Replica method

Free energy of fluid in a matrix of configuration $\{q^{N_0}\}$ in the Canonical ($NVT$) ensemble is given by:

$-\beta F_1(q^{N_0})=\log Z_1(q^{N_0})=\log (\frac{1}{N_1!}\int \exp [-\beta (H_{01}(r^{N_1},q^{N_0})+H_{11}(r^{N_1},q^{N_0}))]d\{r{\}}^{N_1})$

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:

$\log x=\underset{s\to 0}{lim}\frac{\mathrm{d}}{\mathrm{d}s}{x}^s$

one arrives at

$\beta H^{\mathrm{r}\mathrm{e}\mathrm{p}}(r^{N_1},q^{N_0})=\frac{{\beta }_0}{\beta }{H}_{00}(q^{N_0})+\sum _{\lambda =1}^s(H_{01}^{\lambda }(r^{N_1},q^{N_0})+H_{11}^{\lambda }(r^{N_1},q^{N_0}))$

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

$-\beta {\overline{F}}_1=\underset{s\to 0}{lim}\frac{\mathrm{d}}{\mathrm{d}s}[-\beta F^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)]$

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)