Effective Scalar Field Dynamics from Coarse-Grained Kinetic Entropy Distributions

This framework introduces a phenomenological scalar field model in which spatial gra- dients of coarse-grained kinetic entropy in collisionless media act as the source of an emergent force.This force is defined as the gradient of a scalar potential, constructed by smoothing the entropy field with a finite-range kernel, typically of Yukawa type, to ensure localized interactions. The field arises from a Lagrangian formulation designed to cap- ture the spatial structure of entropy gradients. Unlike traditional long-range forces, this force is short-ranged, context-dependent, and vanishes in entropy-equilibrated regions. The formulation is compatible with the collisionless Boltzmann (Vlasov) equation, While the scalar field is introduced on phenomenological grounds, the framework is supported by numerical simulations and yields testable predictions relevant to early structure for- mation. As a speculative effective field theory, It provides a falsifiable lens to explore entropy-driven dynamics in astrophysical and cosmological settings.