An information-geometric framework for nonlocal continuum mechanics

Abstract

We develop a geometric and variational framework for nonlocal continuum mechanics in which nonlocal interaction kernels are reinterpreted as probabilistic transition structures on the configuration space, placing the theory within the setting of statistical manifolds equipped with the Fisher–Rao metric. After normalizing an interaction kernel into a probability transition density, we define the associated Fisher information metric and derive a class of information-geometric nonlocal evolution equations from a free-energy variational principle. The interaction-density flow is the Fisher–Rao (replicator) gradient flow of the free energy; we prove that it preserves positivity and that, for a fixed displacement field, it dissipates the free energy at a rate equal to the squared Fisher norm of the free-energy gradient. We give the corresponding energy balance for the coupled inertial system, in which dissipation originates solely in the interaction flow. We show that classical continuum mechanics emerges as the singular Dirac-delta limit of the information-geometric theory—formally, and rigorously under stated regularity—with rate O ( ℓ 2 ) , and we establish that the probabilistic normalization of the kernel is exactly the spectral-positivity condition guaranteeing real wave frequencies and well-posed elastodynamics; the variance of the interaction density sets the effective elastic modulus, and its degradation governs loss of hyperbolicity, strain localization, and coherent energy focusing. We provide a coarse-graining derivation that gives the kernel and the Gibbs/entropic closure a microscopic origin and the statistical manifold a physical meaning (microstructural descriptors and their identifiability via the Cramér–Rao bound). The framework is developed in detail, and benchmarked numerically, for nonlocal gradient damage and peridynamics; it is specialized formally to anomalous diffusion, phase-field fracture, and viscoelasticity, where established models are shown to occupy identifiable points of the construction, and it is connected, as a proposal, to optimal-transport geometry (the Wasserstein–Fisher–Rao/Hellinger–Kantorovich metrics) and to Fisher-regularized operator learning, in each case carefully positioned relative to existing work. Numerical experiments, including a quantitative benchmark of the damage model against implicit-gradient and phase-field references under mesh refinement, illustrate the theoretical predictions. We are explicit throughout about which results are proved, which are formal, and which remain open.