Renormalization-group theory of dynamics of first-order phase transitions in a field-driven scalar model

We show by a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a $\varphi^{3}$ theory that its instability fixed points with their associated instability exponents are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar $\phi^4$ model below its critical temperature near their instability points. Finite-time scaling and leading corrections to scaling are considered. We also show that the instability exponents of the first-order phase transitions equal those of the Yang--Lee edge singularity and employ the latter to improve our estimates of the former. The outcomes agree well with existent numerical results.