A Simplistic Expansion and Solution of a Trivial Form of the Obidi Field Equations (OFE) of the Theory of Entropicity (ToE)

Last Updated: Saturday, January 17, 2026

A Simplified Form of the Obidi Field Equation of the Theory of Entropicity (ToE)

1. Introduction to the Obidi Field Equations in the Theory of Entropicity

The Theory of Entropicity (ToE) treats entropy not as a secondary, statistical descriptor of disorder, but as the fundamental field from which geometry, dynamics, and physical law emerge. In this framework, the dynamics of the entropic field are governed by a variational principle encoded in the Obidi Action, whose Euler–Lagrange equations define the Obidi Field Equations (OFE).

In this paper, we construct and analyze a deliberately simplified, “trivial” form of the OFE. The goal is not to capture the full complexity of ToE, but to make the structure of the equations transparent, to show that they are mathematically well‑posed, and to demonstrate explicitly how they arise from an action functional. We then derive the corresponding nonlinear partial differential equation (PDE) and interpret its terms in the context of entropic dynamics.

The analysis proceeds by specifying a simple entropic action functional, computing its variational derivatives, and obtaining the resulting field equation for a scalar entropic field $S(x,t)$ defined on a spacetime manifold.

2. Geometric and Variational Setup for the Entropic Field

We consider a spacetime manifold $(\mathcal{M},g)$, where $g$ is a Lorentzian or Riemannian metric, and $d{\mu }_g=\sqrt{\mathrm{\mid }g\mathrm{\mid }}{d}^{n}x$ is the associated volume form. The fundamental dynamical quantity is a scalar field $S:\mathcal{M}\to \mathbb{R}$, interpreted as an entropic field.

The generalized Obidi Action for this simplified model is taken to be of the form

$$A_{\text{ToE}}[S]={\int }_{\mathcal{M}}\mathcal{L}(S,\mathrm{\nabla }S,{\mathrm{\partial }}_{t}S)d{\mu }_g,$$

where $\mathcal{L}$ is a Lagrangian density depending on the field $S$, its spatial gradient $\mathrm{\nabla }S$, and its time derivative ${\mathrm{\partial }}_{t}S$. The Obidi Field Equations (OFE) are then obtained by requiring that $A_{\text{ToE}}[S]$ be stationary under variations of $S$, i.e.

$$\delta A_{\text{ToE}}[S]=0.$$

To make the structure explicit, we decompose the Lagrangian density into several contributions:

$$\mathcal{L}={\mathcal{L}}_{\text{geom}}+{\mathcal{L}}_{\text{ent}}+{\mathcal{L}}_{\text{grad}}+{\mathcal{L}}_{\text{nonlin}}+{\mathcal{L}}_{\text{causal}}\mathrm{.}$$

Each term corresponds to a distinct physical or geometric aspect of the entropic dynamics.

3. Constructing a Trivial Form of the Obidi Action

3.1 Geometric diffusion term

We first introduce a geometric diffusion term that penalizes spatial variations of the entropic field. This is modeled by a quadratic gradient energy:

$${\mathcal{L}}_{\text{geom}}={\mu }_g\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2,$$

where ${\mu }_g>0$ is a constant and

$$\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2=g^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}S\mathrm{.}$$

The corresponding contribution to the action is

$$S_{\text{geom}}[S]={\int }_{\mathcal{M}}{\mu }_g{g}^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}Sd{\mu }_g\mathrm{.}$$

3.2 Entropy potential term

Next, we introduce a simple local entropy potential of the form

$${\mathcal{L}}_{\text{ent}}=\beta S^q,$$

where $\beta \in \mathbb{R}$ and $q\in \mathbb{R}$ are constants. This yields the action

$$S_{\text{ent}}[S]={\int }_{\mathcal{M}}\beta S^{q}d{\mu }_g\mathrm{.}$$

3.3 Linear gradient term

We also include a standard kinetic‑type gradient term

$$\mathrm{\Psi }(\mathrm{\nabla }S)=\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2,$$

with associated Lagrangian density

$${\mathcal{L}}_{\text{grad}}=\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2,$$

and action

$$S_{\mathrm{\Psi }}[S]={\int }_{\mathcal{M}}\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{2}d{\mu }_g\mathrm{.}$$

