metastate

{{short description|Probability measure on the space of all thermodynamic states for a system with quenched randomness}}

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..{{cite journal | last1=Newman | first1=C. M. | last2=Stein | first2=D. L. | title=Spatial Inhomogeneity and Thermodynamic Chaos | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=76 | issue=25 | date=17 June 1996 | issn=0031-9007 | doi=10.1103/physrevlett.76.4821 | pages=4821–4824| pmid=10061389 | arxiv=adap-org/9511001 | bibcode=1996PhRvL..76.4821N | s2cid=871472 }}

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.{{cite journal | last1=Aizenman | first1=Michael | last2=Wehr | first2=Jan | title=Rounding effects of quenched randomness on first-order phase transitions | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=130 | issue=3 | year=1990 | issn=0010-3616 | doi=10.1007/bf02096933 | pages=489–528| bibcode=1990CMaPh.130..489A | s2cid=122417891 | url=http://projecteuclid.org/euclid.cmp/1104200601 }}

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.{{cite journal | last1=Newman | first1=C. M. | last2=Stein | first2=D. L. | title=Metastate approach to thermodynamic chaos | journal=Physical Review E | publisher=American Physical Society (APS) | volume=55 | issue=5 | date=1 April 1997 | issn=1063-651X | doi=10.1103/physreve.55.5194 | pages=5194–5211|arxiv=cond-mat/9612097| bibcode=1997PhRvE..55.5194N | s2cid=14821724 }}{{cite book | last1=Newman | first1=Charles M. | last2=Stein | first2=Daniel L. | title=Mathematical Aspects of Spin Glasses and Neural Networks | chapter=Thermodynamic Chaos and the Structure of Short-Range Spin Glasses | publisher=Birkhäuser Boston | location=Boston, MA | year=1998 | isbn=978-1-4612-8653-0 | doi=10.1007/978-1-4612-4102-7_7 | pages=243–287}}

It was proved 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).Debenedetti, P.G.Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, NJ, USA, 1996.{{cite journal | last1=Imre | first1=Attila | last2=Wojciechowski | first2=Krzysztof | last3=Györke | first3=Gábor | last4=Groniewsky | first4=Axel | last5=Narojczyk | first5=Jakub. | title=Pressure-Volume Work for Metastable Liquid and Solid at Zero Pressure | journal=Entropy | publisher=MDPI AG | volume=20 | issue=5 | date=3 May 2018 | issn=1099-4300 | doi=10.3390/e20050338 | page=338| pmid=33265428 | pmc=7512857 | bibcode=2018Entrp..20..338I |doi-access=free}}