Newton's Laws, Thermodynamics in Theory of Entropicity(ToE)

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The Theory of Entropicity(ToE), first formulated and developed by John Onimisi Obidi,^{[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]} postulates that entropy is not merely a statistical descriptor of disorder, but a fundamental, dynamical field that governs all physical interactions. In this framework, motion itself is interpreted as the redistribution of entropy, unifying classical mechanics and thermodynamics under a single principle.

In the Theory of Entropicity, order is not the antithesis of disorder but its natural offspring. Local structures, patterns, and stability arise as transient configurations that accelerate the universal drive of the entropic field toward equilibrium. What we perceive as “order” is the entropic field’s most efficient pathway for redistributing entropy — a fleeting island of structure in the ever‑rising tide of the second law.

That is, in the Theory of Entropicity (ToE), “order from disorder” doesn’t mean that chaos magically organizes itself. It means that what we call order in nature is actually a local, temporary configuration that emerges as part of the universal drive to redistribute entropy.

The Theory of Entropicity(ToE) therefore teaches us as follows:

- Entropy as the driver — In ToE, entropy isn’t just a measure of disorder; it’s the fundamental field shaping all motion and interactions.

- Local order, global increase — The second law still holds: total entropy increases. But in the process, the entropic field can create localized decreases in entropy (what we perceive as “order”) if that facilitates a greater overall increase elsewhere.

- Self-organization as entropic flow

- Structures like galaxies, hurricanes, living cells, or crystal lattices can form because they are efficient channels for entropy redistribution. They’re not exceptions to the second law — they’re its most effective servants.

- Order is a byproduct — In ToE terms, order is not the opposite of entropy; it’s a patterned state that exists because it accelerates the entropic field’s drive toward equilibrium.

Thus, the Theory of Entropicity(ToE) reframes “order from disorder” as a natural, inevitable outcome of the entropic field’s dynamics:

What looks like structure and stability(seeming order) is therefore actually a transient stage in the deeper process of entropy flow.

So, for every local ordering, there is a global entropy flow and redistribution that enforces entropy increase.

Geometry itself is a local ordering in nature. Therefore:

Where there is no entropy, there can be no geometry.

Hence:

Principle of Entropic Geometry:

• For every local ordering in nature, there exists a corresponding global entropy flow and redistribution that enforces the second law. Geometry itself is a form of local ordering; therefore, in the absence of entropy, geometry cannot exist. In the Theory of Entropicity, spatial and temporal structures are emergent consequences of the entropic field’s dynamics.

We therefore note the following:

- Local ordering — whether it’s a crystal lattice, a planetary orbit, or the curvature of spacetime itself — is a patterned arrangement of matter and energy.

- Global entropy flow — the existence of that pattern is sustained only because it participates in the larger, irreversible redistribution of entropy.

- Geometry as order — geometry is the mathematical language of arrangement; in physical reality, it’s the manifestation of local order.

- Entropic primacy — if entropy is the fundamental field, then geometry is a secondary phenomenon that emerges from it. Without entropy, there is no driver for structure, no gradients to define distances, no “shape” to the universe.

In other words, geometry is not the stage on which entropy plays out — it’s a byproduct of entropy’s play itself.

• Fundamental field: A scalar entropic field ${\displaystyle S(\mathbf{𝐱},t)}$ exists, with gradients driving motion.
• Entropic intensity: A positive field ${\displaystyle \mathrm{\Theta }(\mathbf{𝐱},t)>0}$ (energy per entropy unit) weights entropic gradients; in equilibrium it coincides locally with temperature.
• Matter degrees of freedom: Particles or coarse variables ${\displaystyle q(t)}$ (mass matrix ${\displaystyle M}$) couple to ${\displaystyle S}$ and move in conservative potentials ${\displaystyle U(q)}$.
• Entropy balance:

${\displaystyle {\partial }_{t}s+\mathrm{\nabla }\cdot {\mathbf{𝐣}}_s={\sigma }_s,{\sigma }_s\ge 0}$

• Constitutive coupling:

${\displaystyle {\mathbf{𝐣}}_s=s\mathbf{𝐯}-D_s\mathrm{\nabla }s,{\mathbf{𝐅}}_{\text{ent}}=g\mathrm{\nabla }S}$

The coupling between matter and entropy is encoded in the Obidi Action:

${\displaystyle {𝒜}[q,S]=\int dt[\frac{1}{2}{\dot{q}}^{\mathrm{\top }}M\dot{q}-U(q)+gS(q(t),t)]+\int d^{3}xdt[\frac{\kappa }{2}({\partial }_{t}S{)}^2-\frac{\xi }{2}|\mathrm{\nabla }S{|}^2-V(S)]}$

