The emergence of Newtonian mechanics from the inhomogeneity of an ensemble

TL;DR

This paper derives Newtonian mechanics from statistical principles applied to a dilute gas, showing how Newton's second law and forces emerge from distribution inhomogeneity without assuming classical mechanics.

Contribution

It introduces a statistical framework that derives particle dynamics and forces, including entropic forces, from inhomogeneity in distributions, offering a new perspective beyond classical mechanics.

Findings

Newton's second law emerges at equilibrium.

External force magnitude relates to distribution inhomogeneity.

Entropic force equals statistical force, with deviations out of equilibrium.

Abstract

To address the observation of Max Born (M. Born 1969) that the Newton's second law can emerge from a purely statistical perspective, we derive the evolution equation about the statistical distribution for dilute gas based solely on statistical principles, without invoking Newtonian mechanics, and then obtain the equations of motion for individual particles. Newton's second law for a single particle naturally emerges when the distribution reaches equilibrium. We demonstrate that the magnitude of an external force, traditionally measured by particle acceleration, can be understood as a measure of distribution inhomogeneity. We further show that the entropic force (utilized in current gravity studies) is equivalent to the statistical force and under non-equilibrium conditions, a deviation arises between the entropic force and the Newtonian force. This framework offers a novel perspective distinct from classical Newtonian mechanics and broadens the potential scope of its application.