Metastate

In statistical mechanics, the metastate is a probability measure on the space of all thermodynamic states for a system with quenched randomness. The term metastate, in this context, was first used in by Charles M. Newman and Daniel L. Stein in 1996..^[1]

Two different versions have been proposed:

1) The Aizenman-Wehr construction, a canonical ensemble approach, constructs the metastate through an ensemble of states obtained by varying the random parameters in the Hamiltonian outside of the volume being considered.^[2]

2) The Newman-Stein metastate, a microcanonical ensemble approach, constructs an empirical average from a deterministic (i.e., chosen independently of the randomness) subsequence of finite-volume Gibbs distributions.^[1][3][4]

It was proved^[4] for Euclidean lattices that there always exists a deterministic subsequence along which the Newman-Stein and Aizenman-Wehr constructions result in the same metastate. The metastate is especially useful in systems where deterministic sequences of volumes fail to converge to a thermodynamic state, and/or there are many competing observable thermodynamic states.

As an alternative usage, "metastate" can refer to thermodynamic states, where the system is in a metastable state (for example superheated or undercooled liquids, when the actual temperature of the liquid is above or below the boiling or freezing temperature, but the material is still in a liquid state).^[5][6]

References

1. Newman, C. M.; Stein, D. L. (17 June 1996). "Spatial Inhomogeneity and Thermodynamic Chaos". Physical Review Letters. 76 (25). American Physical Society (APS): 4821–4824. arXiv:adap-org/9511001. Bibcode:1996PhRvL..76.4821N. doi:10.1103/physrevlett.76.4821. ISSN 0031-9007. PMID 10061389. S2CID 871472.
2. Aizenman, Michael; Wehr, Jan (1990). "Rounding effects of quenched randomness on first-order phase transitions". Communications in Mathematical Physics. 130 (3). Springer Science and Business Media LLC: 489–528. Bibcode:1990CMaPh.130..489A. doi:10.1007/bf02096933. ISSN 0010-3616. S2CID 122417891.
3. Newman, C. M.; Stein, D. L. (1 April 1997). "Metastate approach to thermodynamic chaos". Physical Review E. 55 (5). American Physical Society (APS): 5194–5211. arXiv:cond-mat/9612097. Bibcode:1997PhRvE..55.5194N. doi:10.1103/physreve.55.5194. ISSN 1063-651X. S2CID 14821724.
4. Newman, Charles M.; Stein, Daniel L. (1998). "Thermodynamic Chaos and the Structure of Short-Range Spin Glasses". Mathematical Aspects of Spin Glasses and Neural Networks. Boston, MA: Birkhäuser Boston. pp. 243–287. doi:10.1007/978-1-4612-4102-7_7. ISBN 978-1-4612-8653-0.
5. Debenedetti, P.G.Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, NJ, USA, 1996.
6. Imre, Attila; Wojciechowski, Krzysztof; Györke, Gábor; Groniewsky, Axel; Narojczyk, Jakub. (3 May 2018). "Pressure-Volume Work for Metastable Liquid and Solid at Zero Pressure". Entropy. 20 (5). MDPI AG: 338. Bibcode:2018Entrp..20..338I. doi:10.3390/e20050338. ISSN 1099-4300. PMC 7512857. PMID 33265428.