When Stochasticity Resolves into Certainty: Hidden Structure of Deterministic Motion

TL;DR

This paper demonstrates that deterministic behavior in dissipative systems can be rigorously derived from contact geometry, showing how stochastic effects vanish and deterministic motion emerges with exponential convergence.

Contribution

It introduces a geometric framework using contact geometry to rigorously derive deterministic dynamics from stochastic systems, with exact closure theorems and validation on a Duffing oscillator.

Findings

Deterministic motion is a geometric attractor in dissipative systems.

Exponential convergence rate is governed by the drift-field Jacobian spectrum.

Validation on the Duffing oscillator confirms theoretical predictions.

Abstract

We prove that deterministic motion in dissipative systems emerges as a strict geometric attractor of contact flow, not a statistical approximation. Building on the contact geometry of stochastic vector bundles, we develop time-dependent contact potentials with an exact closure theorem ensuring exact satisfaction of the master equation at any finite order. The Contact Locking Theorem shows that exponential gradient amplification of the probability field is precisely counterbalanced by synchronous stiffness decay, forcing the effective macroscopic-microscopic coupling to vanish exponentially. Deterministic dynamics therefore emerges through deterministic focusing with a universal timescale governed by the drift-field Jacobian spectrum. Validation of the damped-driven Duffing oscillator confirms the predicted rate and exponential convergence.