Replica method

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

$-\beta A_{1}(q^{N_{0}})=\log Z_{1}(q^{N_{0}})=\log \left({\frac {1}{N_{1}!}}\int \exp[-\beta (H_{11}(r^{N_{1}})+H_{10}(r^{N_{1}},q^{N_{0}}))]~d\{r\}^{N_{1}}\right)$

where $Z_{1}(q^{N_{0}})$ is the fluid partition function, and $H_{11}$, $H_{10}$ and $H_{00}$ are the pieces of the Hamiltonian corresponding to the fluid-fluid, fluid-matrix and matrix-matrix interactions. Assuming that the matrix is a configuration of a given fluid, with interaction hamiltonian $H_{00}$, we can average over matrix configurations to obtain

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

(see Refs. 1 and 2)

An important mathematical trick to get rid of the logarithm inside of the integral is to use the mathematical identity

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

One can apply this trick to the $\log Z_{1}$ we want to average, and replace the resulting power $(Z_{1})^{s}$ by $s$ copies of the expression for $Z_{1}$ (replicas). The result is equivalent to evaluate ${\overline {A}}_{1}$ as

$-\beta {\overline {A}}_{1}=\lim _{s\to 0}{\frac {d}{ds}}\left({\frac {Z^{\rm {rep}}(s)}{Z_{0}}}\right)$,

where $Z^{\rm {rep}}(s)$ is the partition function of a mixture with Hamiltonian

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

This Hamiltonian describes a completely equilibrated system of $s+1$ components; the matrix the $s$ identical non-interacting replicas of the fluid. Since $Z_{0}=Z^{\rm {rep}}(0)$, then

$\lim _{s\to 0}{\frac {d}{ds}}[-\beta A^{\rm {rep}}(s)]=\lim _{s\to 0}{\frac {d}{ds}}\log Z^{\rm {rep}}(s)=\lim _{s\to 0}{\frac {{\frac {d}{ds}}Z^{\rm {rep}}(s)}{Z^{\rm {rep}}(s)}}=\lim _{s\to 0}{\frac {{\frac {d}{ds}}Z^{\rm {rep}}(s)}{Z_{0}}}=-\beta {\overline {A}}_{1}.$

Thus the relation between the Helmholtz energy function of the non-equilibrium partially frozen system and the replicated (equilibrium) system is given by

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

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)