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.

Figure 1

A double-well potential. Equation (1) describes the dynamics of an overdamped Brownian particle in the potential.

Figure 2

The piecewise-linear, double-well potential that we use as the basis for our analysis. The energy levels of the left well, the barrier, and the right well are $E_1, E_b$, and $E_2$, respectively. The widths of well 1 and well 2 are ${\delta }_1$ and ${\delta }_2$, respectively. $F_1$ and $F_2$ are the deterministic forces driving a particle toward the bottom. The transition rates are $k_{12}$ and $k_{21}$.

Figure 3

A setup to illustrate that epicatalysis in a closed environment is absurd as it leads to the possibility of a perpetuum mobile. A simple monomolecular reaction, $A_1\rightleftharpoons A_2$, is taking place in a sealed container. A weightless and frictionless trapdoor is repeatedly opened and closed by a demon. When the trapdoor is open, the reactants and products in the container are exposed to a catalyst. If the catalyst changes the Boltzmann distribution, then the system moves to a new equilibrium after every opening or closing of the trapdoor. As energy can be extracted from such equilibration, the setup is obviously nonsensical.

Figure 4

From Ref. [39]. The reaction coordinate for one of the studied DNA strands. The wells correspond to the folded and unfolded states.