分支随机力学 I：薛定谔方程分支过程表示中的聚类与连通关联
Branching stochastic mechanics: Clustering and connected correlations within a branching-process representation of the Schrödinger equation
浏览论文内容
中文总结 AI 辅助
该研究将薛定谔方程表示为分支超过程，分析其连通关联的聚类特性，推导关联屏蔽长度，提出扩展玻恩密度与有限关联范围可共存于同一随机核，暗示类粒子组织。
中文摘要 AI 辅助
有限范围的关联是否能隐藏在扩展量子态的统计特性中？薛定谔-长泽变换将波函数表示为正向与反向的正扩散场，二者的乘积即为玻恩密度。我们将其提升为分支超过程Φ_F与Φ_B：扩散过程采样随机路径，而由玻姆/费希尔控制的分支则生成替代延续的谱系。基于规定的玻恩密度评估分支速率，其均值可精确重现薛定谔动力学。连通区域在受限空间中表现出超临界、临界与亚临界聚类，在自由空间中临界维度为d_c=2。稳态本征模保持扩展态；亚临界聚类则呈现出约化德布罗意尺度。随后，我们让分支速率对波动乘积Φ_FΦ_B作出响应。在倒易基下，场等于光滑参考项R加上中心波动项ψ_F、ψ_B，由此定义符号核C_FB(x,y)=E_ω[ψ_F(x)ψ_B(y)]。在反相关分支上，ρ_BSM(x)=-C_FB(x,x)为正配对密度。稳态恢复要求其对角项匹配玻恩分布，而非对角项的衰减则定义了关联范围。配对方程可拆分为集体与相对区域：谱抵消选择玻恩集体模，而主导密度涨落的抑制则选择反相关源通道。结合相对扩散率D_eff与正弛豫率μ_FB，关联的屏蔽长度为ξ_FB=√(D_eff/μ_FB)。因此，扩展的玻恩密度与有限关联范围可在一个随机核中共存，暗示类粒子结构的存在。
英文摘要
Can finite-range correlations hide in the statistics of an extended quantum state? The Schrödinger-Nagasawa transform represents the wave function by positive forward and backward diffusion fields whose product is the Born density. We promote them to branching superprocesses, \(Φ_F\) and \(Φ_B\): diffusion samples stochastic paths, whereas Bohm/Fisher-controlled branching generates genealogies of alternative continuations. With rates evaluated on the prescribed Born density, their means reproduce Schrödinger dynamics exactly. The connected sector exhibits supercritical, critical, and subcritical clustering in confinement and has critical dimension \(d_c=2\) in free space. Stationary eigenmodes remain extended; subcritical clusters acquire the reduced de~Broglie scale. We then let the branching rate respond to the fluctuating product \(Φ_FΦ_B\). In the reciprocal basis, the fields equal a smooth reference \(R\) plus centered fluctuations \(ψ_F,ψ_B\), defining the signed kernel \(C_{\rm FB}(x,y)=\mathbb E_ω[ψ_F(x)ψ_B(y)]\). On the anticorrelated branch, \(ρ_{\rm BSM}(x)=-C_{\rm FB}(x,x)\) is the positive organized connected weight. Stationary matching of this weight to the Born profile is imposed as a closure condition, while off-diagonal decay defines the correlation range. The pair equation splits into collective and relative sectors: spectral cancellation selects the Born collective mode, while suppression of the leading density fluctuation selects the anticorrelated source channel. With relative diffusivity \(D_{\rm eff}\) and positive relaxation rate \(μ_{\rm FB}\), correlations have screening length \(ξ_{\rm FB}=\sqrt{D_{\rm eff}/μ_{\rm FB}}\). Within this reciprocal closure, an extended Born-shaped connected weight and a finite correlation range can coexist in one stochastic kernel.
发表机构
- Université Paris-Saclay, CEA Institut de Recherche sur les Lois Fondamentales de l’Univers（巴黎萨克雷大学，法国原子能与替代能源委员会基础规律研究学院）
- Université Paris-Saclay, Ecole Normale Supérieure Paris-Saclay（巴黎萨克雷大学，巴黎萨克雷高等师范学院）
机构由 AI 辅助整理，请以论文原文为准。
