The lower bound on the energy for bounded systems is equivalent to the Bekenstein upper bound on the entropy to energy ratio for bounded systems

TL;DR

This paper explores the equivalence between the lower energy bounds in bounded systems and Bekenstein's upper entropy-to-energy ratio bound, using quantum, thermodynamic, and information theory approaches.

Contribution

It demonstrates that the lower energy bounds are mathematically equivalent to Bekenstein's entropy-to-energy upper bound through multiple theoretical frameworks.

Findings

Energy bounds derived from eigenmode analysis match Bekenstein's entropy bound

Quantum and thermodynamic approaches yield consistent results

Information theory provides insights into measurement variance and entropy

Abstract

Several approaches were used to proof the assumption that an universal upper bound on the entropy to energy ratio (S/E) exists in bounded systems. In 1981 Jacob D. Bekenstein published his findings that S/E is limited by the effective radius of the system and mentioned various approaches to derive S/E employing quantum statistics or thermodynamics. It can be shown that similar results are obtained considering the energetic difference of longitudinal eigenmodes inside a closed cavity like it was done by Max Planck in 1900 to derive the correct formula for the spectral distribution of the black-body radiation. Considering an information theoretical approach this derivation suggests that the variance of an expectation value is the same like the variance of the probability for measuring this expectation value. Implications of these findings are shortly discussed.