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>\{ {\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 16:21, 10 July 2007
The Helmholtz energy function of fluid in a matrix of configuration in the Canonical ensemble is given by:
![{\displaystyle -\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f788bb8c171a0c754e3aec088d3c269ff7bed6c)
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
![{\displaystyle -\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e92efa81f2a0f6b3f035909dd33eee00c792e0e)
(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 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 \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
- 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 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}_\lambda, q^{N_0}) + H_{11}^\lambda (r^{N_1}_\lambda, q^{N_0})\right).}
This Hamiltonian describes a completely equilibrated system of 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
- 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) ] } .