title: Topography of chance.
abstract: We present a model of multiplicative Langevin dynamics that is based on two foundations: the Langevin equation and the notion of multiplicative evolution. The model is a nonlinear mechanism transforming a white-noise input to a dynamic-equilibrium output, using a single control: an underlying convex U-shaped potential function. The output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories. The model generates each admissible combination of the output's shape and randomness in a universal and robust fashion. Moreover, practically all the probability distributions that are supported on the positive half-line, and that are commonly encountered and applied across the sciences, can be reverse engineered by this model. Hence, this model is a universal equilibrium mechanism, in the context of multiplicative dynamics, for the robust generation of "chance": the model's output. In turn, the properties of the produced "chance," the output's shape and randomness, are determined with mathematical precision by the control's landscape, its topography. Thus, a topographic map of chance is established. As a particular application, probability distributions with power-law tails are shown to be universally and robustly generated by controls on the "edge of convexity": convex U-shaped potential functions with asymptotically linear wings.
authors: School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel.
authors: Iddo I
authors: II
authors: Eliazar
authors: Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA and Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA.
authors: Morrel H
authors: MH
authors: Cohen