Entropy-Driven Adaptive Gradient Flow Systems with Mixed Wasserstein–Euclidean Geometry

Entropy-Driven Adaptive Gradient Flow Systems with Mixed Wasserstein–Euclidean Geometry: A Unified Operator Framework for Dissipative Neural and Thermodynamic Dynamicshttps://orcid.org/0009-0007-9967-5807 We introduce a rigorously structured class of entropy-driven dynamical systems defined on a mixed metric state space combining Wasserstein geometry for distributional variables and Euclidean geometry for finite-dimensional adaptive parameters. The system is formulated as a nonlinear gradient flow of a generalized free-energy functional E(ρ,R)\mathcal{E}(\rho, R)E(ρ,R), where ρ\rhoρ represents a probability density on a bounded domain and R∈RmR \in \mathbb{R}^mR∈Rm encodes internal adaptive degrees of freedom. The evolution is governed by a coupled operator system of the form ∂tρ=∇⋅(ρ∇δEδρ),R˙=−η∇RE,\partial_t \rho = \nabla \cdot \left(\rho \nabla \frac{\delta \mathcal{E}}{\delta \rho}\right), \quad \dot{R} = -\eta \nabla_R \mathcal{E},∂tρ=∇⋅(ρ∇δρδE),R˙=−η∇RE, which generates a nonlinear semigroup in a product space endowed with a Wasserstein–Euclidean hybrid metric. We establish a complete variational and operator-theoretic foundation of the model, including coercivity of the energy functional, convexity properties of its components, and displacement convexity in the sense of optimal transport geometry. Under these structural assumptions, the associated generator is shown to be m-accretive, yielding existence, uniqueness, and contractivity of weak solutions via the Crandall–Liggett framework. Furthermore, the system admits a Lyapunov dissipation structure, ensuring monotone decay of free energy along trajectories. This leads to the existence of absorbing sets and a compact global attractor governing the long-time asymptotic dynamics. Linearization around equilibrium yields a spectral operator with a well-defined gap structure, implying exponential stability and modal decomposition governed by dominant Perron–Frobenius-type eigenmodes. The resulting framework unifies entropy-transport dynamics, nonlinear operator semigroup theory, and adaptive internal state evolution into a single axiomatic structure. It provides a mathematically consistent foundation for modeling dissipative cognitive, physical, or thermodynamic systems where geometry, entropy, and adaptation co-evolve under a common variational principle. Keywords: nonlinear semigroups, Wasserstein gradient flow, entropy dissipation, m-accretive operators, global attractor, spectral gap, variational PDE, adaptive systems, thermodynamic geometry