.

Chemotactic collapse and mesenchymal morphogenesis

We study the effect of chemotactic signaling among mesenchymal cells. We show that the particular physiology of the mesenchymal cells allows one-dimensional collapse in contrast to the case of bacteria, and that the mesenchymal morphogenesis represents thus a more complex type of pattern formation than those found in bacterial colonies. We compare our theoretical predictions with recent in vitro experiments.

Shock-wave formation in Rosenau's extended hydrodynamics

We study the extended hydrodynamics proposed by Rosenau [Phys. Rev. A 40, 7193 (1989)] in the context of a regularization of the Chapman-Enskog expansion. We are able to prove that shock waves appear in finite time in Rosenau's extended Burgers equation, and we discuss the physical implications of this fact and its connection with a possible extension of hydrodynamics to the short-wavelength domain.

Effects of internal fluctuations on the spreading of Hantavirus

We study the spread of Hantavirus over a host population of deer mice using a population dynamics model. We show that taking into account the internal fluctuations in the mouse population due to its discrete character strongly alters the behavior of the system. In addition to the familiar transition present in the deterministic model, the inclusion of internal fluctuations leads to the emergence of an additional deterministically hidden transition. We determine parameter values that lead to maximal propagation of the disease and discuss some implications for disease prevention policies.

Continuum approach to discreteness

We study analytically and numerically continuum models derived on the basis of Padé approximations and their effectiveness in modeling spatially discrete systems. We not only analyze features of the temporal dynamics that can be captured through these continuum approaches (e.g., shape oscillations, radiation effects, and trapping) but also point out ones that cannot be captured (such as Peierls-Nabarro barriers and Bloch oscillations). We analyze the role of such methods in providing an effective "homogenization" of spatially discrete, as well as of heterogeneous continuum equations. Finally, we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations, as well as to compare them with other semicontinuum approximations.