Metastate
Encyclopedia
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 . 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) 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
.
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.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, the metastate is a probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
on the
space of all thermodynamic states for a system with quenched randomness. The term metastate, in this context, was first used in . Two different versions have been proposed:
1) The Aizenman-Wehr construction, a canonical ensemble
Canonical ensemble
The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
approach,
constructs the metastate through an ensemble of states obtained by varying
the random parameters in the Hamiltonian
Molecular Hamiltonian
In atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and nuclei in a molecule...
outside of the volume being
considered .
2) The Newman-Stein metastate, a microcanonical ensemble
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
approach,
constructs an empirical average from a deterministic (i.e., chosen
independently of the randomness) subsequence of finite-volume Gibbs distributions
Gibbs measure
In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is the measure associated with the Boltzmann distribution, and generalizes the notion of the canonical ensemble...
.
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
Thermodynamic state
A thermodynamic state is a set of values of properties of a thermodynamic system that must be specified to reproduce the system. The individual parameters are known as state variables, state parameters or thermodynamic variables. Once a sufficient set of thermodynamic variables have been...
, and/or there are many competing
observable thermodynamic states.