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：复杂结构往往从初始均匀或弱关联的状态中涌现。我们通过一个统一框架——融合半微观相空间动力学、输运几何、信息论与粗粒化有效建模——来解决这种有序性与熵增之间的表面张力。关键在于：熵依赖于描述层次——粗粒化空间场可能在结构形成时变得更有序，而全相空间描述则因壳层交叉、多流叠加以及速度自由度的激活而变得更加复杂。利用拉格朗日-欧拉输运映射，我们展示了密度放大如何由形变的雅可比行列式主导，以及各向异性坍缩如何源于一系列形变张量特征值的层级。长程相互作用或信息流编码于位移场中，使得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线相联系，并展示非线性输运与非局域耦合如何产生尺度耦合、高阶关联及非高斯性。进而，我们构建了朗道-金兹堡描述，其中种子各向异性的增长被解释为低有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，该框架表明非局域潮汐层次在中等过密度条件下已变得相关。尽管宇宙学结构形成是本文考虑的主要实现，但该框架旨在更广泛地作为一类输运、各向异性、非局域性与自组织为核心理念的系统的介观语言。