Derivation of Bose-Einstein statistics from the uncertainty principle

TL;DR

The paper argues that imposing a universal lower bound on microstate-precision, $\,\Delta_Q \Delta_P > h_{?} > 0$, on all classical degrees of freedom drives the energy distribution into the Bose-Einstein form in the low-temperature limit, without requiring quantization of energy. Building on Jaynes' maximum-entropy framework, it introduces unfalsifiable statistical models and a probability-domain quantization that constrains the domain of testable distributions to partitions of phase space. By transforming each degree of freedom's Hamiltonian into an affine form (via action-angle coordinates for oscillators), the Maxwell-Boltzmann distribution is recovered as a baseline, while Bose-Einstein statistics naturally arise for sufficiently cold, weakly interacting systems and can be extended to non-oscillatory cases. The work highlights that BE statistics can emerge purely from information-theoretic considerations and a universal uncertainty bound, with potential implications for interpreting blackbody spectra, zero-point energies, and the boundary between classical and quantum descriptions. It does not claim microstates are quantized; rather, it shows that testable probability distributions acquire BE form under the stated uncertainty principle, linking classical dynamics to quantum-like statistics in a fundamental way.

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.

Figures (1)

• Figure 1: (a) A portion of the phase space $\mathbb{G}_{\eta}$ of mode ${\eta}$. The continuous blue ellipse, ${\partial\mathcal{R}_{\eta}}$, is a particular constant-energy path that the oscillation follows when it is decoupled from other modes. The set of pale blue and green spots is a maximal set of mutually-distinguishable microstates, $\Gamma_{\eta}$, of mode ${\eta}$. In statistical models of the mode's microstates, each spot represents all points within its rectangular neighbourhood. (b) A portion of the microstructure space of modes ${\eta}$ and ${\nu}$. The spots belong to a maximal set of mutually distinguishable points and represent the rectangular regions they inhabit. The pale blue spots mark regions visited during the motion of the modes, assuming that their energies, ${\mathscr{E}_{\eta}}$ and ${\mathscr{E}_{\nu}}$, are constant and that neither of their frequencies, $\omega_{\eta}$ and $\omega_{\nu}$, is an integer multiple of the other. Each of the $15$ pale blue spots represents the four points ${(Q_{\eta},Q_{\nu},P_\eta,P_\nu)= \left(Q_\eta,Q_\nu, \pm\sqrt{2\mathscr{E}_{\eta}-\omega_{\eta}^2Q_{\eta}^2}, \pm\sqrt{2\mathscr{E}_{\nu}-\omega_{\nu}^2Q_{\nu}^2}\right)}$ in their joint phase space ${\mathbb{G}_{\eta}\times\mathbb{G}_{\nu}}$. (c) The pale blue spot is the energy of the trajectory represented by pale blue spots in panels (a) and (b). We cannot calculate the partition function of modes ${\eta}$ and ${\nu}$ as ${\mathcal{Z}_{\eta}\mathcal{Z}_{\nu} =\sum_{\mathscr{E}_{\eta}}\sum_{\mathscr{E}_{\nu}} e^{-\beta\left(\mathscr{E}_{\eta}+\mathscr{E}_{\nu}\right)}}$ if the double summation is over a square grid in ${(\mathscr{E}_{\eta},\mathscr{E}_{\nu})}$-space. The numbers of energies sampled along each axis are only in the same ratio as the numbers of mutually-distinguishable mode coordinates along each axis in ${(Q_{\eta},Q_{\nu})}$-space, and the numbers of mutually-distinguishable points in ${\mathbb{G}_{\eta}}$ and ${\mathbb{G}_{\nu}}$, if the spacings of sampled values along the mode's energy axes are their frequencies times the same constant.