Quantum-Statistical Metastability

Abstract

Consider a system rendered unstable by both quantum tunneling and thermodynamic fluctuation. The tunneling rate $\ensuremath{\Gamma}$, at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$, is related to the free energy $F$ by $\ensuremath{\Gamma}=(\frac{2}{\ensuremath{\hbar}})\mathrm{Im}F$. However, the classical escape rate is $\ensuremath{\Gamma}=(\frac{\ensuremath{\omega}\ensuremath{\beta}}{\ensuremath{\pi}})\mathrm{Im}F$, $\ensuremath{-}{\ensuremath{\omega}}^{2}$ being the negative eigenvalue at the saddle point. A general theory of metastability is constructed in which these formulas are true for temperatures, respectively, below and above $\frac{\ensuremath{\omega}\ensuremath{\hbar}}{2\ensuremath{\pi}}$ with a narrow transition region of $O({\ensuremath{\hbar}}^{\frac{3}{2}})$.