Replica method: Difference between revisions

Revision as of 13:05, 24 May 2007

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

\log x=\lim _{s\rightarrow 0}{\frac {\rm {d}}{{\rm {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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -\beta {\overline {A}}_{1}={\frac {Z^{\rm {rep}}(s)}{Z_{0}}}} ,

where $Z^{\rm {rep}}(s)$ 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 and 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 copies (replicas) of the fluid. Thus the relation between the Helmholtz energy function of the non-equilibrium partially frozen and the replica (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

@@ Line 4: / Line 4: @@
 :<math>- \beta A_1 (q^{N_0}) = \log Z_1  (q^{N_0})
 = \log \left( \frac{1}{N_1!}
-\int \exp [- \beta (H_{01}(r^{N_1}, q^{N_0}) + H_{11}(r^{N_1}, q^{N_0}) )]~d \{ r \}^{N_1} \right)</math>
+\int \exp [- \beta (H_{11}(r^{N_1}) + H_{10}(r^{N_1}, q^{N_0}) )]~d \{ r \}^{N_1} \right)</math>
-where <math>Z_1  (q^{N_0})</math> is the fluid [[partition function]], and <math>H_{00}</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>
-is the Hamiltonian of the matrix.
+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
-Taking an average over matrix configurations gives
 :<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>
-(Ref.s 1 and 2) An important mathematical trick to get rid of the logarithm inside of the integral is
+(see Refs. 1 and 2)
-:<math>\log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s</math>
+:An important mathematical trick to get rid of the logarithm inside of the integral is to use the mathematical identity
-one arrives at
+:<math>\log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s</math>.
+One can apply this trick to the <math>\log Z_1</math> we want to average, and replace the resulting power <math>(Z_1)^s</math> by <math>s</math> copies of the expression for <math>Z_1</math> ''(replicas)''. The result is equivalent to evaluate <math>\overline{A}_1</math> as
+:<math> -\beta\overline{A}_1=\frac{Z^{\rm rep}(s)}{Z_0} </math>,
+where <math>Z^{\rm rep}(s)</math> is the partition function of a mixture with Hamiltonian
 :<math>\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}, q^{N_0}) +  H_{11}^\lambda (r^{N_1}, q^{N_0})\right)</math>
+\left( H_{01}^\lambda (r^{N_1}_\lambda, q^{N_0}) +  H_{11}^\lambda (r^{N_1}_\lambda, q^{N_0})\right).</math>
-The Hamiltonian written in this form describes a completely equilibrated system
+This Hamiltonian describes a completely equilibrated system of <math>s+1</math> components; the matrix and <math>s</math> identical non-interacting copies (''replicas'') of the fluid. Thus the relation between the [[Helmholtz energy function]] of the non-equilibrium partially frozen
-of <math>s+1</math> components; the matrix and <math>s</math> identical non-interacting copies (''replicas'') of the fluid.
-Thus the relation between the [[Helmholtz energy function]] of the non-equilibrium partially frozen
 and the replica (equilibrium) system is given by

Replica method: Difference between revisions

Revision as of 13:05, 24 May 2007

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