Dissipation is introduced via the Rayleigh functional:

${\displaystyle {ℛ}=\frac{1}{2}{\gamma }_m{\dot{q}}^{\mathrm{\top }}\dot{q}+\frac{1}{2}{\gamma }_S({\partial }_{t}S{)}^2}$

The Euler–Lagrange–Rayleigh equations are:

${\displaystyle M\ddot{q}=-{\mathrm{\nabla }}_{q}U(q)+g{\mathrm{\nabla }}_{q}S(q,t)-{\gamma }_m\dot{q}}$

${\displaystyle \kappa {\partial }_{tt}S-\xi {\mathrm{\nabla }}^{2}S+V^{\prime }(S)+{\gamma }_S{\partial }_{t}S=g\sum _a\delta (\mathbf{𝐱}-{\mathbf{𝐪}}_a(t))}$

• Law I: If ${\displaystyle \mathrm{\nabla }S=0}$ and ${\displaystyle \mathrm{\nabla }U=0}$, then ${\displaystyle M\ddot{q}=0}$ (uniform motion).
• Law II:

${\displaystyle M\ddot{q}=-{\mathrm{\nabla }}_{q}U+g{\mathrm{\nabla }}_{q}S-{\gamma }_m\dot{q}}$

• Law III: For pairwise ${\displaystyle S({\mathbf{𝐫}}_1,{\mathbf{𝐫}}_2)=\tilde{S}(|{\mathbf{𝐫}}_1-{\mathbf{𝐫}}_2|)}$:

${\displaystyle {\mathbf{𝐅}}_{12}=g{\mathrm{\nabla }}_{{\mathbf{𝐫}}_1}\tilde{S}(r),{\mathbf{𝐅}}_{21}=-{\mathbf{𝐅}}_{12}}$

Total energy: ${\displaystyle E_{\text{mech}}=\frac{1}{2}{\dot{q}}^{\mathrm{\top }}M\dot{q}+U(q),E_S=\int d^{3}x[\frac{\kappa }{2}({\partial }_{t}S{)}^2+\frac{\xi }{2}|\mathrm{\nabla }S{|}^2+V(S)]}$ Energy balance: ${\displaystyle \frac{d}{dt}(E_{\text{mech}}+E_S)=-{\gamma }_m{\dot{q}}^{\mathrm{\top }}\dot{q}-{\gamma }_S\int d^{3}x({\partial }_{t}S{)}^2\le 0}$ Entropy production: ${\displaystyle {\sigma }_s=\frac{{\gamma }_m}{\mathrm{\Theta }}{\dot{q}}^{\mathrm{\top }}\dot{q}\delta (\mathbf{𝐱}-q(t))+\frac{{\gamma }_S}{\mathrm{\Theta }}({\partial }_{t}S{)}^2\ge 0}$

The GENERIC form: ${\displaystyle \dot{y}=L(y)\mathrm{\nabla }E(y)+M(y)\mathrm{\nabla }{S}_{\text{tot}}(y)}$ with ${\displaystyle y=(q,p,S)}$, ${\displaystyle L=-L^{\mathrm{\top }}}$, ${\displaystyle M=M^{\mathrm{\top }}⪰0}$, and degeneracy conditions: ${\displaystyle M\mathrm{\nabla }E=0,L\mathrm{\nabla }{S}_{\text{tot}}=0}$

${\displaystyle {𝒜}[q]\propto \int {𝒟}q\exp \{\frac{i}{\hslash }\int (\frac{1}{2}m{\dot{q}}^2-U)dt+\alpha \int S(q(t),t)dt\}}$ Stationary phase yields: ${\displaystyle m\ddot{q}=-{\mathrm{\nabla }}_{q}U+\alpha {\mathrm{\nabla }}_{q}S}$

Discrete dynamics with step ${\displaystyle {\tau }_E}$: ${\displaystyle M\frac{q_{n+1}-2q_n+q_{n-1}}{{\tau }_E^2}=-\mathrm{\nabla }U(q_n)+g\mathrm{\nabla }S(q_n,t_n)-{\gamma }_m\frac{q_n-q_{n-1}}{{\tau }_E}}$ ${\displaystyle \frac{S_{n+1}-S_n}{{\tau }_E}+\mathrm{\nabla }\cdot {\mathbf{𝐣}}_{s,n}={\sigma }_{s,n}\ge 0}$

• Temperature-dependent entropic forces.
• Relaxation with overdamped or weakly propagating entropic waves.
• Gravity as a static entropic field in the Poisson limit.
• Arrow-of-time bound via ${\displaystyle {\tau }_E}$.

• Entropy
• Newton's laws of motion
• Second law of thermodynamics
• Entropic gravity

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