  We consider the cosmological model of a self-interacting $\phi^4 - \phi^2$
quantum scalar field and extend our previous results, [3], on resonant
tunneling and consequent particle production, to the case of finite
temperature. Using the mathematical equivalence between, the Euclidean path
integral of a $\phi^4 - \phi^2$ quantum field theory (in the saddle point
approximation), on one hand, and the partition function of a 4-dimensional
ferromagnet (in the Ising model approximation), on the other, we derive the
following results. Tunneling is a first order phase transition. The creation of
metastable bound states of instanton-antinstanton pairs under the barrier
,(i.e. resonant tunneling), is the seed that gives rise to particle production.
Through the application of the Lee-Yang theorem for phase transitions, (as well
as demonstrating the underlying connection this has with the poles of the
S-matrix element in the quantum scattering theory), we show that the
fluctuations around the dominant escape paths of instantons (i.e. fluctuations
of the bubble wall) with momenta comparable to the scale curvature of the
bubble, drive the mechanism for resonant tunneling in false vacuum decay. We
also identify the temperature dependence of the parameters in the potential
term, (or equivalently, of the instanton bubbles), for a wide range of
temperatures Finally, we show that the picture of a dilute instanton
gas,remains valid even at finite temperatures, as this gas becomes more and
more dilute with the increase of the temperature. This suppression continues
until we reach the critical temperature, at which point there is only one
instanton left, with an infinitely thick wall.
