Physics:The Theory of Entropicity(ToE) Formulated by John Onimisi Obidi

Theory of Entropicity(ToE)

Theory of Entropicity (abbreviated ToE), first formulated and developed by John Onimisi Obidi, is a proposed unified framework that seeks to connect thermodynamics, information theory, and gravitation under a single entropic principle. It introduces the concept of entropicity as a fundamental physical quantity driving the evolution of macroscopic and microscopic systems.

Overview

The Theory of Entropicity postulates that:

1. All physical interactions can be recast as entropy exchange or redistribution.
2. Spacetime curvature, information flow, and thermodynamic irreversibility are manifestations of the same underlying entropic field.
3. The Einstein field equations, quantum statistical mechanics, and black‑hole thermodynamics are limiting cases of the more general entropic framework.

Mathematical Formulation

The core postulate is expressed via the generalized entropy functional:

[math]\displaystyle{ I[p] = \int d^4x\,\sqrt{-g}\, \Bigl[ -p\,\ln p + \lambda\,p\,g^{\mu\nu}(\nabla_{\mu}\ln p)(\nabla_{\nu}\ln p) \Bigr] }[/math]

Here:

• [math]\displaystyle{ p }[/math] represents the normalized probability density associated with microstates.
• [math]\displaystyle{ \lambda }[/math] is a coupling parameter linking informational and geometric terms.
• [math]\displaystyle{ g_{\mu\nu} }[/math] is the spacetime metric.

A related action functional involving auxiliary fields [math]\displaystyle{ S }[/math] and [math]\displaystyle{ \Lambda }[/math] is given by:

[math]\displaystyle{ J[p,S,\Lambda] = I[p] + \int d^4x\,\sqrt{-g}\,\Lambda(x)\,\bigl[S(x) + k_{B}\,\ln p(x)\bigr] }[/math]

Connection to Gravitational Theories

By variation with respect to the metric and information fields, the theory recovers:

• An effective stress‑energy tensor:

[math]\displaystyle{ T_{\mu\nu}^{(\mathrm{eff})} = T_{\mu\nu}^{(m)} + \frac{1}{8\pi G}(\nabla_{\mu}\nabla_{\nu} - g_{\mu\nu}\Box)f }[/math]

• A generalized Einstein equation with entropic source terms.

Physical Implications

Potential predictions and consequences of the ToE include:

• Emergent gravity from statistical mechanics of microscopic degrees of freedom.
• Corrections to black hole entropy:

[math]\displaystyle{ S_{\mathrm{gen}} = \frac{4G\hbar}{k_{B}}\,A + \alpha\frac{\hbar}{k_{B}} \int R\,dA + \dots }[/math]

• Entropic bounds on information processing rates in curved spacetime.

Applications

Research avenues:

• Cosmology: explaining dark energy as entropic vacuum pressure.
• Quantum gravity: linking holographic principle to probabilistic field dynamics.
• Statistical field theory: extending Boltzmann‑Gibbs statistics to curved manifolds.

See also

• Entropic gravity
• Bekenstein–Hawking entropy
• Information geometry