Boltzmann-distribution-equivalent for Lévy noise and how it leads to thermodynamically consistent epicatalysis

Nonequilibrium systems commonly exhibit Lévy noise. This means that the distribution for the size of the Brownian fluctuations has a "fat" power-law tail. Large Brownian kicks are then more common as compared to the ordinary Gaussian distribution. We consider a two-state system, i.e., two wells and a barrier in between. The barrier is sufficiently high for a barrier crossing to be a rare event. When the noise is Lévy, we do not get a Boltzmann distribution between the two wells. Instead we get a situation where the distribution between the two wells also depends on the height of the barrier that is in between. Ordinarily, a catalyst, by lowering the barrier between two states, speeds up the relaxation to an equilibrium, but does not change the equilibrium distribution. In an environment with Lévy noise, on the other hand, we have the possibility of epicatalysis, i.e., a catalyst effectively altering the distribution between two states through the changing of the barrier height. After deriving formulas to quantitatively describe this effect, we discuss how this idea may apply in nuclear reactors and in the biochemistry of a living cell.