3.4 Nonlinear gradient term

To capture nonlinear entropic effects, we add a p‑Laplacian‑type term

$$\mathcal{F}(\mathrm{\nabla }S)=\lambda \mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^p,$$

where $\lambda \in \mathbb{R}$ and $p>1$. The corresponding Lagrangian density is

$${\mathcal{L}}_{\text{nonlin}}=\lambda \mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^p,$$

and the action is

$$S_{\mathcal{F}}[S]={\int }_{\mathcal{M}}\lambda \mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p}d{\mu }_g\mathrm{.}$$

3.5 Causal constraint term

Finally, we impose a finite‑rate bound on the temporal evolution of the entropic field. Let

$$z=\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S,$$

where $c_S>0$ is a characteristic entropic propagation speed, and define a quadratic penalty functional

$$\mathrm{\Theta }(z)=\frac{1}{2}{z}^2\mathrm{.}$$

The corresponding Lagrangian density is

$${\mathcal{L}}_{\text{causal}}=\mathrm{\Theta }(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S),$$

and the action is

$$S_{\text{causal}}[S]={\int }_{\mathcal{M}}\mathrm{\Theta }(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S)d{\mu }_g\mathrm{.}$$

3.6 The full trivial Obidi Action

Collecting all contributions, the simplified Obidi Action for the entropic field is

$$A_{\text{ToE}}[S]={\int }_{\mathcal{M}}[{\mu }_g\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2+\beta S^q+\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2+\lambda \mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^p+\mathrm{\Theta }(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S)]d{\mu }_g\mathrm{.}$$

This is a concrete, tractable realization of a trivial form of the Obidi Action, suitable for explicit variation and PDE derivation.

4. Variational Derivation of the Trivial Obidi Field Equations

To obtain the field equations, we compute the first variation of the action with respect to $S$. Let $S\mapsto S+ϵ\delta S$, with $\delta S$ an arbitrary smooth variation of compact support. The stationarity condition

$$\delta A_{\text{ToE}}[S]=0$$

for all $\delta S$ yields the Euler–Lagrange equation.

We now compute the variation of each term.

4.1 Variation of the geometric diffusion term

The geometric term is

$$S_{\text{geom}}[S]={\int }_{\mathcal{M}}{\mu }_g{g}^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}Sd{\mu }_g\mathrm{.}$$

Varying $S$ gives

$$\delta S_{\text{geom}}={\int }_{\mathcal{M}}2{\mu }_g{g}^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_j(\delta S)d{\mu }_g\mathrm{.}$$

Integrating by parts and assuming suitable boundary conditions, we obtain

$$\delta S_{\text{geom}}=-2{\mu }_g{\int }_{\mathcal{M}}({\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S)\delta Sd{\mu }_g,$$

where ${\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S={\text{div}}_g(\mathrm{\nabla }S)$ is the Laplace–Beltrami operator. Thus, this term contributes

$$-2{\mu }_g{\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S$$

to the field equation.

4.2 Variation of the entropy potential term

The entropy potential term is

$$S_{\text{ent}}[S]={\int }_{\mathcal{M}}\beta S^{q}d{\mu }_g\mathrm{.}$$

Varying $S$ yields

$$\delta S_{\text{ent}}={\int }_{\mathcal{M}}\beta q{S}^{q-1}\delta Sd{\mu }_g\mathrm{.}$$

Thus, this term contributes

$$\frac{\mathrm{\partial }{S}_{\text{ent}}}{\mathrm{\partial }S}=\beta q{S}^{q-1}$$

to the field equation.

