thermodynamics - Non-extensivity in a few body system

Thermodynamics finds application in many areas of physics, many of them sharing the feature of fluid-like or many-body like behavior.

However, small systems or few body systems have been studied too using thermodynamics. But generally it is accepted that these systems behave differently from macroscopic systems: they are non-extensive.

Non-extensivity generally is deduced from the fact that extensive properties in these systems don't add up the way they would in macroscopic ones. But I am struggling to find hard evidence of this, because merging two systems, at least in a thermodynamics view, implies that we should check the equilibrium cases only.

I mean, when I say that for two non extensive systems with entropies $S_a$ and $S_b$ it doesn't hold $S_a + S_b = S_{a+b}$, I should be checking that all these systems are under the same conditions regarding intensive properties, right? If not, then not even macroscopic systems would exhibit extensivity.

My question is: Are there publications checking this aspects for certain small systems?

I am searching for a papers where this is checked, because I have not seen this done and find it strange.

Answer

A recent work that uses nanothermodynamics and includes a computational investigation of the kind you are asking about for an Ising lattice:

R.V. Chamberlin, The Big World of Nanothermodynamics

Sec.5 of the following paper makes a reference to another paper that appears to have tested the limits of usual thermodynamics in single polymer stretching experiments:

J. M. Rubi, D. Bedeaux, and S. Kjelstrup, Thermodynamics for Single-Molecule Stretching Experiments, J. Phys. Chem. B 2006, 110, 12733-12737

You may also find some clues in discussion and refs from Secs. 2.3-2.5 of this paper:

T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens, Dynamics and Thermodynamics of Systems with Long-Range Interactions: An Introduction

It is actually the Intro to this volume: Dynamics and Thermodynamics of Systems with Long Range Interactions, Eds. T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens (Google Books)