Editing Replica method
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:''This article is about integral equations. For other the simulation method, see [[Replica-exchange simulated tempering]] or [[Replica-exchange molecular dynamics]]''. | :''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 | The [[Helmholtz energy function]] of fluid in a matrix of configuration | ||
<math>\{ | <math>\{ q^{N_0} \}</math> in the Canonical (<math>NVT</math>) ensemble is given by: | ||
:<math>- \beta A_1 ( | :<math>- \beta A_1 (q^{N_0}) = \log Z_1 (q^{N_0}) | ||
= \log \left( \frac{1}{N_1!} | = \log \left( \frac{1}{N_1!} | ||
\int \exp [- \beta (H_{11}( | \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 ( | 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> | ||
are the pieces of the | 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> |