Memory of Initial Conditions in Self-Similar Diffusion: A Renormalization-Group Perspective

Universality is usually associated with asymptotic behavior becoming independent of details of the initial state. Using a unified renormalization-group (RG) framework for self-similar dynamics, we show that retaining relevant length scales leads naturally to distinct classes of memory-retaining fixed points. A modified density-dependent diffusion model reveals a general scaling structure in which initial-condition information remains asymptotically relevant, with the Barenblatt equation emerging as a special case. These results provide a new perspective on universality and anomalous scaling.