Finite Temperature Theory of Metastable Anharmonic Potentials

TL;DR

This paper develops a finite temperature quantum theory for decay rates in metastable anharmonic potentials, revealing how decay behavior transitions from quantum tunneling to classical activation as temperature increases.

Contribution

It introduces a semiclassical Euclidean path integral approach to analyze metastable decay at finite temperature, deriving classical paths and fluctuation spectra analytically.

Findings

Decay rate increases with temperature, especially near the crossover.

Soft modes dominate the temperature dependence of decay.

Longer lifetime in quartic potentials compared to cubic ones.

Abstract

The decay rate for a particle in a metastable cubic potential is investigated in the quantum regime by the Euclidean path integral method in semiclassical approximation. The imaginary time formalism allows one to monitor the system as a function of temperature. The family of classical paths, saddle points for the action, is derived in terms of Jacobian elliptic functions whose periodicity sets the energy-temperature correspondence. The period of the classical oscillations varies monotonically with the energy up to the sphaleron, pointing to a smooth crossover from the quantum to the activated regime. The softening of the quantum fluctuation spectrum is evaluated analytically by the theory of the functional determinants and computed at low $T$ up to the crossover. In particular, the negative eigenvalue, causing an imaginary contribution to the partition function, is studied in detail by solving the Lam\`{e} equation which governs the fluctuation spectrum. For a heavvy particle mass, the decay rate shows a remarkable temperature dependence mainly ascribable to a low lying soft mode and, approaching the crossover, it increases by a factor five over the predictions of the zero temperature theory. Just beyond the peak value, the classical Arrhenius behavior takes over. A similar trend is found studying the quartic metastable potential but the lifetime of the latter is longer by a factor ten than in a cubic potential with same parameters. Some formal analogies with noise-induced transitions in classically activated metastable systems are discussed.