Eur. Phys. J. B 62, 221-231 (2008)
https://doi.org/10.1140/epjb/e2008-00148-3

Finite temperature theory of metastable anharmonic potentials

Istituto Nazionale Fisica della Materia, Dipartimento di Fisica Universitá di Camerino, 62032 Camerino, Italy

Corresponding author: ^a marco.zoli@unicam.it

Received: 29 December 2007
Revised: 1 March 2008
Published online: 11 April 2008

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 Lamè 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.