Combined use of “two-step” and “field-theoretic” renormalization group theories for deterministic partial differential equations

The renormalization group (RG) theory, which revealed the origin of universality in critical phenomena, was further developed for non-thermodynamic systems, in particular, for partial differential equations (PDEs) without noise in two different ways: one by physicists, which is “field-theoretic”, and the other by mathematicians, which is “two-step”, while both approaches seem difficult to understand without the basic knowledge on each field. Here, we combine the two approaches with the aid of a stability analysis, in a way understandable for non-specialists, to bridge the gap between the fields. As a result, we show that the self-similar solution appears as a stable RG fixed point and is shared by a wide class of PDEs called “the universality class”, that the scale invariance is the origin of universality, and that a critical exponent is obtained from a divergent series, which is made finite by the renormalization technique in quantum field theory.