Unified theory for the scaling of the crossover between strong and weak disorder behaviors of optimal paths and directed or undirected polymers in disordered media.

In this paper, we are concerned with the crossover between strong disorder (SD) and weak disorder (WD) behaviors in three well-known problems that involve minimal paths: directed polymers (directed paths with fixed starting point and length), optimal paths (undirected paths with a fixed end-to-end or spanning distance), and undirected polymers (undirected paths with a fixed starting point and length). We present a unified theoretical framework from which we can easily deduce the scaling of the crossover point of each problem in an arbitrary dimension. Our theory is based on the fact that the SD limit behavior of these systems is closely related to the corresponding percolation problem.

Unified scaling for the optimal path length in disordered lattices.

In recent decades, much attention has been focused on the topic of optimal paths in weighted networks due to its broad scientific interest and technological applications. In this work we revisit the problem of the optimal path between two points and focus on the role of the geometry (size and shape) of the embedding lattice, which has received very little attention. This role becomes crucial, for example, in the strong disorder (SD) limit, where the mean length of the optimal path (ℓ[over ¯]_{opt}) for a fixed end-to-end distance r diverges as the lattice size L increases.

Information in feedback ratchets.

Feedback control uses the state information of the system to actuate on it. The information used implies an effective entropy reduction of the controlled system, potentially increasing its performance. How to compute this entropy reduction has been formally shown for a general system and has been explicitly computed for spatially discrete systems.