Temperature fluctuations in finite systems: Application to the one-dimensional Ising chain

TL;DR

This paper investigates temperature fluctuations in a finite one-dimensional Ising chain, demonstrating limitations of superstatistics and proposing a new framework for describing microcanonical regions with positive heat capacity.

Contribution

It shows that superstatistics cannot describe temperature fluctuations in the Ising chain and introduces a potential new framework for microcanonical systems with positive heat capacity.

Findings

Superstatistics fails to describe the Ising chain subsystem.

A relation between fundamental and microcanonical inverse temperatures is verified.

Suggests a new class of statistical ensembles beyond superstatistics.

Abstract

The theory of superstatistics, originally proposed for the study of complex nonequilibrium systems, has recently been extended to studies of small systems interacting with a finite environment, because such systems display interestingly similar statistical behavior. In both situations there are several applicable definitions of inverse temperature, either intrinsic or dependent of the statistical ensemble. In this work we develop these concepts focusing our attention on a region of an isolated, one-dimensional Ising chain as an example of a subsystem that does not follow the canonical Gibbs distribution. For this example, we explicitly show that superstatistics cannot describe the behavior of the subsystem, and verify a recently reported relation between the fundamental and microcanonical inverse temperatures. Our results hint at a new framework for dealing with regions of microcanonical systems with positive heat capacity, which should be described by some new class of statistical ensembles outside superstatistics but still preserving the notion of temperature fluctuations.