Uncertainty Relations in Statistical Mechanics: a numerical study in small lattice systems

TL;DR

This paper numerically investigates the applicability of uncertainty relations in small lattice systems within statistical mechanics, focusing on Ising models in contact with reservoirs, and identifies conditions where these relations hold or break down.

Contribution

It provides a numerical analysis of uncertainty relations in small lattice systems, highlighting their limitations in systems with high boundary state probabilities.

Findings

Uncertainty relations are valid for small lattice systems under certain conditions.

High probability of boundary states leads to invalidity of the relations.

The study applies to systems with discrete variables like Ising models.

Abstract

We have analyzed the validity of uncertainty relations between the fluctuations of thermodynamically conjugated extensive and intensive variables within the field of statistical mechanics. Analysis is presented for two particular examples of small lattice systems that are in contact with reservoirs of comparable sizes: an Ising paramagnet and an Ising ferromagnet. The numerical results enable determination of the range of applicability of the proposed relations. Due to the fact that the examples correspond to systems described by discrete variables, the uncertainty relations are not valid if the probability of boundary states is too high.