Effect of dissipation on the decay-rate phase transition

A general condition for sharp transition of decay rate from quantum to thermal regimes is derived in dissipative tunneling models when position-dependent mass is involved. It is shown that the effect of dissipation in general changes the order of the phase transition. Especially, for the models with constant mass the Ohmic dissipation enlarges the range of parameters for first-order phase transitions. In the case of second-order phase transition the Ohmic dissipation suppresses the decay rate near the transition temperature ${(T}_c).\mathrm{}$ For the super-Ohmic case the dissipation yields an opposite effect to the Ohmic dissipation within exponential approximation.