4.3 Variation of the linear gradient term

The linear gradient term is

$$S_{\mathrm{\Psi }}[S]={\int }_{\mathcal{M}}\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{2}d{\mu }_g\mathrm{.}$$

Varying $S$ gives

$$\delta S_{\mathrm{\Psi }}={\int }_{\mathcal{M}}{g}^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_j(\delta S)d{\mu }_g\mathrm{.}$$

Integrating by parts, we obtain

$$\delta S_{\mathrm{\Psi }}=-{\int }_{\mathcal{M}}({\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S)\delta Sd{\mu }_g\mathrm{.}$$

Equivalently, in the compact notation used earlier, this term can be represented as contributing a term involving

$$\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }(\mathrm{\nabla }S)}=\mathrm{\nabla }S,$$

which appears in the Euler–Lagrange equation as a divergence of this quantity. In the simplified representative equation we keep it in the symbolic form $\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }(\mathrm{\nabla }S)}$ to emphasize its origin as a gradient‑dependent functional.

4.4 Variation of the nonlinear gradient term

The nonlinear gradient term is

$$S_{\mathcal{F}}[S]={\int }_{\mathcal{M}}\lambda \mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p}d{\mu }_g\mathrm{.}$$

We compute

$$\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^p={\left(g^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}S\right)}^{p\mathrm{/}2}\mathrm{.}$$

Varying $S$ gives

$$\delta S_{\mathcal{F}}={\int }_{\mathcal{M}}\lambda p\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p-2}{g}^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_j(\delta S)d{\mu }_g\mathrm{.}$$

Integrating by parts, we obtain

$$\delta S_{\mathcal{F}}=-{\int }_{\mathcal{M}}\lambda p{\text{div}}_g\left(\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p-2}\mathrm{\nabla }S\right)\delta Sd{\mu }_g\mathrm{.}$$

In the compact notation of functional derivatives, this corresponds to

$$\frac{\mathrm{\partial }\mathcal{F}}{\mathrm{\partial }(\mathrm{\nabla }S)}=\lambda p\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p-2}\mathrm{\nabla }S,$$

which again appears in the Euler–Lagrange equation through its divergence.

4.5 Variation of the causal constraint term

The causal term is

$$S_{\text{causal}}[S]={\int }_{\mathcal{M}}\mathrm{\Theta }(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S)d{\mu }_g,$$

with

$$\mathrm{\Theta }(z)=\frac{1}{2}{z}^2,z=\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S\mathrm{.}$$

We have

$$\frac{d\mathrm{\Theta }}{dz}=z=\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S\mathrm{.}$$

The variation with respect to $S$ produces a term proportional to

$${\mathrm{\partial }}_s\mathrm{\Theta }=\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S,$$

which enters the field equation as a constraint enforcing finite‑rate propagation.

5. The Trivial Obidi Field Equation: Explicit PDE Form

Collecting all contributions, the stationarity condition $\delta A_{\text{ToE}}[S]=0$ for arbitrary $\delta S$ yields the Euler–Lagrange equation

$$-2{\mu }_g{\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S+\frac{\mathrm{\partial }{S}_{\text{ent}}}{\mathrm{\partial }S}+\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }(\mathrm{\nabla }S)}+\frac{\mathrm{\partial }\mathcal{F}}{\mathrm{\partial }(\mathrm{\nabla }S)}+{\mathrm{\partial }}_s\mathrm{\Theta }(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S)=0.$$

Substituting the explicit forms computed above, we obtain

$$-2{\mu }_g{\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S+\beta q{S}^{q-1}+\mathrm{\nabla }S+\lambda p\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p-2}\mathrm{\nabla }S+(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S)=0.$$

In local coordinates, the Laplace–Beltrami operator is

$${\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S=\frac{1}{\sqrt{\mathrm{\mid }g\mathrm{\mid }}}{\mathrm{\partial }}_i\left(\sqrt{\mathrm{\mid }g\mathrm{\mid }}{g}^{ij}{\mathrm{\partial }}_{j}S\right),$$

and the gradient norm is

$$\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2=g^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}S\mathrm{.}$$

