# 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] 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.

## References

1. ^ a b Newman, C. M.; Stein, D. L. (1996), Spatial inhomogeneity and thermodynamic chaos, Phys. Rev. Lett., Vol. 76: 4821-4824.
2. ^ Aizenman, M.; Wehr, J. (1990), Rounding effects of quenched randomness on first-order phase transitions, Commun. Math. Phys., Vol. 130: 489-528.
3. ^ Newman, C. M.; Stein, D. L. (1997), Metastate approach to thermodynamic chaos, Phys. Rev. E, Vol. 55: 5194-5211.
4. ^ a b Newman, C. M.; Stein, D. L. (1998). "Thermodynamic chaos and the structure of short-range spin glasses." in A. Bovier and P. Picco. Mathematics of Spin Glasses and Neural Networks. Boston: Birkhauser. pp. 243-247.