Document Type


Publication Date


Publication Source

Journal of Chemical Physics


Increasing interest in the thermodynamics of small and/or isolated systems, in combination with recent observations of negative temperatures of atoms in ultracold optical lattices, has stimulated the need for estimating the conventional, canonical temperature Tconvc of systems in equilibrium with heat baths using eigenstate-specific temperatures (ESTs). Four distinct ESTs—continuous canonical, discrete canonical, continuous microcanonical, and discrete microcanonical—are accordingly derived for two-level paramagnetic spin lattices (PSLs) in external magnetic fields. At large N, the four ESTs are intensive, equal to Tconvc, and obey all four laws of thermodynamics. In contrast, for N < 1000, the ESTs of most PSL eigenstates are non-intensive, differ from Tconvc, and violate each of the thermodynamic laws. Hence, in spite of their similarities to Tconvc at large N, the ESTs are not true thermodynamic temperatures. Even so, each of the ESTs manifests a unique functional dependence on energy which clearly specifies the magnitude and direction of their deviation from Tconvc; the ESTs are thus good temperature estimators for small PSLs. The thermodynamic uncertainty relation is obeyed only by the ESTs of small canonical PSLs; it is violated by large canonical PSLs and by microcanonical PSLs of any size. The ESTs of population-inverted eigenstates are negative (positive) when calculated using Boltzmann (Gibbs) entropies; the thermodynamic implications of these entropically induced differences in sign are discussed in light of adiabatic invariance of the entropies. Potential applications of the four ESTs to nanothermometers and to systems with long-range interactions are discussed.



Document Version



The document available for download is the authors' accepted manuscript, provided in compliance with the publisher's policy on self-archiving. Permission documentation is on file. Supplemental information is available using the link provided. For the article's version of record, use the DOI provided.

Some mathematical and chemical symbols may not appear correctly in the metadata for this item; please see the document available for download for the correct display.


AIP Publishing



Peer Reviewed




Supplemental materials.pdf (811 kB)
Appendices and references

Link to published version