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

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 nu s$\nu _s$, describing the exploration of the phase space. As a particular case, this statistical oscillation frequency nu s$\nu _s$ can be defined for an arbitrary convex potential energy well. This value is compared here with the deterministic oscillation frequency nu d$\nu _d$ 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. For an arbitrary anharmonic oscillator, the deterministic frequency depending on the initial energy and the statistical frequency induced by thermal fluctuations are introduced. A mathematical relationship between these quantities is demonstrated, which is exact for both classical and relativistic systems. The conceptual link between this result and the transition states theory is explained, along with several examples of application.image