Nondifferentiable potentials for nonequilibrium steady states
In the weak noise limit of Fokker-Planck models, the stationary probability density can be expressed in the form \(P\sim \exp (-1/\epsilon \phi (x))\), where \(\epsilon\) measures the strength of the noise. We show firstly that, although in general \(\phi\) is not differentiable, it is regular enough to allow its use as a Lyapunov function for the underlying deterministic dynamical system. This justifies the interpretation of \(\phi\) as a generalized thermodynamic potential for systems away from equilibrium. Secondly, we show that the nondifferentiability of \(\phi\) does not presuppose any kind of irregular behavior in the underlying deterministic system, nor the presence of several stationary points. We construct a simple model with one single attractor and trivial deterministic dynamics, which has a nondifferentiable quasipotential \(\phi\). We relate the nondifferentiability to the existence of caustics and shock lines in the associated variational problem.
weak noise limit of Fokker-Planck models
scientific article; zbMATH DE number 4096453
scientific article