Probing the extensive nature of entropy

Abstract

We have devised a general numerical scheme applied to a system of independent, distinguishable, non-interacting particles, to demonstrate in a direct manner the extensive nature of statistical entropy. Working within the microcanonical ensemble, our methods enable one to directly monitor the approach to the thermodynamic limit (N ! 1) in a manner that has not been known before. We show that (sN − s∞) ! N− where sN is the entropy per particle for N particles and s∞ is the entropy per particle in the thermodynamic limit. We demonstrate universal behaviour by considering a number of different systems each defined by its unique single-particle spectrum. Various thermodynamic quantities as a function of N may be computed using our methods; in this paper, we focus on the entropy, the chemical potential and the temperature. Our results are applicable to systems of finite size, e.g. nano-particle systems. Furthermore, we demonstrate a new phenomenon, referred to as entropic interference, which manifests as a cancellation of terms in the thermodynamic limit and which results in the additive nature of entropy.