Semiclassical theory explains stochastic ghosts scaling

Abstract

Slowing down phenomena occur in both deterministic and stochastic dynamical systems at the vicinity of phase transitions or bifurcations. An example is found in systems exhibiting a saddle-node bifurcation, which undergo a dramatic time delay towards equilibrium. Specifically the duration of the transient, $\tau$, close to this bifurcation in deterministic systems follows scaling laws of the form $\tau \sim |\epsilon - \epsilon_c|^{-1/2}$, where $\epsilon$ is the bifurcation or control parameter, and $\epsilon_c$ its critical value. For systems undergoing a saddle-node bifurcation, the mechanism involves transients getting trapped by a so-called ghost. In a recent article we explored how intrinsic noise affected the deterministic picture. Extensive numerical simulations showed that, although scaling behaviour persisted in the presence of noise, the scaling law was more complicated than a simple power law. In order to gain deeper insight into this scaling behaviour, we resort to the WKB asymptotic approximation of the Master Equation. According to this approximation, the behaviour of the system is given as the weighted sum of \emph{trajectories} within the phase space of the Hamiltonian associated to the corresponding Hamilton-Jacobi equation. By analysing the flight time of the Hamilton equations, we show that the statistically significant paths follow a scaling function that exactly matches the one observed in the stochastic simulations. We therefore put forward that the properties of the flight times of the Hamiltonian system underpin the scaling law of the underlying stochastic system, and that the same properties should extend in a universal way to all stochastic systems whose associated Hamiltonian exhibits the same behaviour.