AI 中文总结
该研究构建了有限记忆生成输运几何的框架，通过依赖记忆的输运联络产生非对易性等，为不可逆拉格朗日输运提供纯运动学机制，数值模拟验证了相关预测，揭示因果历史是曲率等的微观起源。
AI 中文摘要
在传统输运理论中，记忆通常作为本构修正项被纳入，作用于已预先给定的运动学结构。本文构建了一个不同的框架，其中有限记忆本身生成输运的几何结构。通过从因果输运历史中重构形变，瞬时速度梯度被一个依赖于记忆的输运联络取代，该联络的有序演化会在输运历史空间中产生非对易性、曲率和和乐。研究表明，即使在时间周期、无旋流中，有限记忆也会对输运产生非零的几何贡献，为不可逆拉格朗日输运提供了一种纯运动学机制，且无需引入涡度、本构非线性、随机强迫或显式对称性破缺。本文引入了一个与输运表示无关的固有曲率不变量，用于测量输运历史生成的累积几何结构。该框架预测了由无量纲参数(ωτₘ)控制的普适标度，识别出一个特征记忆尺度(τ_c)，其将快速几何积累与渐近饱和区分开，并揭示了单调递减的记忆 susceptibility 以及全局凹的积累动力学。数值模拟证实了这些预测，表明几何不可逆性通过逐步的曲率积累而非共振驱动的放大产生。这些结果确立了有限记忆作为固有几何结构的生成者，而非仅仅是动力学演化的修饰者，揭示了因果历史是广泛非马尔可夫系统中曲率、和乐与不可逆输运的微观起源。
英文摘要
Memory is traditionally incorporated into transport theory as a constitutive correction acting on an already prescribed kinematic structure. Here we develop a different framework in which finite memory itself generates the geometry of transport. By reconstructing deformation from causal transport histories, the instantaneous velocity gradient is replaced by a memory-dependent transport connection whose ordered evolution gives rise to noncommutativity, curvature, and holonomy in transport-history space. We show that finite memory generates a nonvanishing geometric contribution to transport even in time-periodic, irrotational flows, providing a purely kinematic mechanism for irreversible Lagrangian transport without invoking vorticity, constitutive nonlinearities, stochastic forcing, or explicit symmetry breaking. We introduce an intrinsic curvature invariant, independent of the transport representation that measures the accumulated geometric structure generated by transport history. The framework predicts universal scaling governed by the dimensionless parameter (ωτ_m), identifies a characteristic memory scale (tau_c) separating rapid geometric accumulation from asymptotic saturation, and reveals a monotonically decreasing memory susceptibility with globally concave accumulation dynamics. Numerical simulations confirm these predictions and show that geometric irreversibility emerges through progressive curvature accumulation rather than resonance-driven amplification. These results establish finite memory as a generator of an intrinsic geometric structure rather than merely a modifier of dynamical evolution, revealing causal history as the microscopic origin of curvature, holonomy, and irreversible transport across a broad class of non-Markovian systems.
Comments49 pages, 11 figures. The submission includes 15 pages of Supplemental Material
