Number Fluctuation and the Fundamental Theorem of Arithmetic

TL;DR

This paper explores how the fundamental theorem of arithmetic influences number fluctuations in quantum systems, revealing a case where ground state fluctuations vanish, contrasting with standard statistical mechanics predictions.

Contribution

It introduces a novel quantum spectrum based on prime number logarithms, showing exact ground state fluctuation suppression due to the fundamental theorem of arithmetic.

Findings

Ground state fluctuation vanishes exactly for the prime logarithm spectrum.

Standard ensembles predict substantial fluctuations, unlike the prime spectrum case.

Microcanonical and canonical ensembles yield fundamentally different fluctuation results.

Abstract

We consider N bosons occupying a discrete set of single-particle quantum states in an isolated trap. Usually, for a given excitation energy, there are many combinations of exciting different number of particles from the ground state, resulting in a fluctuation of the ground state population. As a counter example, we take the quantum spectrum to be logarithms of the prime number sequence, and using the fundamental theorem of arithmetic, find that the ground state fluctuation vanishes exactly for all excitations. The use of the standard canonical or grand canonical ensembles, on the other hand, gives substantial number fluctuation for the ground state. This difference between the microcanonical and canonical results cannot be accounted for within the framework of equilibrium statistical mechanics.