Replica method

Free energy of fluid in a matrix of configuration $\{q^{N_{0}}\}$ in the Canonical ($NVT$) ensemble is given by:

$-\beta F_{1}(q^{N_{0}})=\log Z_{1}(q^{N_{0}})=\log \left({\frac {1}{N_{1}!}}\int \exp[-\beta (H_{01}(r^{N_{1}},q^{N_{0}})+H_{11}(r^{N_{1}},q^{N_{0}}))]~d\{r\}^{N_{1}}\right)$

where $Z_{1}(q^{N_{0}})$ is the fluid partition function, and $H_{00}$ is the Hamiltonian of the matrix. Taking an average over matrix configurations gives

$-\beta {\overline {F}}_{1}={\frac {1}{N_{0}!Z_{0}}}\int \exp[-\beta _{0}H_{00}(q^{N_{0}})]~\log Z_{1}(q^{N_{0}})~d\{q\}^{N_{0}}$

\cite{JPFMP_1975_05_0965,JPAMG_1976_09_01595} Important mathematical trick to get rid of the logarithm inside of the integral:

$\log x=\lim _{s\rightarrow 0}{\frac {\rm {d}}{{\rm {d}}s}}x^{s}$

one arrives at

$\beta H^{\rm {rep}}(r^{N_{1}},q^{N_{0}})={\frac {\beta _{0}}{\beta }}H_{00}(q^{N_{0}})+\sum _{\lambda =1}^{s}\left(H_{01}^{\lambda }(r^{N_{1}},q^{N_{0}})+H_{11}^{\lambda }(r^{N_{1}},q^{N_{0}})\right)$

The Hamiltonian written in this form describes a completely equilibrated system of $s+1$ components; the matrix and $s$ identical non-interacting copies (replicas) of the fluid. Thus the relation between the free energy of the non-equilibrium partially frozen and the replica (equilibrium) system is given by

$-\beta {\overline {F}}_{1}=\lim _{s\rightarrow 0}{\frac {\rm {d}}{{\rm {d}}s}}[-\beta F^{\rm {rep}}(s)]$

References

1. S F Edwards and P W Anderson "Theory of spin glasses",Journal of Physics F: Metal Physics 5 pp. 965-974 (1975)
2. 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)