The Obidi Actions as Variational Principles in the Theory of Entropicity (ToE)

The Obidi Actions are variational principles within John Onimisi Obidi's proposed Theory of Entropicity (ToE), a framework in theoretical physics that posits entropy as the fundamental field of nature. The theory presents two main expressions for the Obidi Action: a local, continuous field form, and a global spectral form.The two primary mathematical expressions are:1. The Local Obidi Action (Lagrangian Density Form)This is a local variational form that describes the dynamics of a continuous scalar entropy field \(S(\lambda )\). It is expressed as:\(A[S]=\int d^{4}x\sqrt{-g}\left((\partial S)(\partial S)-V(S)+J(\lambda )S\right)\text{\ or\ }A[S]=\int d\lambda \sqrt{-g}\left((\partial S)(\partial S)-V(S)+J(\lambda )S\right)\text{\ (as\ found\ in\ sources)}\)\(\mathbfit{A[S]}\) represents the action, which is a functional of the entropy field \(S\).\(\int \mathbfit{d}^{\mathbfit{4}}\mathbfit{x}\sqrt{\mathbfit{-g}}\) is the integral over spacetime, incorporating the metric tensor's determinant \(g\) to ensure general covariance.\(\mathbfit{(\partial S)(\partial S)}\) is a kinetic term involving the gradients of the entropy field, governing its propagation.\(\mathbfit{V(S)}\) is an entropic potential term that accounts for localized entropic condensation (mass).\(\mathbfit{J(\lambda )S}\) is a source term, possibly representing coupling to other fields or matter.2. The Spectral Obidi Action (Global Form)This formulation provides a global perspective on the dynamics, expressed through spectral operator algebra, and is stated as a trace-log operation over a differential operator \(\Delta \):\(S=-Tr\ln (\Delta )\text{\ or\ }S=-Tr\ln (\Delta ),\Delta =G[S]g[S]\text{\ (as\ found\ in\ sources)}\)\(\mathbfit{S}\) here represents the action in the spectral context (note: the symbol \(S\) is also used for the entropy field in the local action).\(\mathbfit{Tr}\ln \mathbfit{(\Delta )}\) is the trace logarithm of the differential operator \(\Delta \).\(\mathbfit{\Delta }\) is an operator that incorporates \(G[S]\), the equilibrium entropy geometry, and \(g[S]\), the matter-deformed entropy geometry. This form highlights the duality between local dynamics and global spectral coherence, and expanding it via heat-kernel regularization is claimed to reproduce the terms of the local action.These expressions serve as the foundation for the Master Entropic Equation (MEE), the ToE's equivalent of Einstein's field equations, which describes how entropy gradients influence geometry, matter, and information.For more details, you can consult the publications available on platforms like ResearchGate or the Theory of Entropicity website.