# 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.[1][clarification needed] 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 thermodynamic states, where the system is in metastable state (for example superheating or undercooling liquids, when the actual temperature are above or below the boiling or freezing temperature, but the material is still in liquid state).[5][6]

## References

1. ^ a b Newman, C. M.; Stein, D. L. (17 June 1996). "Spatial Inhomogeneity and Thermodynamic Chaos". Physical Review Letters. American Physical Society (APS). 76 (25): 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. Springer Science and Business Media LLC. 130 (3): 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. American Physical Society (APS). 55 (5): 5194–5211. arXiv:cond-mat/9612097. Bibcode:1997PhRvE..55.5194N. doi:10.1103/physreve.55.5194. ISSN 1063-651X. S2CID 14821724.
4. ^ a b 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. MDPI AG. 20 (5): 338. Bibcode:2018Entrp..20..338I. doi:10.3390/e20050338. ISSN 1099-4300.