The Obidi Actions
A central innovation of the Theory of Entropicity is the introduction of two variational principles — the Local Obidi Action (LOA) and the Spectral Obidi Action (SOA) — which govern the dynamics of the entropic field. Together they yield a Master Entropic Equation that plays a role analogous to Einstein’s field equations in general relativity【525666122072993†L115-L124】.
Local Obidi Action (LOA)
The LOA reframes entropy as the architect of reality by treating variations of entropy as fundamental degrees of freedom【61771916848198†L34-L41】. It integrates diverse geometric structures from information theory — the Fisher–Rao and Fubini–Study metrics and Amari–Čencov α‑connections — into a unified action principle【61771916848198†L42-L59】. In this formulation, the entropic metric becomes the stage on which physics unfolds: in classical limits it reduces to the Fisher–Rao form, quantum‑coherent regimes align with the Fubini–Study metric, and irreversible processes are captured by α‑connections【61771916848198†L63-L72】. Generalised entropies such as Tsallis, Rényi and Araki’s relative entropy appear as deformations of the entropic geometry【61771916848198†L72-L75】.
Spectral Obidi Action (SOA)
The SOA pushes the entropic programme further by translating Araki’s relative entropy into a global variational principle【61771916848198†L94-L101】. While the LOA is local in nature, the SOA introduces non‑local and global constraints through spectral data and modular operators. This brings together bosonic actions, fermionic bilinears and familiar geometric actions (Einstein–Hilbert and Yang–Mills) as projections of a common entropic action【922363920161669†L54-L60】. The result is a unified entropic–spectral variational principle in which gravity, matter and quantum structures arise from a single underlying field【922363920161669†L60-L64】.
Master Entropic Equation (MEE)
Variation of the Obidi Actions yields the Master Entropic Equation (MEE). This equation governs how entropy gradients evolve and couple to geometry, matter and information【525666122072993†L115-L124】. Solutions to the MEE are generally approached through iterative methods rather than closed forms【525666122072993†L128-L133】, reflecting the probabilistic and information‑theoretic nature of the entropic field.
Entropic Geodesics and Forces
From the MEE follow secondary structures: entropic geodesics, which describe natural paths in the entropic manifold, and an entropy potential equation that governs how entropic forces manifest【525666122072993†L115-L124】. In this picture, motion results from systems following entropic geodesics rather than trajectories in a fixed spacetime.