Lifschitz Tails for Random Schr\"{o}dinger Operator in Bernoulli Distributed Potentials

This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schr\"{o}dinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite

Phototactic Robot Tunable by Sensorial Delays

The presence of a delay between sensing and reacting to a signal can determine the long-term behavior of autonomous agents whose motion is intrinsically noisy. In a previous work [M. Mijalkov, A. McDaniel, J. Wehr, and G. Volpe, Phys. Rev. X 6, 011008 (2016)], we have shown that sensorial delay can alter the drift and the position probability distribution of an autonomous agent whose speed depends on the illumination intensity it measures. Here, using theory, simulations, and experiments with a phototactic robot, we generalize this effect to an agent for which both speed and rotational diffusion depend on the illumination intensity and are subject to two independent sensorial delays. We show that both the drift and the probability distribution are influenced by the presence of these sensorial delays. In particular, the radial drift may have positive as well as negative sign, and the position probability distribution peaks in different regions depending on the delay. Furthermore, the presence of multiple sensorial delays permits us to explore the role of the interaction between them.Comment: 13 pages, 6 figure

The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.Comment: This paper has been corrected from a previous version. Author Austin McDaniel has been added. Lemma 2 has been rewritten, Lemma 3 added, previous version's Lemma 3 moved to Lemma 4. 20 pages, 1 figur

Engineering sensorial delay to control phototaxis and emergent collective behaviors

Collective motions emerging from the interaction of autonomous mobile individuals play a key role in many phenomena, from the growth of bacterial colonies to the coordination of robotic swarms. For these collective behaviours to take hold, the individuals must be able to emit, sense and react to signals. When dealing with simple organisms and robots, these signals are necessarily very elementary, e.g. a cell might signal its presence by releasing chemicals and a robot by shining light. An additional challenge arises because the motion of the individuals is often noisy, e.g. the orientation of cells can be altered by Brownian motion and that of robots by an uneven terrain. Therefore, the emphasis is on achieving complex and tunable behaviors from simple autonomous agents communicating with each other in robust ways. Here, we show that the delay between sensing and reacting to a signal can determine the individual and collective long-term behavior of autonomous agents whose motion is intrinsically noisy. We experimentally demonstrate that the collective behaviour of a group of phototactic robots capable of emitting a radially decaying light field can be tuned from segregation to aggregation and clustering by controlling the delay with which they change their propulsion speed in response to the light intensity they measure. We track this transition to the underlying dynamics of this system, in particular, to the ratio between the robots' sensorial delay time and the characteristic time of the robots' random reorientation. Supported by numerics, we discuss how the same mechanism can be applied to control active agents, e.g. airborne drones, moving in a three-dimensional space.Comment: 8 pages, 5 figure