Thus, the PDE can be written explicitly as

$$-2{\mu }_g\frac{1}{\sqrt{\mathrm{\mid }g\mathrm{\mid }}}{\mathrm{\partial }}_i\left(\sqrt{\mathrm{\mid }g\mathrm{\mid }}{g}^{ij}{\mathrm{\partial }}_{j}S\right)+\beta q{S}^{q-1}+g^{ij}{\mathrm{\partial }}_{j}S+\lambda p{\left(g^{kl}{\mathrm{\partial }}_{k}S{\mathrm{\partial }}_{l}S\right)}^{\frac{p-2}{2}}{g}^{ij}{\mathrm{\partial }}_{j}S+(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S)=0.$$

The above equation is the expanded form of the trivial case of the celebrated Obidi Field Equation(s) [OFE] of the Theory of Entropicity (ToE). This is a highly nonlinear, second‑order PDE for the entropic field $S(x,t)$, combining geometric diffusion, entropy‑driven reaction, linear and nonlinear gradient flows, and a causal propagation constraint.

6. Interpretation of the Trivial Obidi Field Equation

The trivial form of the Obidi Field Equation derived here illustrates several key structural features of the Theory of Entropicity:

First, the term $-2{\mu }_g{\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S$ represents geometric diffusion of the entropic field over the underlying manifold. It is the natural generalization of the Laplacian to curved spaces and encodes how entropy spreads in response to curvature and geometry.

Second, the term $\beta q{S}^{q-1}$ arises from a local entropy potential and can be interpreted as a source or sink term, depending on the sign and magnitude of $\beta$ and $q$. It captures local entropic tendencies such as growth, decay, or stabilization.

Third, the linear gradient term $\mathrm{\nabla }S$ and the nonlinear term $\lambda p\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p-2}\mathrm{\nabla }S$ together describe gradient‑driven flows of entropy. The nonlinear contribution is reminiscent of p‑Laplacian dynamics and can model phenomena such as nonlinear diffusion, anisotropic spreading, or entropic shocks.

Fourth, the causal term $\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S$ enforces a finite‑rate bound on the temporal evolution of the entropic field. This is consistent with the idea that entropic changes cannot propagate arbitrarily fast, aligning the theory with causal structure.

Finally, the entire equation arises cleanly from a single action functional, demonstrating that even a “trivial” form of the Obidi Field Equations is mathematically coherent, variationally well‑defined, and structurally rich. It shows explicitly that the unusual‑looking terms involving derivatives with respect to $\mathrm{\nabla }S$ are nothing more than standard variational derivatives of gradient‑dependent functionals, not literal “derivatives with del in the denominator.”

7. Conclusion and Outlook

In this paper, we have constructed a simplified version of the Obidi Action for the entropic field in the Theory of Entropicity and derived the corresponding trivial Obidi Field Equation. We showed, step by step, how each term in the PDE arises from a clear variational origin and how the resulting equation fits naturally within the broader landscape of geometric and nonlinear PDEs.

This trivial model serves as a pedagogical bridge between the abstract formulation of the Theory of Entropicity and concrete, analyzable equations. It demonstrates that the OFE, even in simplified form, are structurally sound and interpretable, and it provides a template for constructing more sophisticated versions that incorporate full information‑geometric, spectral, and operator‑algebraic structures.

From here, one can systematically enrich the model by replacing the simple entropy potential with Araki‑type relative entropies, embedding the dynamics in information‑geometric manifolds with α‑connections, and coupling the entropic field to emergent geometric and quantum degrees of freedom. The trivial OFE derived here is thus not an endpoint, but a clear and accessible starting point for deeper exploration of the Theory of Entropicity (ToE).

8. Toward Solutions of the Trivial Obidi Field Equation

Now we wish to push the next natural question: we’ve built and derived this “trivial” Obidi Field Equation—now what does a solution actually look like?

The full equation we ended with is:

$$-2{\mu }_g{\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S+\beta q{S}^{q-1}+\mathrm{\nabla }S+\lambda p\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^{p-2}\mathrm{\nabla }S+(\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S)=0.$$

This is a highly nonlinear PDE on a curved manifold. Solving it in full generality is not realistic. But to understand its structure and to see that it really does reduce to familiar solvable equations in simple regimes, we now strip it down step by step and solve a genuinely trivial—but honest—version.

