Semantic Gravitation II: Semantic Field Dynamics and Free Energy

Semantic Gravitation is an open research programme that explores whether “meaning” can be treated as a dynamical object with its own geometry, stability structure, and flow laws. This second paper develops the probabilistic layer of the programme by lifting semantic dynamics from individual states to probability measures. Starting from a state space S modelled as a separable Hilbert space equipped with a semantic potential, a semantic field is represented as a probability measure \rho on S with finite second moment, i.e. an element of the Wasserstein space \mathcal{P}_2(S). On this space, the paper formulates continuity- and Fokker–Planck-type evolution equations describing semantic field dynamics. A central result is the identification of a family of free-energy functionals as natural Lyapunov functions governing these dynamics. This establishes explicit connections to optimal transport, stochastic dynamics, and non-equilibrium statistical mechanics, and provides the probabilistic and thermodynamic completion of the Hilbert-space gradient framework introduced in Paper I.