Replica method - Revision history

Carl McBride: Added disambiguation

2010-02-11T16:56:37Z

Added disambiguation

← Older revision		Revision as of 18:56, 11 February 2010
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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]]''.
	The [[Helmholtz energy function]] of fluid in a matrix of configuration		The [[Helmholtz energy function]] of fluid in a matrix of configuration
	<math>\{ {\mathbf q}^{N_0} \}</math> in the [[Canonical ensemble]] is given by:		<math>\{ {\mathbf q}^{N_0} \}</math> in the [[Canonical ensemble]] is given by:

Carl McBride at 12:57, 22 January 2008

2008-01-22T12:57:37Z

← Older revision		Revision as of 14:57, 22 January 2008
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	:<math>- \beta \overline{A}_1 = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s} [- \beta A^{\rm rep} (s) ]		:<math>- \beta \overline{A}_1 = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s} [- \beta A^{\rm rep} (s) ]
	</math>.		</math>.
			==Interesting reading==
			*Viktor Dotsenko "Introduction to the Replica Theory of Disordered Statistical Systems", Collection Alea-Saclay: Monographs and Texts in Statistical Physics, Cambridge University Press (2000)
	==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)]
	[[category: integral equations]]		[[category: integral equations]]

Carl McBride at 14:05, 2 August 2007

2007-08-02T14:05:06Z

← Older revision		Revision as of 16:05, 2 August 2007
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	#[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)]
			[[category: integral equations]]

Nice and Tidy at 15:21, 10 July 2007

2007-07-10T15:21:51Z

← Older revision		Revision as of 17:21, 10 July 2007
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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>NVT</math>)~~ ensemble is given by:		<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>

Cuesta at 15:02, 26 May 2007

2007-05-26T15:02:09Z

← Older revision		Revision as of 17:02, 26 May 2007
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	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		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>,		:<math> -\beta\overline{A}_1=\lim_{s\to 0}\frac{d}{ds}\left(\frac{Z^{\rm rep}(s)}{Z_0}\right) </math>,

	where <math>Z^{\rm rep}(s)</math> is the partition function of a mixture with Hamiltonian		where <math>Z^{\rm rep}(s)</math> is the partition function of a mixture with Hamiltonian
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	\left( H_{01}^\lambda (r^{N_1}_\lambda, q^{N_0}) + H_{11}^\lambda (r^{N_1}_\lambda, 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		This Hamiltonian describes a completely equilibrated system of <math>s+1</math> components; the matrix the <math>s</math> identical non-interacting replicas of the fluid. Since <math>Z_0=Z^{\rm rep}(0)</math>, then
	and the ~~replica~~ (equilibrium) system is given by
			:<math>\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.</math>

			Thus the relation between the [[Helmholtz energy function]] of the non-equilibrium partially frozen system and the replicated (equilibrium) system is given by

	:<math>- \beta \overline{A}_1 = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s} [- \beta A^{\rm rep} (s) ]		:<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)]

80.38.96.35 at 11:05, 24 May 2007

2007-05-24T11:05:27Z

← Older revision		Revision as of 13:05, 24 May 2007
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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_{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_{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>
	is the Hamiltonian of the matrix.		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
	~~Taking an~~ average over matrix configurations ~~gives~~

	:<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>

	(~~Ref~~.s 1 and 2) ~~An important mathematical trick to get rid of the logarithm inside of the integral is~~		(see Refs. 1 and 2)

	:~~<math>\log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s</math>~~		:An important mathematical trick to get rid of the logarithm inside of the integral is to use the mathematical identity

	~~one arrives at~~		:<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>

	~~The~~ Hamiltonian ~~written in this form~~ describes a completely equilibrated system		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

Carl McBride at 10:07, 22 May 2007

2007-05-22T10:07:35Z

← Older revision		Revision as of 12:07, 22 May 2007
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	~~Free~~ energy 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>NVT</math>) ensemble is given by:		<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})		:<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{F}_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>

	~~\cite{JPFMP_1975_05_0965,JPAMG_1976_09_01595}~~		(Ref.s 1 and 2) An important mathematical trick to get rid of the logarithm inside of the integral is
	~~Important~~ mathematical trick to get rid of the logarithm inside of the integral:

	:<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 ~~free~~ energy of the non-equilibrium partially frozen		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{F}_1 = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s} [- \beta F^{\rm rep} (s) ]		:<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)]

Carl McBride: New page: Free energy of fluid in a matrix of configuration $\{ q^{N_0} \}$ in the Canonical ( $NVT$ ) ensemble is given by: : $- \beta F_1 (q^{N_0}) = \log Z_1 (q^{N_0}) ...$

2007-05-22T10:02:53Z

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}) ...

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})
= \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>

where <math>Z_1 (q^{N_0})</math> is the fluid partition function, and <math>H_{00}</math>
is the Hamiltonian of the matrix.
Taking an average over matrix configurations gives

:<math>- \beta \overline{F}_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>

\cite{JPFMP_1975_05_0965,JPAMG_1976_09_01595}
Important mathematical trick to get rid of the logarithm inside of the integral:

:<math>\log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s</math>

one arrives at

:<math>\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)</math>

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.
Thus the relation between the free energy of the non-equilibrium partially frozen
and the replica (equilibrium) system is given by

:<math>- \beta \overline{F}_1 = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s} [- \beta F^{\rm rep} (s) ]
</math>
==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/10.1088/0305-4470/9/10/011 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)]