Replica method

This article is about integral equations. For other the simulation method, see Replica-exchange simulated tempering or Replica-exchange molecular dynamics.

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

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

where $Z_{1}({\mathbf {q} }^{N_{0}})$ 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 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 ${\overline {A}}_{1}$ 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

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

This Hamiltonian describes a completely equilibrated system of 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

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

Interesting reading

• Viktor Dotsenko "Introduction to the Replica Theory of Disordered Statistical Systems", Collection Alea-Saclay: Monographs and Texts in Statistical Physics, Cambridge University Press (2000)

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)