Replica method: Difference between revisions
Carl McBride (talk | contribs) mNo edit summary |
Carl McBride (talk | contribs) m (Added disambiguation) |
||
Line 1: | Line 1: | ||
:''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: |
Latest revision as of 18:56, 11 February 2010
- 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 in the Canonical ensemble is given by:
where 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_{10}} and 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
(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
- .
One can apply this trick to the we want to average, and replace the resulting power 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 is the partition function of a mixture with Hamiltonian
This Hamiltonian describes a completely equilibrated system of components; the matrix the identical non-interacting replicas of the fluid. Since , then
Thus the relation between the Helmholtz energy function of the non-equilibrium partially frozen system and the replicated (equilibrium) system is given by
- .
Interesting reading[edit]
- Viktor Dotsenko "Introduction to the Replica Theory of Disordered Statistical Systems", Collection Alea-Saclay: Monographs and Texts in Statistical Physics, Cambridge University Press (2000)