Properties of diffusive systems near a saddle point: application to a quartic double well

Abstract

This paper aims at the analysis of diffusive properties of unidimensional mechanical systems in the environment of maxima and minima of the potential. It begins with a study of the properties of the singular solutions of the Hamilton–Jacobi–Yasue equation in the above-mentioned environment, in both strong or very small frictional forces. For the quartic symmetrical double-well potential, approximate solutions are found for local validity and the diffusion operator is then calculated in the limits of deep wells and small temperature, the regime being supposed to be aperiodic, with high or moderate values of frictional coefficient. This equation is proved to be nonunique. This operator is then reduced to second order by imposing suitable boundary conditions. Thus an appropriate eigenvalue equation is obtained to describe stationary states in the environment of extremal points of the potential energy function. The main interest of this work relies upon the fact that transition times between wells mainly depend upon fluctuations near the saddle point.