Complex structures often emerge from initially homogeneous or weakly correlated states. We address the apparent tension between this ordering and entropy growth through a unified framework combining semi-microscopic phase-space dynamics, transport geometry, information theory, and coarse-grained effective modeling. The key point is that entropy depends on the level of description: a coarse-grained spatial field may become more ordered as structure forms, even while the full phase-space description becomes more complex through shell crossing, multistreaming, and the activation of velocity degrees of freedom. Using a Lagrangian--Eulerian transport map, we show how density amplification is governed by the Jacobian of the deformation and how anisotropic collapse arises from the eigenvalues of a hierarchy of deformation tensors. Long-range interaction or information flow is encoded in the displacement field, so that nonlocality enters directly through transport. We connect this geometric description to a maximum-entropy Gaussian baseline and show how nonlinear transport and nonlocal coupling generate scale coupling, higher-order correlations, and non-Gaussianity. We then formulate a Landau--Ginzburg description in which the growth of seed anisotropies is interpreted as the activation of lower effective free-energy branches, providing a coarse-grained realization of self-organization. Applied to generated cosmological fields, this framework indicates that the nonlocal tidal level becomes relevant already at moderate overdensity. Although cosmological structure formation is the main realization considered here, the framework is intended more broadly as a mesoscopic language for systems in which transport, anisotropy, nonlocality, and self-organization are central.

翻译：复杂结构常常从初始均匀或弱关联状态涌现。我们通过一个统一框架来解决这种有序性与熵增之间的表面矛盾，该框架结合了半微观相空间动力学、输运几何、信息论以及粗粒化有效建模。关键在于熵依赖于描述层级：当结构形成时，粗粒化的空间场可能会变得更有序，而完整的相空间描述则因壳层交叉、多流以及速度自由度的激活而变得更加复杂。借助拉格朗日-欧拉输运映射，我们展示了密度增长如何受变形雅可比矩阵支配，以及各向异性坍缩如何源于一个变形张量层级结构的特征值。长程相互作用或信息流被编码在位移场中，因此非局域性直接通过输运引入。我们将这一几何描述与最大熵高斯基准联系起来，并展示了非线性输运和非局域耦合如何生成尺度耦合、高阶关联以及非高斯性。然后，我们构建了一个朗道-金兹堡描述，其中种子各向异性的增长被解释为低效自由能分支的激活，从而提供了自组织的粗粒化实现。应用于生成的宇宙学场时，这一框架表明，非局域潮汐层级在中等过密度时就已经变得重要。尽管宇宙结构形成是本框架考虑的主要实现，但该框架旨在更广泛地作为一种介观语言，适用于以输运、各向异性、非局域性和自组织为核心的系统。