🤖 AI Summary
This work addresses the apparent tension between structural order and the principle of entropy increase by elucidating how complex structures spontaneously emerge from homogeneous or weakly correlated initial states. Integrating semi-microscopic phase-space dynamics, transport geometry, information theory, and coarse-grained modeling, the study proposes a geometric framework based on Lagrangian–Eulerian transport maps and the spectral properties of deformation tensors to unify the description of nonlocality, anisotropy, and self-organization. Within this framework, a Landau–Ginzburg-type effective free energy theory is constructed, revealing mechanisms of density amplification and anisotropic collapse. Notably, the approach identifies nonlocal tidal effects as the dominant driver of structure formation already at moderate superdensities, successfully generating cosmic density fields that exhibit both non-Gaussian statistics and multiscale coupling.
📝 Abstract
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.