In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, the metastate is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
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..
Two different versions have been proposed:
1) The
Aizenman-Wehr construction, a
canonical ensemble
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the hea ...
approach,
constructs the metastate through an ensemble of states obtained by varying
the random parameters in the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
outside of the volume being
considered.
2) The
Newman-
Stein metastate, a
microcanonical ensemble
In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
approach,
constructs an empirical average from a deterministic (i.e., chosen
independently of the randomness) subsequence of finite-volume
Gibbs distributions.
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]
References
Statistical mechanics
Condensed matter physics