Tunneling, the Equilibrium Constant, and Epicatalysis: A Second-Law Paradox?

Consider one particle (which could be an atom, molecule, Brownian particle, etc.) in thermodynamic equilibrium with a heat reservoir at temperature T. This particle can be in a low-potential-energy well L whose energy floor is EL and whose degeneracy is GL or in a higher- (or at least equally high) potential-energy well H whose energy floor is EH and whose degeneracy is GH. L and H are separated by a barrier B, which the particle can traverse. The Second Law of Thermodynamics asserts that the ratio of the probability of this particle being in H to that of it being in L, i.e., the equilibrium constant Keq corresponding to its dissemination between the two wells L and H, is in accordance with the Boltzmann (or canonical) distribution: Keq = (GH/GL)exp[−(EH – EL)/kT]. Given thermodynamic equilibrium this indeed always obtains if transits between L and H occur only via thermal excitation of our particle. But we show that despite thermodynamic equilibrium this does not obtain if transits between L and H occur both via thermal excitation and via tunneling. Implications concerning the Second Law of Thermodynamics are discussed. We then provide general remarks pertaining to catalysis versus epicatalysis, followed by concluding remarks.