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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s} .

One can apply this trick to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log Z_1} we want to average, and replace the resulting power Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Z_1)^s} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} copies of the expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_1} (replicas). The result is equivalent to evaluate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{A}_1} as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\beta\overline{A}_1=\lim_{s\to 0}\frac{d}{ds}\left(\frac{Z^{\rm rep}(s)}{Z_0}\right) } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z^{\rm rep}(s)} is the partition function of a mixture with Hamiltonian

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}_\lambda, q^{N_0}) + H_{11}^\lambda (r^{N_1}_\lambda, q^{N_0})\right).}

This Hamiltonian describes a completely equilibrated system of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s+1} components; the matrix the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} identical non-interacting replicas of the fluid. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0=Z^{\rm rep}(0)} , then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \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)