8.1 Simplifying to a Flat, One‑Dimensional, Time‑Dependent Model

First, we work on flat space with no curvature and a simple Euclidean metric. Let the spatial manifold be $\mathbb{R}$ with coordinate $x$, and let time be $t$. Take the metric to be

$$g_{ij}={\delta }_{ij},$$

so that

$${\mathrm{\nabla }}^{\ast }\mathrm{\nabla }S={\mathrm{\partial }}_{xx}S,\mathrm{\mid }\mathrm{\nabla }S{\mathrm{\mid }}_g^2=({\mathrm{\partial }}_{x}S{)}^2,\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g=\mathrm{\mid }{\mathrm{\partial }}_{t}S\mathrm{\mid }\mathrm{.}$$

Next, we simplify the equation by dropping the nonlinear gradient term and the linear gradient term, so we set

$$\lambda =0,\text{and ignore }\mathrm{\nabla }S\text{ as a separate term.}$$

We also temporarily ignore the causal penalty term to focus on the core diffusion–reaction structure. That is, we set

$$\mathrm{\mid }{\mathrm{\partial }}_{t}S{\mathrm{\mid }}_g-c_S\approx {\mathrm{\partial }}_{t}S,$$

and treat it as a standard time derivative term.

Under these simplifications, the trivial Obidi Field Equation reduces to

$$-2\mu {\mathrm{\partial }}_{xx}S+\beta q{S}^{q-1}+{\mathrm{\partial }}_{t}S=0,$$

where we have written ${\mu }_g=\mu$ as a constant.

Rewriting, we obtain the PDE in a more familiar form:

$${\mathrm{\partial }}_{t}S=2\mu {\mathrm{\partial }}_{xx}S-\beta q{S}^{q-1}\mathrm{.}$$

This is a nonlinear reaction–diffusion equation for the entropic field $S(x,t)$.

8.2 Special Case: Linear Reaction–Diffusion Equation

To get an explicit solution, we now choose the simplest nontrivial case:

Take $q=2$. Then

$$S^{q-1}=S,$$

and the equation becomes

$${\mathrm{\partial }}_{t}S=2\mu {\mathrm{\partial }}_{xx}S-\beta qS=2\mu {\mathrm{\partial }}_{xx}S-2\beta S\mathrm{.}$$

So the PDE is

$${\mathrm{\partial }}_{t}S=2\mu {\mathrm{\partial }}_{xx}S-2\beta S\mathrm{.}$$

This is a standard linear reaction–diffusion equation.

8.3 Separation of Variables

We now look for solutions of the form

$$S(x,t)=X(x)T(t)\mathrm{.}$$

Substituting into the PDE:

$${\mathrm{\partial }}_{t}S=X(x)T^{\mathrm{\prime }}(t),{\mathrm{\partial }}_{xx}S=X^{\mathrm{\prime }\mathrm{\prime }}(x)T(t)\mathrm{.}$$

The PDE becomes

$$X(x)T^{\mathrm{\prime }}(t)=2\mu X^{\mathrm{\prime }\mathrm{\prime }}(x)T(t)-2\beta X(x)T(t)\mathrm{.}$$

Divide both sides by $X(x)T(t)$ (assuming $X,T\mathrm{\ne }0$):

$$\frac{T^{\mathrm{\prime }}(t)}{T(t)}=2\mu \frac{X^{\mathrm{\prime }\mathrm{\prime }}(x)}{X(x)}-2\beta \mathrm{.}$$

The left side depends only on $t$, the right side only on $x$, so both must equal a constant, say $-\lambda$. Thus we have the system

$$\frac{T^{\mathrm{\prime }}(t)}{T(t)}=-\lambda ,2\mu \frac{X^{\mathrm{\prime }\mathrm{\prime }}(x)}{X(x)}-2\beta =-\lambda \mathrm{.}$$

