David A. Kessler

TL;DR: Through detailed simulations of anisotropic fingering, it is demonstrated conclusively that no selection independent of the small-scale cutoff (macroscopic selection) occurs in this system, and ordered patterns are dynamically selected only for not too small surface tensions.

Abstract: A Bessel excursion is a Bessel process that begins at the origin and first returns there at some given time $$T$$ . We study the distribution of the area under such an excursion, which recently found application in the context of laser cooling. The area $$A$$ scales with the time as $$A \sim T^{3/2}$$ , independent of the dimension, $$d$$ , but the functional form of the distribution does depend on $$d$$ . We demonstrate that for $$d=1$$ , the distribution reduces as expected to the distribution for the area under a Brownian excursion, known as the Airy distribution, deriving a new expression for the Airy distribution in the process. We show that the distribution is symmetric in $$d-2$$ , with nonanalytic behavior at $$d=2$$ . We calculate the first and second moments of the distribution, as well as a particular fractional moment. We also analyze the analytic continuation from $$d<2$$ to $$d>2$$ . In the limit where $$d\rightarrow 4$$ from below, this analytically continued distribution is described by a one-sided Levy $$\alpha $$ -stable distribution with index $$2/3$$ and a scale factor proportional to $$[(4-d)T]^{3/2}$$ .

TL;DR: In this paper, an overdamped Brownian particle subject to an asymptotically flat potential with a trap of depth $U_0$ around the origin is considered, and the standard Boltzmann-Gibbs statistical framework and thermodynamic relations can still be applied through proper regularization.

TL;DR: In this paper, the authors presented the analytic solution of the Mode III steady-state crack in a square lattice with piecewise linear springs and Kelvin viscosity, and showed how the results simplify in the limit of large width.

TL;DR: In this article, the level-crossing densities along two orthogonal directions in an isotropic two-dimensional Gaussian random wave field are discussed for the real and the imaginary parts of the wave function, for the intensity, for phase, and for all the first and second-order spatial derivatives of these functions.