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:

$-\beta A_1({\mathbf{q}}^{N_0})=\log Z_1({\mathbf{q}}^{N_0})=\log (\frac{1}{N_1!}\int \exp [-\beta (H_{11}({\mathbf{r}}^{N_1})+H_{10}({\mathbf{r}}^{N_1},{\mathbf{q}}^{N_0}))]d\{\mathbf{r}{\}}^{N_1})$

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

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

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

$-\beta {\overline{A}}_1=\underset{s\to 0}{lim}\frac{d}{ds}(\frac{Z^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)}{Z_0})$,

where $Z^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)$ is the partition function of a mixture with Hamiltonian

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

This Hamiltonian describes a completely equilibrated system of $s+1$ components; the matrix the $s$ identical non-interacting replicas of the fluid. Since $Z_0=Z^{\mathrm{r}\mathrm{e}\mathrm{p}}(0)$, then

$\underset{s\to 0}{lim}\frac{d}{ds}[-\beta A^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)]=\underset{s\to 0}{lim}\frac{d}{ds}\log Z^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)=\underset{s\to 0}{lim}\frac{\frac{d}{ds}{Z}^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)}{Z^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)}=\underset{s\to 0}{lim}\frac{\frac{d}{ds}{Z}^{\mathrm{r}\mathrm{e}\mathrm{p}}(s)}{Z_0}=-\beta {\overline{A}}_1.$

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

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

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)