Replica method: Difference between revisions
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Carl McBride (talk | contribs) (New page: Free energy of fluid in a matrix of configuration <math>\{ q^{N_0} \}</math> in the Canonical (<math>NVT</math>) ensemble is given by: :<math>- \beta F_1 (q^{N_0}) = \log Z_1 (q^{N_0}) ...) |
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The [[Helmholtz energy function]] of fluid in a matrix of configuration | |||
<math>\{ q^{N_0} \}</math> in the Canonical (<math>NVT</math>) ensemble is given by: | <math>\{ q^{N_0} \}</math> in the Canonical (<math>NVT</math>) ensemble is given by: | ||
:<math>- \beta | :<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_{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_{01}(r^{N_1}, q^{N_0}) + H_{11}(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_{00}</math> | ||
is the Hamiltonian of the matrix. | is the Hamiltonian of the matrix. | ||
Taking an average over matrix configurations gives | Taking an average over matrix configurations gives | ||
:<math>- \beta \overline{ | :<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 | |||
:<math>\log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s</math> | :<math>\log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s</math> | ||
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The Hamiltonian written in this form describes a completely equilibrated system | The Hamiltonian written in this form 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. | 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 | 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 | ||
:<math>- \beta \overline{ | :<math>- \beta \overline{A}_1 = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s} [- \beta A^{\rm rep} (s) ] | ||
</math> | </math>. | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1088/0305-4608/5/5/017 S F Edwards and P W Anderson "Theory of spin glasses",Journal of Physics F: Metal Physics '''5''' pp. 965-974 (1975)] | #[http://dx.doi.org/10.1088/0305-4608/5/5/017 S F Edwards and P W Anderson "Theory of spin glasses",Journal of Physics F: Metal Physics '''5''' pp. 965-974 (1975)] | ||
#[http://dx.doi.org/10.1088/0305-4470/9/10/011 S F Edwards and R C Jones "The eigenvalue spectrum of a large symmetric random matrix", Journal of Physics A: Mathematical and General '''9''' pp. 1595-1603 (1976)] | #[http://dx.doi.org/10.1088/0305-4470/9/10/011 S F Edwards and R C Jones "The eigenvalue spectrum of a large symmetric random matrix", Journal of Physics A: Mathematical and General '''9''' pp. 1595-1603 (1976)] | ||
Revision as of 11:07, 22 May 2007
The Helmholtz energy function of fluid in a matrix of configuration in the Canonical () ensemble is given by:
![{\displaystyle -\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57f27ece5946ea5fb66b8f62c187daa36e942ab1)
where is the fluid partition function, and is the Hamiltonian of the matrix. Taking an average over matrix configurations gives
- 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}}
(Ref.s 1 and 2) An important mathematical trick to get rid of the logarithm inside of the integral is
- 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 arrives at
- 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}, q^{N_0}) + H_{11}^\lambda (r^{N_1}, q^{N_0})\right)}
The Hamiltonian written in this form 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
- .
![{\displaystyle -\beta {\overline {A}}_{1}=\lim _{s\rightarrow 0}{\frac {\rm {d}}{{\rm {d}}s}}[-\beta A^{\rm {rep}}(s)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7654516da01b5e9fd609dc75dfbfd4ece3a3d3c6)