Memory Retention and the Classification of Renormalization-Group Fixed Points in Self-Similar Dynamics

Abstract

Universality is usually associated with the loss of information about initial conditions under repeated coarse-graining or renormalization-group (RG) transformations. We show that the unified RG framework for nonlinear partial differential equations can accommodate a broader class of asymptotic fixed points in which relevant scales remain encoded in the asymptotic state. By retaining relevant length scales within the RG description, asymptotic self-similar solutions, identified with RG fixed points, become functions of all scales that survive the RG flow. A modified density-dependent diffusion model yields a memory-retaining RG fixed point with scaling form {\eta}^{{\alpha}}F({\xi}/{\eta}^{\b{eta}}), where {\eta} represents a surviving relevant scale, while the Barenblatt equation emerges as the special case {\eta}^{{\alpha}}F({\xi}) with \b{eta}=0. These examples suggest that asymptotic RG fixed points may be classified according to whether relevant scales survive repeated RG transformations. We refer to the fixed points retaining such information as memory-retaining fixed points. Within this perspective, anomalous scaling can be interpreted as a manifestation of information retention. The resulting classification distinguishes fixed points according to how initial-condition information survives RG flow and remains encoded in the asymptotic fixed-point function itself. The results suggest that memory retention constitutes an organizing principle of RG fixed points complementary to the conventional classification by universality classes. In contrast with conventional Hamiltonian-based RG, the surviving information appears as an explicit variable of the asymptotic RG fixed-point function itself.