自相似扩散中初始条件的记忆：重整化群视角
Memory Retention and the Classification of Renormalization-Group Fixed Points in Self-Similar Dynamics
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中文总结 AI 辅助
研究自相似扩散中初始条件的记忆，利用统一重整化群框架，通过修正密度依赖扩散模型揭示标度结构，初始条件信息渐近相关，Barenblatt方程是特例，为普遍性和反常标度提供新视角。
中文摘要 AI 辅助
普遍性通常与渐近行为变得独立于初始状态的细节相关。我们使用一个用于自相似动力学的统一重整化群（RG）框架表明，保留相关长度尺度自然会导致不同类别的保留记忆的不动点。一个修正的密度依赖扩散模型揭示了一种一般的标度结构，其中初始条件信息渐近相关，而Barenblatt方程是一个特殊情况。这些结果为普遍性和反常标度提供了新视角。
英文摘要
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 η^{α}F(ξ/η^{\b{eta}}), where η represents a surviving relevant scale, while the Barenblatt equation emerges as the special case η^{α}F(ξ) 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.
