Derivation of Bose-Einstein statistics from the uncertainty principle

TL;DR

This paper demonstrates that assuming a finite lower bound on phase space area for classical systems leads to energy distributions that match Bose-Einstein statistics, bridging classical uncertainty and quantum behavior.

Contribution

It derives Bose-Einstein statistics from classical principles by imposing a finite phase space area constraint based on the uncertainty principle.

Findings

Energy distribution follows Bose-Einstein form under the bounded phase space assumption.

Connects classical uncertainty limits to quantum statistical distributions.

Provides a classical derivation of quantum statistical behavior.

Abstract

The microstate of any degree of freedom of any classical dynamical system can be represented by a point in its two dimensional phase space. Since infinitely precise measurements are impossible, a measurement can, at best, constrain the location of this point to a region of phase space whose area is finite. This paper explores the implications of assuming that this finite area is bounded from below. I prove that if the same lower bound applied to every degree of freedom of a sufficiently cold classical dynamical system, the distribution of the system's energy among its degrees of freedom would be a Bose-Einstein distribution.