Replica method: Difference between revisions
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:<math>- \beta A_1 (q^{N_0}) = \log Z_1 (q^{N_0}) | :<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_{ | \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> | ||
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> | ||
( | (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 | ||
:<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}) | :<math>\beta H^{\rm rep} (r^{N_1}, q^{N_0}) | ||
= \frac{\beta_0}{\beta}H_{00} (q^{N_0}) + \sum_{\lambda=1}^s | = \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> | ||
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 | and the replica (equilibrium) system is given by | ||
Revision as of 12:05, 24 May 2007
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 \{ q^{N_0} \}} in the Canonical (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 NVT} ) 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 (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 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 (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 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}} , 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 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 Z_1} 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
- 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{Z^{\rm rep}(s)}{Z_0} } ,
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 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) ] } .