Claim: The N-body limit is not the emergence of randomness, but the saturation of transport non-closure.
Classical Framing
- Many interacting particles
- Exact dynamics infeasible
- Statistical mechanics introduced
Probability replaces trajectory.
Why This Is Treated as Fundamental
- Phase space dimension grows as $6N$
- Sensitive dependence on initial conditions
- Microscopic reversibility vs macroscopic irreversibility
Transport Reinterpretation
Each body defines a local transport frame.
As $N \to \infty$:
- No privileged frame exists
- All loops are open
- Transport order dominates outcome
Path A — Geometric Saturation
Primitive: Motion relative to evolving frames
The cumulative transport around many bodies:
$$ \sum_{i=1}^N \oint (\mathbf{v} - \mathbf{u}_i) \cdot d\mathbf{r} $$
diverges as $N$ increases.
No global closure condition survives.
↓
Path B — Phase Mixing
Primitive: Phase accumulation under transport
Each interaction shifts phase.
Total phase drift:
$$ \Delta \Phi = \sum_{i=1}^N \delta \phi_i $$
Phases decorrelate.
Only coarse-grained invariants remain.
↓
Agreement Condition
$$ \text{Geometric saturation} \iff \text{Phase ergodicity} $$
Why Thermodynamics Emerges
- Energy is conserved locally
- Transport history is lost globally
- Statistics replace trajectories
Entropy Reinterpreted
Entropy measures transport degeneracy:
$$ S \sim \log(\text{number of transport histories}) $$
Not disorder, but untrackable frame order.
Connection to Earlier Results
- Three-body chaos → onset of saturation
- Basel / Gaussian → invariant remnants
- Transport invariants survive averaging
Transport-First Summary
- The N-body limit is deterministic but non-closable
- Statistics emerge from frame overload
- Entropy is bookkeeping loss
- No randomness is required