Semiclassical theory predicts stochastic ghosts scaling

Abstract

Slowing down occurs in dynamical systems close to bifurcations or phase transitions. The time length ( τ ) of ghost transients close to a saddle-node bifurcation in deterministic systems follows τ ∼ | ϵ − ϵ c | − 1 / 2 , ϵ being the bifurcation parameter and ϵ c its critical value. Recently, we numerically investigated how intrinsic noise affected the deterministic picture, finding a more complicated scaling law. We here provide a theoretical basis for this new law with two models of cooperation using a Wentzel–Kramers–Brillouin asymptotic approximation of the Master Equation. A study of the phase space of the Hamiltonian derived from the Hamilton–Jacobi equation shows that the statistically significant orbits (paths) reproduce the scaling function observed in the stochastic simulations. The flight times tied to these orbits underpin the scaling law of the stochastic system, and the same properties should extend in a universal way to all stochasticsystems whose associated Hamiltonian exhibits the same behaviour. Our approach allows to make useful theoretical predictions of transient times in stochastic systems close to a bifurcation.