The Bekenstein Bound in Asymptotically Free Field Theory

TL;DR

This paper investigates whether the Bekenstein bound on specific entropy holds in an asymptotically free scalar field theory, analyzing thermodynamic properties at different temperatures using the effective potential.

Contribution

It provides a detailed analysis of the Bekenstein bound within the $( ext{lambda}\, ext{phi}^4)_d$ theory, including thermodynamic calculations and conditions for the bound's validity.

Findings

The system exhibits condensates at low and high temperatures.

The renormalized energy and entropy are computed explicitly.

Conditions under which the Bekenstein bound is satisfied are identified.

Abstract

For spatially bounded free fields, the Bekenstein bound states that the specific entropy satisfies the inequality $\frac{S}{E} \leq 2 \pi R$, where $R$ stands for the radius of the smallest sphere that circumscribes the system. The validity of the Bekenstein bound on the specific entropy in the asymptotically free side of the Euclidean $(\lambda\,\phi^{\,4})_{d}$ self-interacting scalar field theory is investigated. We consider the system in thermal equilibrium with a reservoir at temperature $\beta^{\,-1}$ and defined in a compact spatial region without boundaries. Using the effective potential, we presented an exhaustive study of the thermodynamic of the model. For low and high temperatures the system presents a condensate. We obtain also the renormalized mean energy $E$ and entropy $S$ for the system. With these quantities, we shown in which situations the specific entropy satisfies the quantum bound.