Free-energy structure of the entropy–extropy ratio on the probability simplex

Background:
Information measures such as Shannon entropy and its complementary quantity, extropy, play a central role in statistical physics, information theory, and eco-evolutionary dynamics. However, the geometric and thermodynamic interpretation of entropy–extropy relationships on the probability simplex remains largely unexplored.

Objective:
This work aims to establish a variational and thermodynamic structure for the normalised entropy–extropy ratio, and to determine whether it induces a dissipative gradient flow compatible with nonequilibrium relaxation.

Methods:
Using the Fisher–Shahshahani metric on the probability simplex, the Riemannian gradient flow is derived and expressed in replicator form. The relationship with Kullback–Leibler dissipation is analysed, and a generalised free energy functional is constructed to compare gradient dynamics.

Results:
The gradient flow is shown to be equivalent, up to time reparametrization, to the Fisher–Shahshahani gradient flow of the generalised free energy. The functional decreases monotonically along trajectories, the uniform distribution emerges as the unique equilibrium, and the evolution corresponds to nonequilibrium thermodynamic relaxation governed by the KL-divergence dissipation.

Conclusions:
The entropy–extropy ratio admits a free energy interpretation linking information geometry, replicator dynamics, and thermodynamic dissipation. This framework provides a structural measure of the balance between dispersion and organisation in probabilistic systems and suggests its relevance in quasi-stationary nonequilibrium regimes and to high dimensional limits.