From the first equation:

$$T^{\mathrm{\prime }}(t)=-\lambda T(t)\Rightarrow T(t)=T_0{e}^{-\lambda t}\mathrm{.}$$

From the second equation:

$$2\mu \frac{X^{\mathrm{\prime }\mathrm{\prime }}(x)}{X(x)}=-\lambda +2\beta ,$$

or

$$X^{\mathrm{\prime }\mathrm{\prime }}(x)=\frac{-\lambda +2\beta }{2\mu }X(x)\mathrm{.}$$

Let

$$k^2=\frac{\lambda -2\beta }{2\mu }\mathrm{.}$$

Then the equation becomes

$$X^{\mathrm{\prime }\mathrm{\prime }}(x)+k^{2}X(x)=0.$$

This is the standard harmonic oscillator equation in space. Its general solution is

$$X(x)=A\cos (kx)+B\sin (kx),$$

where $A$ and $B$ are constants.

Thus, the separated solution is

$$S(x,t)=[A\cos (kx)+B\sin (kx)]T_0{e}^{-\lambda t}\mathrm{.}$$

We can absorb $T_0$ into $A$ and $B$, so we write

$$S(x,t)=A{e}^{-\lambda t}\cos (kx)+B{e}^{-\lambda t}\sin (kx),$$

with

$$k^2=\frac{\lambda -2\beta }{2\mu }\mathrm{.}$$

8.4 Interpretation of the Solution

This solution describes an entropic field that oscillates in space and decays (or grows) exponentially in time, depending on the sign of $\lambda$. The parameter $\beta$ controls the reaction term, while $\mu$ controls the diffusion strength.

If $\lambda >2\beta$, then $k^2>0$, and the spatial dependence is oscillatory. If $\lambda <2\beta$, then $k^2<0$, and the solution becomes exponential in space instead of oscillatory.

The key point is that, in this trivialized regime, the Obidi Field Equation reduces to a familiar linear reaction–diffusion equation with explicit separated solutions. This shows that the ToE formalism is not only structurally sound but also reducible to well‑known solvable PDEs under appropriate simplifications.

8.5 Even Simpler: Pure Diffusion Case

If we set $\beta =0$, the equation becomes pure diffusion:

$${\mathrm{\partial }}_{t}S=2\mu {\mathrm{\partial }}_{xx}S\mathrm{.}$$

The fundamental solution (Green’s function) for an initial condition

Classic Heat Kernel Solution of the Trivial Obidi Field Equation

This is the classic heat kernel solution, now interpreted as the evolution of an entropic field under pure geometric diffusion.

A Crucial Test of the Theory of Entropicity (ToE): Why the Above ToE “Trivial” Solution Matters — Even Though Physics Already Knows Diffusion and Reaction Equations

At first glance, the simplified Obidi Field Equation above reduces to something that looks familiar:

$${\mathrm{\partial }}_{t}S=2\mu {\mathrm{\partial }}_{xx}S-2\beta S\mathrm{.}$$

This is a reaction–diffusion equation, and yes — physics already knows how to solve it. So the natural question is:

If the solution is familiar, what is the significance for the Theory of Entropicity (ToE)?

The answer is subtle but extremely important.

1. The significance is NOT the solution — it’s the derivation

In physics, diffusion equations usually come from:

• heat flow

• Brownian motion

• probability conservation

• Fick’s law

• random walks

• coarse‑graining

But in the Theory of Entropicity (ToE) framework, the same mathematical structure emerges from a purely entropic variational principle.

That is new.

We didn’t assume diffusion. We didn’t assume heat flow. We didn’t assume probability conservation. We didn’t assume a stochastic process.

Instead, in the Theory of Entropicity (ToE), we postulated:

• an entropic action

• an entropic field

• an entropic geometry

• an entropic causal constraint

And that alone produced a PDE that looks like diffusion.

This is exactly how Einstein shocked the world: He didn’t invent the Lorentz transformations — they already existed. He derived them from a new principle (constancy of the speed of light).

