Replica method
The Helmholtz energy function of fluid in a matrix of configuration 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 \{ {\mathbf q}^{N_0} \}} in the Canonical ensemble 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 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)}
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_1 ({\mathbf q}^{N_0})} is the fluid partition function, 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 H_{11}} , 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 H_{10}} 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 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 , we can average over matrix configurations to obtain
- 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 = \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
- 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 \log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s} .
One can apply this trick to the we want to average, and replace the resulting power 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_1)^s} 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 s} copies of the expression for 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_1} (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
- ,
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 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 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) ] } .