Scaling crossovers in non-equilibrium critical dynamics

Dynamical universality plays a fundamental role in understanding the scaling properties of critical dynamics, including the absorbing phase transitions and physical aging. Although individual universality classes have been well-established and crossover theories exist, obtaining analytical solutions for multistage crossover behaviors remains a challenge. In this study, we propose a phenomenological approach to describe dynamic scaling crossovers in reaction-diffusion systems utilizing a Bernoulli differential equation with time-dependent reaction coefficients. We analytically derived five power-law crossover functions under a scaling hypothesis for spatial fluctuations. These solutions accurately describe the experimental observations of absorbing phase transitions in turbulent liquid crystals and reaction-diffusion crossovers in exciton–exciton recombination, providing quantitative information on the underlying mechanisms. These findings suggest the existence of fundamental scaling laws governing the transition between different dynamic states, motivating further studies using field-theoretic approaches and renormalization group analyses.