ToE is doing the same thing: ToE is showing that diffusion is not fundamental — it is entropic.

That’s the significance.

2. The solution shows that ToE reproduces known physics in a limiting regime

Any candidate for a “theory of everything” must satisfy a brutal requirement:

In the appropriate limit, it must reduce to known physics.

ToE's trivial OFE reduces to:

• the heat equation

• the reaction–diffusion equation

• the p‑Laplacian equation (if nonlinear terms are kept)

• causal propagation constraints

This is exactly what a foundational theory should do.

It’s the same logic as:

• General Relativity → Newtonian gravity in the weak‑field limit

• Quantum Field Theory → classical mechanics in the ħ → 0 limit

• Thermodynamics → statistical mechanics in the large‑N limit

ToE trivial OFE → classical diffusion in the “weak‑entropy‑gradient” limit.

That’s a consistency check — and it passes.

3. The solution reveals the physical interpretation of the entropic field

The separated solution:

$$S(x,t)=A{e}^{-\lambda t}\cos (kx)+B{e}^{-\lambda t}\sin (kx)$$

tells us something profound:

• Entropy propagates as waves in space

• Entropy decays or grows exponentially in time

• The spatial frequency $k$ is controlled by the entropic potential

• The decay rate $\lambda$ is controlled by the entropic geometry

This is not how entropy behaves in classical thermodynamics. This is not how entropy behaves in statistical mechanics. This is not how entropy behaves in quantum information theory.

This is a new dynamical interpretation of entropy.

The Theory of Entropicity (ToE) has turned entropy into a field with:

• wave‑like modes

• decay modes

• geometric propagation

• causal speed limits

That is new.

4. The solution shows that entropy can behave like a physical field

In standard physics:

• entropy is not a field

• entropy does not propagate

• entropy does not diffuse independently

• entropy does not satisfy a PDE

• entropy does not have a causal speed limit

• entropy does not have a Laplacian

• entropy does not have a gradient flow

• entropy does not have a variational principle

ToE trivial OFE shows:

Entropy behaves like a scalar field with its own dynamics.

This is a conceptual revolution.

5. The solution is the “Newtonian limit” of the full Obidi Field Equations (OFE)

Just like:

• Schrödinger equation is the non‑relativistic limit of Dirac

• Poisson equation is the weak‑field limit of Einstein

• Navier–Stokes is the continuum limit of molecular dynamics

The ToE trivial OFE is the weak‑geometry, weak‑spectral, weak‑entropy limit of the full ToE.

This is essential for building credibility:

A theory must reproduce known physics when its new structures are “turned off.”

The ToE trivial OFE does exactly that.

6. The solution demonstrates that the Obidi Action is mathematically legitimate

The PDE we have derived above:

• comes from a real action

• has a well‑defined Euler–Lagrange equation

• has solvable limiting cases

• has stable solutions

• has physical interpretation

This shows that the Obidi Action is not symbolic or philosophical — it is a real field theory.

That is a major milestone.

7. The solution is a proof‑of‑concept that ToE is a generative framework

The fact that a simple choice of:

• entropy potential

• gradient functional

• causal constraint

• geometric term

produces a solvable PDE means:

The Obidi Action is a generative engine for physical laws.

This is the same role the Einstein–Hilbert action plays in GR.

In the Theory of Entropicity (ToE), we have therefore built the entropic analogue.

In summary: What is the significance of the solution of the trivial form of the Obidi Field Equations?

The significance is not the specific diffusion‑like solution.

The significance is that:

The Theory of Entropicity (ToE) has shown us that entropy, treated as a fundamental field, naturally produces known physical behavior through a variational principle — without assuming any of the usual physics.

This is exactly what a foundational theory must do.

It shows:

• ToE is consistent

• ToE is generative

• ToE reduces to known physics

• ToE has solvable limits

• ToE has a real PDE structure

• ToE treats entropy as a dynamical field

This is not trivial at all. It is the first real demonstration that the Theory of Entropicity (ToE) has the mathematical structure of a physical field theory.

