A Brief Introduction to the Obidi Field Equation (OFE) of the Theory of Entropicity (ToE): Its Universal Applications and Implications in Modern Theoretical Physics

A Brief Introduction to the Obidi Field Equations (OFE) of the Theory of Entropicity (ToE): Universal Applications and Implications in Modern Theoretical Physics

This is the Master Entropic Field Equation (also referred to as the trivial form of the Obidi Field Equations or OFE):

$$-2\chi^2 \nabla_\mu \!\left( e^{S/k_B}\nabla^\mu S \right) + \chi^2 e^{S/k_B} k_B (\nabla S)^2 - V'(S) + \lambda \frac{\delta R^{IG}}{\delta S}$$
$$+ \frac{1}{2}\frac{\partial \ln\!\big(-g(S)\big)}{\partial S} \left[ \chi^2 e^{S/k_B}(\nabla S)^2 - V(S) + \lambda R^{IG} \right] = 0$$

It represents the fundamental law of motion for the universe within the Theory of Entropicity (ToE). While Einstein’s Field Equations describe how matter curves space, this equation describes how the Entropy Field (S) generates both matter and space.

1. What is it mathematically?

It is the Euler-Lagrange equation derived from the Obidi Action Principle - OAP (I_{Semergent}). In physics, when you "vary" an action with respect to a field, you get the equations that tell you how that field behaves. This equation tells the entropic substrate how to "flow" and "structure" itself to create the physical world we see.

2. Breakdown of the Terms

The equation is a complex balance of several distinct "pressures" or "forces" within the entropy field:

• The Diffusion/Kinetic Term (−2χ^2∇μ(e^(S/kB) (∇^μ)S) +...): 

  This describes the "flow" of entropy. The presence of the Boltzmann constant (k_B) and the exponential e^{S/k_B} indicates that the rate at which entropy changes is scaled by the amount of entropy already present. This creates a feedback loop where high-entropy regions (like black holes) behave fundamentally differently than low-entropy regions.

• The Potential Term −V′(S):

  This represents the "energy density" or "cost" of having a certain level of entropy at a point in space. It acts like a restorative force, similar to how a spring wants to return to equilibrium.

• The Information-Geometric Term  λ(δRIG / δS):

  This is unique to ToE. It accounts for the curvature of the information space (the Fisher-Rao/Amari-Čencov geometry). It essentially says that the "logic" or "information structure" of the system exerts a physical force on the field.

• The Geometric Coupling Term  ½ [∂ ln√(−g(S)) / ∂S][...]:

  This is the "bridge" to General Relativity. It measures how the spacetime metric (g) changes as the entropy field changes. Because the metric depends on S, the very shape of space is "sucked into" the dynamics of the entropic field.

3. Why is this significant?

This equation has "no analogue in existing physics." Here is why:

1. Monism: In standard physics, you have one equation for the field (like Maxwell's equations) and another for the background (Einstein's). This equation combines them. The entropy field is the background.

2. Resolution of the "Bianconi Mismatch": As discussed earlier, Bianconi had to compare a body to a vacuum. In this equation, there is no vacuum—only varying densities of the S field. Gravity emerges naturally from the entropic gradients (∇S) in this equation, rather than from a comparison of two "unlike" metrics.

3. Non-Linearity: The term ½ [∂ ln√(−g(S)) / ∂S] makes this equation incredibly non-linear. It implies that at very high entropy scales (the early universe or black holes), the laws of physics might "shift" because the metric is so tightly coupled to the entropic state.

Summary

If Einstein's field equation is G₍μν₎ = 8π T₍μν₎ (Geometry = Matter), then the Obidi Field Equation is essentially saying:

Entropy Dynamics + Information Curvature = Geometry & Matter.

It is the mathematical expression of a universe that is not made of "stuff," but of dynamically evolving information arising from the entropic field.

Some Gravitational and Cosmological Applications of ToE

4. Weak-Field (Low-Entropy Gradient) Limit

Assume small entropic deviations, we can write:

$$S = S_0 + \delta S, \qquad |\nabla \delta S| \ll 1$$

Next, we expand the exponential term as follows:

$$e^{S/k_B} \approx e^{S_0/k_B} \left(1 + \frac{\delta S}{k_B}\right)$$

Now, keeping only the first-order terms, the Obidi Field Equation (OFE) readily reduces to:

$$\chi^2 e^{S_0/k_B} \nabla^2 \delta S - V''(S_0)\,\delta S + \lambda \frac{\delta^2 R^{IG}}{\delta S^2}\,\delta S = 0$$

This is a Poisson-type equation governing entropic fluctuations.

By identifying

$$\Phi \propto \delta S$$

we immediately recover Newtonian gravity as an emergent effect of entropy gradients in the Theory of Entropicity (ToE), rather than as a fundamental spacetime interaction.

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5. Cosmological (Homogeneous and Isotropic) Limit

Assume a spatially homogeneous entropy field:

$$S = S(t)$$

with Friedmann–Robertson–Walker (FRW) metric:

$$ds^2 = -dt^2 + a(t)^2 d\vec{x}^{\,2}$$

Then,

$$(\nabla S)^2 = -\dot{S}^2$$

and the Obidi Field Equation (OFE) reduces to:

$$2\chi^2 \frac{d}{dt}\!\left(e^{S/k_B}\dot{S}\right) + \chi^2 e^{S/k_B} k_B \dot{S}^2 + V'(S) - \lambda \frac{dR^{IG}}{dS} = 0$$

This equation governs cosmic evolution as an entropy-driven process.

Accelerated expansion arises naturally from increasing informational curvature, without introducing an ad-hoc cosmological constant.

References

1)

https://theoryofentropicity.blogspot.com/2026/01/a-brief-introduction-to-obidi-field.html