Physics:Newton's Laws of Motion Recast in the Theory of Entropicity(ToE)
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Obidi’s First Law of Entropic Motion: Recasting Newton’s First Law within the Theory of Entropicity (ToE)
We start from Newton’s First Law and recast it within the Theory of Entropicity(ToE), consolidating the concepts its concepts thereby (on entropic gradients, least entropic resistance, entropic geodesics, minimal dynamics, and links to observations).
Abstract
This article recasts Newton’s First Law of Motion through the lens of the Theory of Entropicity (ToE), which posits entropy as the sole fundamental physical principle governing nature. In ToE, motion is not primarily driven by “forces” but by the redistribution of entropy under constraints. We formulate Obidi’s First Law of Entropic Motion, define the entropic gradient and least entropic resistance principle, introduce a minimal dynamic law:
[math]\displaystyle{ \mathbf{a}=-\eta,\nabla\Sigma }[/math],
and connect these ideas to geodesic motion, central-field dynamics (Binet-type forms called the Binet-Obidi Equation), and empirical signatures. This reframing positions “force” and even spacetime curvature as emergent manifestations of underlying entropic flow.
1. Motivation
Classically, Newton’s First Law states that a body at rest remains at rest, and a body in motion persists in uniform motion unless acted upon by an external force. In ToE, inertia is reinterpreted as equilibrium in entropy flow; deviations from uniform motion arise from entropic imbalances (gradients) rather than fundamental external forces. Thus, “force’’ is an effective descriptor of how constraints and boundaries mediate entropy’s drive to redistribute.
2. Foundational Postulate of ToE
Entropy is the unique, underlying physical principle.
All observable dynamics (mechanical, electromagnetic, gravitational, quantum, and informational) are expressions of entropy redistribution under constraints.
Geometry and “force” are emergent from the configuration and flow of the entropic field.
3. Obidi’s First Law of Entropic Motion
Law (ToE–L1). In the absence of an entropic gradient, a system’s macroscopic state (rest or uniform motion) persists. When an entropic gradient exists, the system evolves along the path of least entropic resistance, thereby maximizing entropy redistribution under the prevailing constraints.
Here, the “external force’’ of classical mechanics is replaced by the entropic gradient and constraints that shape feasible redistribution.
4. Core Definitions
Entropic potential [math]\displaystyle{ \Sigma(\mathbf{x},t) }[/math]: a scalar field encoding the local “drive” for entropy to redistribute.
Entropic field [math]\displaystyle{ \mathbf{E}{\Sigma} }[/math]: the gradient of this potential,
- [math]\displaystyle{ \mathbf{E}{\Sigma} \equiv -\nabla\Sigma. }[/math]
Entropic resistance [math]\displaystyle{ R_e(\mathbf{x},\dot{\mathbf{x}}) }[/math]: the local opposition (materials, boundaries, information bottlenecks) to redistributing entropy; dynamics follow paths that minimize cumulative [math]\displaystyle{ R_e }[/math] while increasing entropy.
5. Minimal Entropic Dynamics (kinematic closure)
A practical working law for single-particle kinematics consistent with ToE–L1 is : [math]\displaystyle{ \displaystyle \mathbf{a}(t) = -\eta,\nabla\Sigma(\mathbf{x}(t),t), }[/math] where [math]\displaystyle{ \eta }[/math] is an entropic coupling constant (to be fixed empirically or by deeper ToE field equations). Immediate consequences:
If [math]\displaystyle{ \nabla\Sigma=0 }[/math] then [math]\displaystyle{ \mathbf{a}=0 }[/math] and the system remains in its current state of motion (rest or uniform velocity).
Constant [math]\displaystyle{ \nabla\Sigma }[/math] yields constant acceleration.
Curved trajectories arise when [math]\displaystyle{ \nabla\Sigma }[/math] bends in space and/or time.
This law reframes:
[math]\displaystyle{ \mathbf{F}=m\mathbf{a} }[/math] as an effective statement emerging from the entropic coupling between [math]\displaystyle{ \nabla\Sigma }[/math] and the system.
6. Entropic Geodesics (least-resistance principle)
In ToE, “free” motion follows entropic geodesics: : [math]\displaystyle{ \displaystyle \delta \int R_e \big(\mathbf{x},\dot{\mathbf{x}}\big) dt=0 }[/math] subject to [math]\displaystyle{ \dot{\Sigma}\ge 0, }[/math] i.e., the path that minimizes entropic resistance while permitting net entropy increase. In smooth regimes where [math]\displaystyle{ R_e }[/math] is regular and constraints are mild, the Euler–Lagrange equations reduce to the minimal model [math]\displaystyle{ \mathbf{a}=-\eta\nabla\Sigma }[/math]. In strongly constrained media (dissipation, boundaries, information bottlenecks), [math]\displaystyle{ R_e }[/math] alters the effective trajectory.
