Stochastic foundations in nonlinear density-regulation growth

Waiting-time dependent non-equilibrium phase diagram of simple glass- and gel-forming liquids

Under numerous circumstances, many soft and hard materials are present in a puzzling wealth of non-equilibrium amorphous states, whose properties are not stationary and depend on preparation. They are often summarized in unconventional "phase diagrams" that exhibit new "phases" and/or "transitions" in which time, however, is an essential variable. This work proposes a solution to the problem of theoretically defining and predicting these non-equilibrium phases and their time-evolving phase diagrams, given the underlying molecular interactions. We demonstrate that these non-equilibrium phases and the corresponding non-stationary (i.e., aging) phase diagrams can indeed be defined and predicted using the kinetic perspective of a novel non-equilibrium statistical mechanical theory of irreversible processes. This is illustrated with the theoretical description of the transient process of dynamic arrest into non-equilibrium amorphous solid phases of an instantaneously quenched simple model fluid involving repulsive hard-sphere plus attractive square well pair interactions.

Demographic stochasticity and extinction in populations with Allee effect

We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with the Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with the strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics. We study the conditions for the validity of the WKB solution and analyze the boundary between the weak and strong Allee effect by comparing exact solutions with numerical simulations.