RESUMEN
Despite having been studied for decades, first passage processes remain an active area of research. In this article, we examine a particle diffusing in an annulus with an inner absorbing boundary and an outer reflective boundary. We obtain analytic expressions for the joint distribution of the hitting time and the hitting angle in two and three dimensions. For certain configurations, we observe a "diffusive echo," i.e., two well-defined maxima in the first passage time distribution to a targeted position on the absorbing boundary. This effect, which results from the interplay between the starting location and the environmental constraints, may help to significantly increase the efficiency of the random search by generating a high, sustained flux to the targeted position over a short period. Finally, we examine the corresponding one-dimensional system for which there is no well-defined echo. In a confined system, the flux integrated over all target positions always displays a shoulder. This does not, however, guarantee the presence of an echo in the joint distribution.
RESUMEN
We introduce the Iris billiard that consists of a point particle enclosed by a unit circle around a central scattering ellipse of fixed elongation (defined as the ratio of the semi-major to the semi-minor axes). When the ellipse degenerates to a circle, the system is integrable; otherwise, it displays mixed dynamics. Poincaré sections are presented for different elongations. Recurrence plots are then applied to the long-term chaotic dynamics of trajectories launched from the unstable period-2 orbit along the semi-major axis, i.e., one that initially alternately collides with the ellipse and the circle. We obtain numerical evidence of a set of critical elongations at which the system undergoes a transition to global chaos. The transition is characterized by an endogenous escape event, E, which is the first time a trajectory launched from the unstable period-2 orbit misses the ellipse. The angle of escape, θesc, and the distance of the closest approach, dmin, of the escape event are studied and are shown to be exquisitely sensitive to the elongation. The survival probability that E has not occurred after n collisions is shown to follow an exponential distribution.
RESUMEN
Self-pulsating sessile drops are a striking example of the richness of far-from-equilibrium liquid/liquid systems. The complex dynamics of such systems is still not fully understood, and simple models are required to grasp the mechanisms at stake. In this article, we present a simple mass-spring mechanical model of the highly regular drop pulsations observed in Pimienta, V.; Brost, M.; Kovalchuk, N.; Bresch, S.; Steinbock, O. Complex shapes and dynamics of dissolving drops of dichloromethane. Angew. Chem., Int. Ed. 2011, 50, 10728-10731. We introduce an effective time-dependent spreading coefficient that sums up all of the forces (due to evaporation, solubilization, surfactant transfer, coffee ring effect, solutal and thermal Marangoni flows, drop elasticity, etc.) that pull or push the edge of a dichloromethane liquid lens, and we show how to account for the periodic rim breakup. The model is examined and compared against experimental observations. The spreading parts of the pulsations are very rapid and cannot be explained by a constant positive spreading coefficient or superspreading.