The Helmholtz energy function of fluid in a matrix of configuration
in the Canonical () ensemble is given by:
where is the fluid partition function, and
is the Hamiltonian of the matrix.
Taking an average over matrix configurations gives
(Ref.s 1 and 2) An important mathematical trick to get rid of the logarithm inside of the integral is
one arrives at
The Hamiltonian written in this form describes a completely equilibrated system
of components; the matrix and identical non-interacting copies (replicas) of the fluid.
Thus the relation between the Helmholtz energy function of the non-equilibrium partially frozen
and the replica (equilibrium) system is given by
- .
References
- S F Edwards and P W Anderson "Theory of spin glasses",Journal of Physics F: Metal Physics 5 pp. 965-974 (1975)
- S F Edwards and R C Jones "The eigenvalue spectrum of a large symmetric random matrix", Journal of Physics A: Mathematical and General 9 pp. 1595-1603 (1976)