7. Central entropic potentials and Binet-type orbital form
For planar motion in a central entropic potential [math]\displaystyle{ \Sigma(r) }[/math], let [math]\displaystyle{ u(\theta)=1/r }[/math] and [math]\displaystyle{ \ell = |\mathbf{x}\times\dot{\mathbf{x}}| }[/math] be the specific angular momentum. A Binet-type (Binet-Obidi) equation consistent with the minimal law is:
[math]\displaystyle{ \displaystyle \frac{d^2 u}{d\theta^2} + u = \frac{\eta}{\ell^2 u^2} \frac{d\Sigma}{dr}\Bigg|_{r=1/u}. }[/math]
For [math]\displaystyle{ \Sigma(r)=-\kappa/r }[/math], the right-hand side [math]\displaystyle{ \propto \kappa/r^2 }[/math];
ToE recovers inverse-square-type dynamics as a special case of entropic curvature, not as a fundamental force.
8. Relation to classical mechanics and GR
Newton I (inertia). ToE–L1: no entropic gradient ⇒ no change in state; identical empirical prediction in the free case.
Newton II (dynamics). Classical force becomes an effective projection of [math]\displaystyle{ \nabla\Sigma }[/math] through constraints; the familiar forms arise when [math]\displaystyle{ \Sigma }[/math] induces inverse-power fields.
Geodesics and curvature (GR). In ToE, the apparent “spacetime curvature’’ is the observable imprint of entropy flow and constraints. Geodesics are the least-resistance entropic paths; metric curvature is emergent.
9. Variational foundation: Vuli-Ndlela Integral (ToE standard form)
ToE dynamics are anchored in the entropy-constrained path integral: [math]\displaystyle{ Z_{\text{ToE}} = \int_{\mathbb{S}} \mathcal{D}[\phi] \exp \Big(\tfrac{i}{\hbar} S[\phi]\Big) \exp\Big(-\tfrac{\mathcal{S}G[\phi]}{k_B}\Big) \exp \Big(-\tfrac{\mathcal{S}{\text{irr}}[\phi]}{\hbar_{\text{eff}}}\Big), }[/math] with the admissible domain [math]\displaystyle{ \mathbb{S}={\phi \big| \Lambda(\phi)\gt \Lambda_{\min}}, }[/math] where [math]\displaystyle{ S[\phi] }[/math] is the classical action, [math]\displaystyle{ \mathcal{S}G }[/math] encodes gravitational/entropic geometry, [math]\displaystyle{ \mathcal{S}{\text{irr}} }[/math] enforces irreversibility, and [math]\displaystyle{ \Lambda }[/math] is the entropy-density functional that gates allowed histories. The minimal kinematic law [math]\displaystyle{ \mathbf{a}=-\eta\nabla\Sigma }[/math] is the near-equilibrium, single-degree-of-freedom limit of this variational structure.
10. Worked toy cases
Zero gradient (free motion). [math]\displaystyle{ \nabla\Sigma=0 \Rightarrow \mathbf{a}=0 }[/math] → uniform motion.
Uniform gradient. [math]\displaystyle{ \nabla\Sigma=\mathbf{g}\Sigma }[/math] (constant) → [math]\displaystyle{ \mathbf{a}=-\eta \mathbf{g}\Sigma }[/math] (constant acceleration).
Central potential. [math]\displaystyle{ \Sigma(r)=-\kappa/r }[/math] → Binet-type dynamics reproduce inverse-square phenomenology as an entropic limit.
11. Empirical signatures and tests
Unification of conservative and dissipative dynamics. Dissipation and “forces’’ are both expressions of entropy flow through constraints ([math]\displaystyle{ R_e }[/math]).
Free-fall as free-flow. Trajectories in gravity labs/astronomical settings should correspond to least-resistance entropic paths; light-bending and orbital precession arise from [math]\displaystyle{ \nabla\Sigma }[/math] structure.
Attosecond constraints & no-simultaneity. Measurement/interaction latencies reflect entropy-constrained interaction times (ToE’s irreversibility).
12. Discussion
The entropic reinterpretation preserves classical successes in the appropriate limits while elevating entropy to first principle. It replaces fundamental “forces’’ with entropic gradients, replaces postulated geometrical curvature with entropic curvature, and yields a single driver for diverse phenomena—from mechanics and gravitation to quantum processes and information flow.
13. Conclusion
Obidi’s First Law of Entropic Motion states that systems persist in their current state unless an entropic gradient compels evolution along the path of least entropic resistance. With the minimal closure [math]\displaystyle{ \mathbf{a}=-\eta\nabla\Sigma }[/math], Binet-type central dynamics, and the Vuli-Ndlela Integral as the variational backbone, ToE reframes motion as the visible expression of entropy’s imperative to redistribute under constraints. Classical laws emerge as special-case limits of this entropic foundation.
See also
References
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