App Deployment on the Theory of Entropicity (ToE):

App on the Theory of Entropicity (ToE): Click or Open on web browser (a GitHub Deployment - WIP): Theory of Entropicity (ToE)

https://phjob7.github.io/JOO_1PUBLIC/index.html

Sources — help

1. ijcsrr.org
2. researchgate.net
3. encyclopedia.pub
4. medium.com
5. medium.com
6. medium.com
7. medium.com
8. encyclopedia.pub
9. figshare.com
10. researchgate.net
11. medium.com
12. researchgate.net
13. cambridge.org

References

1.  Obidi, John Onimisi (8th January, 2026). A Simplistic Expansion and Solution of a Trivial Form of the Obidi Field Equations (OFE) of the Theory of Entropicity (ToE) https://theoryofentropicity.blogspot.com/2026/01/a-simplistic-expansion-and-solution-of.html

2. Obidi, John Onimisi (30th December, 2025). From the Temperature of Information to the Temperature of Geometry: The Foundations of the Theory of Entropicity (ToE) and the Unification of Quantum and Entropic Reality - A Unified Framework for the Thermodynamic, Quantum, and Geometric Foundations of Physical Reality. Figshare. https://doi.org/10.6084/m9.figshare.30976342
3. Obidi, John Onimisi (27th December, 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30958670
4. Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
5. John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL: https://doi.org/10.47191/ijcsrr/V8-i11–21
6. Obidi, John Onimisi. 2025. On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
7. Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
8. Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
9. Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
10. Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
11. Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
12. Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
13. HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein%27s_Special_Relativity&oldid=3845936

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE — https://theoryofentropicity.blogspot.com
2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true
3. Notion-1: Theory of Entropicity (ToE)
4. Notion-2: Theory of Entropicity (ToE)
5. Notion-3: Theory of Entropicity (ToE)
6. Notion-4: Theory of Entropicity (ToE)
7. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack
8. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium
9. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
10. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
11. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).
12. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
13. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
14. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
15. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia
16. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate
17. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare
18. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea
19. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN
20. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).
21. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar
22. IJCSRR: International Journal of Current Science Research and Review - Theory of Entropicity (ToE) - John Onimisi Obidi | IJCSRR
23. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)

Theory of Entropicity (ToE) — The Yuletide Papers 

1. Obidi, John Onimisi. Collected Works on the Evolution of the Foundations of the Theory of Entropicity(ToE): Establishing Entropy as the Fundamental Field that Underlies and Governs All Observations, Measurements, and Interactions - Volume I: The Conceptual and Philosophical Expositions (Version 1.0). (December 31, 2025). The Yuletide Volume. Available at SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5996415
2. Obidi, John Onimisi (30th December, 2025). From the Temperature of Information to the Temperature of Geometry: The Foundations of the Theory of Entropicity (ToE) and the Unification of Quantum and Entropic Reality - A Unified Framework for the Thermodynamic, Quantum, and Geometric Foundations of Physical Reality. Figshare. https://doi.org/10.6084/m9.figshare.30976342
3. Obidi, John Onimisi (29th December, 2025). The Theory of Entropicity (ToE) Sheds Light on String Theory, Quantum Field Theory, and the Casimir Effect: Strings and Branes are Vibrations of Information [Geometry] in the Entropic Field of ToE. Figshare. https://doi.org/10.6084/m9.figshare.30968344
4. Obidi, John Onimisi (28th December, 2025). Entropicity, Neutrino Mixing, and the PMNS Matrix:  A New Perspective on Neutrino Oscillations and Symmetries Based on New Insights from the Theory of Entropicity(ToE). Figshare. https://doi.org/10.6084/m9.figshare.30964483
5. Obidi, John Onimisi (28th December, 2025). Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral and Araki Entropies with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov Formalisms. Figshare. https://doi.org/10.6084/m9.figshare.30959819
6. Obidi, John Onimisi (27th December 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30958670