Eigenstate-specific temperatures in two-level paramagnetic spin lattices

Mark B Masthay; Calley N Eads; Amber N Johnson; Robert G Keil; Philip Miller; Ross E Jones; Joe D Mashburn; Harry B Fannin

doi:10.1063/1.5012914

Eigenstate-specific temperatures in two-level paramagnetic spin lattices

J Chem Phys. 2017 Dec 7;147(21):214306. doi: 10.1063/1.5012914.

Authors

Mark B Masthay¹, Calley N Eads², Amber N Johnson¹, Robert G Keil¹, Philip Miller³, Ross E Jones³, Joe D Mashburn⁴, Harry B Fannin³

Affiliations

¹ Department of Chemistry, University of Dayton, 300 College Park, Dayton, Ohio 45469-2357, USA.
² Department of Chemistry and Biochemistry, The University of Arizona, P.O. Box 210041 1306 East University Boulevard, Tucson, Arizona 85721-0041, USA.
³ Department of Chemistry, Murray State University, 1201 Jesse D. Jones Hall, Murray, Kentucky 42071, USA.
⁴ Department of Mathematics, University of Dayton, 300 College Park, Dayton, Ohio 45469-2316, USA.

PMID: 29221376
DOI: 10.1063/1.5012914

Abstract

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 T_c^conv 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 T_c^conv, and obey all four laws of thermodynamics. In contrast, for N < 1000, the ESTs of most PSL eigenstates are non-intensive, differ from T_c^conv, and violate each of the thermodynamic laws. Hence, in spite of their similarities to T_c^conv 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 T_c^conv; 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.