On the One‐Dimensional Transition State Theory and the Relation between Statistical and Deterministic Oscillation Frequencies of Anharmonic Energy Wells

Abstract

The transition state theory allows the development of approximated models useful to study the non‐equilibrium evolution of systems undergoing transformations between two states (e.g., chemical reactions). In a simplified 1D setting, the characteristic rate constants are typically written in terms of a temperature‐dependent characteristic oscillation frequency , describing the exploration of the phase space. As a particular case, this statistical oscillation frequency can be defined for an arbitrary convex potential energy well. This value is compared here with the deterministic oscillation frequency of the corresponding anharmonic oscillator. It is proved that there is a universal relationship between statistical and deterministic frequencies, which is the same for classical and relativistic mechanics. The independence of this relationship from the adopted physical laws gives it an interesting thermodynamic and pedagogical meaning. Several examples clarify the meaning of this relationship from both physical and mathematical viewpoints.

Statistical Mechanics Approaches for Studying Temperature and Rate Effects in Multistable Systems

“…As a matter of fact, when we study the out-of-equilibrium dynamics, the representation based on the sequence of basins and spin discrete variables is not sufficient since the relaxation times of the system depend on the energy barriers between the potential wells [61,98]. This is consistent, e.g., with the Kramers rate formula, originally formulated to deal with chemical reaction rates [99][100][101][102][103]. The consideration of the possible rate effects is particularly important when the stretching of the macromolecular chains, or other nanosystems, is performed with a large traction velocity [104,105].…”