Replica method: Difference between revisions

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The [[Helmholtz energy function]] of fluid in a matrix of configuration  
The [[Helmholtz energy function]] of fluid in a matrix of configuration  
<math>\{ q^{N_0} \}</math> in the Canonical (<math>NVT</math>) ensemble is given by:
<math>\{ {\mathbf q}^{N_0} \}</math> in the [[Canonical ensemble]] is given by:


:<math>- \beta A_1 (q^{N_0}) = \log Z_1  (q^{N_0})
:<math>- \beta A_1 ({\mathbf q}^{N_0}) = \log Z_1  ({\mathbf q}^{N_0})
= \log \left( \frac{1}{N_1!}  
= \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)</math>
\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)</math>


where <math>Z_1  (q^{N_0})</math> is the fluid [[partition function]], and <math>H_{11}</math>, <math>H_{10}</math> and <math>H_{00}</math>
where <math>Z_1  ({\mathbf q}^{N_0})</math> is the fluid [[partition function]], and <math>H_{11}</math>, <math>H_{10}</math> and <math>H_{00}</math>
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 <math>H_{00}</math>, we can average over matrix configurations to obtain
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 <math>H_{00}</math>, we can average over matrix configurations to obtain


:<math>- \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}</math>
:<math>- \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}</math>

Revision as of 17:21, 10 July 2007

The Helmholtz energy function of fluid in a matrix of configuration in the Canonical ensemble is given by:

where is the fluid partition function, and , and 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 , we can average over matrix configurations to obtain

(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 we want to average, and replace the resulting power by copies of the expression for (replicas). The result is equivalent to evaluate as

,

where is the partition function of a mixture with Hamiltonian

This Hamiltonian describes a completely equilibrated system of components; the matrix the identical non-interacting replicas of the fluid. Since , then

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

.

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)