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

{{Physics}}

== Theory of Entropicity(ToE) ==
'''[https://doi.org/10.33774/coe-2025-n4n45 Theory of Entropicity]''' (abbreviated '''ToE'''),<ref name="obidi_addendum">Obidi, John Onimisi. ''A Critical Review of the Theory of Entropicity (ToE) on Original Contributions, Conceptual Innovations, and Pathways towards Enhanced Mathematical Rigor: An Addendum to the Discovery of New Laws of Conservation and Uncertainty''. Cambridge University.(2025-06-30). https://doi.org/10.33774/coe-2025-hmk6n</ref> first formulated and developed by [https://doi.org/10.33774/coe-2025-7lvwh John Onimisi Obidi],<ref name="obidi2025reconcile">Obidi, John Onimisi. ''Einstein and Bohr Finally Reconciled on Quantum Theory: The Theory of Entropicity (ToE) as the Unifying Resolution to the Problem of Quantum Measurement and Wave Function Collapse''. Cambridge University.  (14 April 2025). https://doi.org/10.33774/coe-2025-vrfrx</ref><ref>Obidi, John Onimisi (2025). Master Equation of the Theory of Entropicity (ToE). Encyclopedia. https://encyclopedia.pub/entry/58596</ref> 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:
# All physical interactions can be recast as entropy exchange or redistribution.
# Spacetime curvature, information flow, and thermodynamic irreversibility are manifestations of the same underlying entropic field.
# 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>
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>p</math> represents the normalized probability density associated with microstates.
* <math>\lambda</math> is a coupling parameter linking informational and geometric terms.
* <math>g_{\mu\nu}</math> is the spacetime metric.

A related action functional involving auxiliary fields <math>S</math> and <math>\Lambda</math> is given by:

<math>
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>
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>
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]]

== References ==
<references/>

[[Category:Theoretical physics]]
[[Category:Thermodynamics]]
[[Category:Quantum gravity]]

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