Temperature distribution in finite systems: application to the one-dimensional Ising chain

Thermodynamic studies of small systems interacting with a finite environment display an interesting statistical behavior, similar to complex non-equilibrium systems. In both situations there are several applicable definitions of inverse temperature, either intrinsic or dependent of the statistical ensemble, and uncertainty in these quantities has to be taken into account. In this work we develop these concepts using as an example an isolated one-dimensional Ising chain subsystem that does not follow the canonical distribution. In the context of this example, we explicitly show that the theory of superstatistics cannot describe the behavior of the subsystem, and verify a recently reported relation between the ensemble 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 uncertainty.