Replica method
From SklogWiki
The Helmholtz energy function of fluid in a matrix of configuration
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)](/SklogWiki/images/math/2/1/2/2126f831921d535f8cc5296fe22d4b64.png)
where 
is the fluid partition function, and H11, H10 and H00
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 H00, 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}](/SklogWiki/images/math/2/9/e/29e2f88a9424377f26c8448c3864cb76.png)
(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
.
One can apply this trick to the logZ1 we want to average, and replace the resulting power (Z1)s by s copies of the expression for Z1 (replicas). The result is equivalent to evaluate 
as
,
where Zrep(s) is the partition function of a mixture with Hamiltonian
This Hamiltonian describes a completely equilibrated system of s + 1 components; the matrix the s identical non-interacting replicas of the fluid. Since Z0 = Zrep(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.](/SklogWiki/images/math/d/5/9/d59098a9bf6aeef1f7b144c063e72b98.png)
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) ]](/SklogWiki/images/math/e/9/2/e92727c64703c451f23841184fc003ed.png)
.
[edit] 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)
