Gibbs ensemble: Difference between revisions

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Phase separation is one of the topics for which simulation techniques have preferentially been focused in the recent past. Different procedures have been used for this purpose. Thus, for the particular case of chain systems, we can perform simulations in the semi-grand canonical ensemble, histogram reweighting, or characterization of the spinodal curve from the study of computed collective scattering function. <p>
Here we have the ''N-particle distribution function''
The Gibbs ensemble Monte Carlo method has been specificly designed to characterize phase transitions. It was mainly developed by Panagiotopoulos to avoid the problem of finite size interfacial effects. In this method, a NVT (or NPT)ensemble containing two (or more) species is divided into two (or) boxes. In addition to the usual particle moves in each one of the boxes, the algorithm includes moves steps to change the volume and composition of the boxes at mechanical and chemical equilibrium. Transferring a chain molecule from a box to the other requires the use of an efficient method to insert chains. The configurational bias method, is specially recommended for this purpose.
(Ref. 1 Eq. 2.2)
 
:<math>\mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}</math>
 
where <math>\Gamma_{(N)}^{(0)}</math> is a normalized constant with the dimensions
of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>.
 
:<math>{\mathbf X}_{(N)} = \{ {\mathbf r}_1 , ..., {\mathbf r}_N ; {\mathbf p}_1 , ...,  {\mathbf p}_N \}</math>
 
Normalization condition (Ref. 1 Eq. 2.3):
 
:<math>\frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1</math>
 
it is convenient to set (Ref. 1 Eq. 2.4)
 
:<math>\Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}</math>
 
where <math>V</math> is the volume of the system and <math>\mathcal{P}</math> is the characteristic momentum
of the particles (Ref. 1 Eq. 3.26),
 
:<math>\mathcal{P} = \sqrt{2 \pi m \Theta}</math>
 
Macroscopic mean values are given by (Ref. 1 Eq. 2.5)
 
:<math>\langle \psi ({\mathbf r},t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}}
\int_{\Gamma_{(N)}}  \psi  ({\mathbf X}_{(N)}) \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t) {\rm d}\Gamma_{(N)}
</math>
 
===[[Ergodic hypothesis |Ergodic theory]]===
Ref. 1 Eq. 2.6
 
:<math>\langle \psi \rangle = \overline \psi</math>
 
===[[Entropy]]===
Ref. 1 Eq. 2.70
 
:<math>S_{(N)}= - \frac{k_B}{ V^N \mathcal{P}^{3N}} \int_\Gamma  \Omega_1,... _N  \mathcal{G}_1,... _N {\rm d}\Gamma_{(N)}</math>
 
where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12)
 
:<math>\Omega_{(N)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)</math>
 
==References==
# G. A. Martynov  "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
[[category: statistical mechanics]]

Latest revision as of 15:45, 21 November 2007

Here we have the N-particle distribution function (Ref. 1 Eq. 2.2)

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 \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}}

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 \Gamma_{(N)}^{(0)}} is a normalized constant with the dimensions of the phase space 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 \left. \Gamma_{(N)} \right.} .

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 X}_{(N)} = \{ {\mathbf r}_1 , ..., {\mathbf r}_N ; {\mathbf p}_1 , ..., {\mathbf p}_N \}}

Normalization condition (Ref. 1 Eq. 2.3):

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 \frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1}

it is convenient to set (Ref. 1 Eq. 2.4)

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 \Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}}

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 V} is the volume of the system 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 \mathcal{P}} is the characteristic momentum of the particles (Ref. 1 Eq. 3.26),

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 \mathcal{P} = \sqrt{2 \pi m \Theta}}

Macroscopic mean values are given by (Ref. 1 Eq. 2.5)

Ergodic theory[edit]

Ref. 1 Eq. 2.6

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 \langle \psi \rangle = \overline \psi}

Entropy[edit]

Ref. 1 Eq. 2.70

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_{(N)}= - \frac{k_B}{ V^N \mathcal{P}^{3N}} \int_\Gamma \Omega_1,... _N \mathcal{G}_1,... _N {\rm d}\Gamma_{(N)}}

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 \Omega} is the N-particle thermal potential (Ref. 1 Eq. 2.12)

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 \Omega_{(N)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)}

References[edit]

  1. G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
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