Physics:Concept of Non-Vanishing Entropy in the Theory of Entropicity (ToE)

Entropy is Never Zero: Foundations from the Theory of Entropicity (ToE)

=== Overview === The Theory of Entropicity (ToE) presents a radical reformulation of physical law by elevating entropy, traditionally viewed as a scalar measure of disorder, into a dynamic, observer-linked, interaction-constraining field that governs the evolution of the universe. Within this framework, the concept of zero entropy becomes not just physically unattainable but logically incoherent. This article formalizes and expands the implications of the ToE perspective on the impossibility of zero entropy across all domains—classical, quantum, cosmological, and informational.

We propose and derive the following principle:

This assertion has deep implications for thermodynamics, quantum theory, cosmology, black hole physics, information theory, and the nature of time itself.

1. Classical Thermodynamic Arguments

=== 1.1 The Third Law of Thermodynamics === According to classical thermodynamics, the Third Law states:

[math]\displaystyle{ {{{1}}} }[/math]

This applies only to a perfect crystalline substance at absolute zero temperature. However, three caveats challenge this idealization:

No physical system can be perfectly isolated.

Absolute zero is thermodynamically unreachable (per Nernst's postulate).

Real-world substances exhibit zero-point motion and residual disorder.

Therefore, even classically:

2. Quantum Statistical Mechanics: Von Neumann Entropy

In quantum theory, entropy is defined for a system with density matrix as:

[math]\displaystyle{ {{{1}}} }[/math]

2.1 Entropy of Pure vs Mixed States

A pure state:

A mixed state:

However, real systems are constantly interacting with their environments. Decoherence drives systems toward mixed states:

[math]\displaystyle{ {{{1}}} }[/math]

\left( |\Psi(t)\rangle \langle\Psi(t)| \right)}}

Thus, physical subsystems never remain pure, even in vacuum.

3. Entropic Field Framework in ToE

=== 3.1 The Entropic Field === In ToE, entropy is a continuous, dynamic scalar field defined on spacetime, sourcing and constraining all physical processes. This is captured in the Obidi Action:

[math]\displaystyle{ {{{1}}} }[/math]

[\Phi, g_{\mu\nu}] }}

This yields the entropic field equation:

[math]\displaystyle{ {{{1}}} }[/math]

This PDE governs how entropy propagates and couples to matter.

=== 3.2 Entropic Time Limit and Non-Markovianity === ToE also establishes the No-Rush Theorem, which asserts that no physical interaction can occur instantaneously. There is always a finite time interval for entropy to propagate:

[math]\displaystyle{ \Delta t \geq \frac{\delta S}{\dot{S }[/math]

}}

This finite entropic latency introduces memory into dynamics—systems are non-Markovian:

4. Implications of Non-Zero Entropy in ToE

=== 4.1 Existence and Measurement === ToE posits that existence is constrained by entropy thresholds:

Thus:

If does not exist.

Hence, non-existence.

=== 4.2 Spacetime Curvature and Motion === In ToE, spacetime curvature is emergent from entropy gradients:

[math]\displaystyle{ {{{1}}} }[/math]

If , then , implying zero acceleration, zero interaction, and total stasis—again, physically impossible.

=== 4.3 Irreversibility and Time Asymmetry === Entropy provides the arrow of time:

[math]\displaystyle{ \frac{dS}{dt} \gt 0 }[/math]

A vanishing entropy field would erase the arrow of time—contradicting all physical observation.

5. Cosmological Considerations

=== 5.1 Early Universe === In standard cosmology, the early universe had "low entropy," but ToE refines this:

Inflation massively amplified entropy:

[math]\displaystyle{ S_{\text{post-inflation }[/math]

\gg S_{\text{pre-inflation}} > 0}}

=== 5.2 Entropic Cosmological Expansion === In ToE, the universe expands due to entropy flux:

[math]\displaystyle{ \frac{\ddot{a }[/math]

{a} \propto -\nabla^2 S + f(\dot{S})}}

A zero-entropy universe would imply no expansion, no thermodynamic direction, and no metric evolution.

6. Black Hole Physics

=== 6.1 Bekenstein--Hawking Entropy === The entropy of a black hole is given by:

[math]\displaystyle{ S_{\text{BH }[/math]

= \frac{k_B c^3 A}{4\hbar G}}}

Where is the area of the event horizon.

Thus, even the most compact known systems exhibit maximal entropy—not zero.

=== 6.2 Entropic Pressure and Hawking Radiation in ToE === ToE interprets Hawking radiation as a redistribution of entropic curvature:

[math]\displaystyle{ \frac{dM}{dt} \sim -\frac{1}{c^2} \frac{dS}{dt }[/math]

}

Hence, black hole evaporation is itself a function of persistent, evolving entropy.

7. Information Theory and AI Systems

Entropy in information theory is defined as:

[math]\displaystyle{ {{{1}}} }[/math]

In ToE, this is linked to the Self-Referential Entropy (SRE) index, governing internal consistency of intelligent systems.

7.1 Entropic Interpretation of Intelligence

AI systems (e.g., transformers, RNNs, LSTMs) behave as entropic processors:

They store memory from previous states.

They compare present inputs to prior structure.

They explore future outputs under entropy-constrained backpropagation.

Thus, ToE explains their learning dynamics as fundamentally non-Markovian entropic systems.

8. Philosophical and Foundational Consequences

No true vacuum exists: Even regions with no matter contain entropy via quantum vacuum fluctuations.

Non-zero entropy as a prerequisite for perception: Consciousness itself may require a minimum entropy flux (cf. psychentropy).

Existence equals entropic presence: The ontology of the universe is structured by entropy.

10. Future Research Directions

Quantify experimentally.

Test ToE predictions in extreme vacuum conditions (e.g., Casimir cavities).

Apply Obidi’s Entropic Axiom to redefine vacuum energy and cosmological constant.

Develop entropic quantum logic gates for entropy-based AI processors.

11. Conclusion

The claim that entropy has never been and can never be zero is not merely a thermodynamic truism—it is a deep ontological and dynamical principle within the Theory of Entropicity (ToE). This principle binds together gravitational theory, quantum mechanics, cosmology, and artificial intelligence under a unified entropic field. Far from being a peripheral effect, entropy is the very foundation of interaction, time, motion, and even being itself. Its non-vanishing nature is not a limitation, but the very precondition for a universe that evolves, remembers, and reflects.

References

Bekenstein, J. D. (1973). Black holes and entropy. Physical Review D, 7(8), 2333.

Hawking, S. W. (1975). Particle creation by black holes. Communications in Mathematical Physics, 43(3), 199-220.

Obidi, J. O. (2025). The Theory of Entropicity: Obidi Action and Entropic Field Equations. Cambridge Open Engage.

von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics.

Susskind, L. (2008). The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics.