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| Here we have the ''N-particle distribution function''
| | 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. |
| (Ref. 1 Eq. 2.2)
| | The Gibbs ensemble Monte Carlo method mainly developed by Panagiotopoulos5 avoids the problem of finite size interfacial effects. In this method, a NVT ensemble containing two species is divided into two 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 Monte Carlo method6,7 is specially recommended for this purpose. |
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| :<math>\mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}</math>
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| where <math>\Gamma_{(N)}^{(0)}</math> is a normalized constant with the dimensions
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| of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>. | |
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| :<math>{\mathbf X}_{(N)} = \{ {\mathbf r}_1 , ..., {\mathbf r}_N ; {\mathbf p}_1 , ..., {\mathbf p}_N \}</math>
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| Normalization condition (Ref. 1 Eq. 2.3):
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| :<math>\frac{1}{\Gamma_{(N)}^{(0)}} \int_{\Gamma_{(N)}} \mathcal{G}_{(N)} {\rm d}\mathcal{N} =1</math>
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| it is convenient to set (Ref. 1 Eq. 2.4)
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| :<math>\Gamma_{(N)}^{(0)} = V^N \mathcal{P}^{3N}</math>
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| where <math>V</math> is the volume of the system and <math>\mathcal{P}</math> is the characteristic momentum
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| of the particles (Ref. 1 Eq. 3.26), | |
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| :<math>\mathcal{P} = \sqrt{2 \pi m \Theta}</math>
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| Macroscopic mean values are given by (Ref. 1 Eq. 2.5)
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| :<math>\langle \psi ({\mathbf r},t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}}
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| \int_{\Gamma_{(N)}} \psi ({\mathbf X}_{(N)}) \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t) {\rm d}\Gamma_{(N)}
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| </math>
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| ===[[Ergodic hypothesis |Ergodic theory]]===
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| Ref. 1 Eq. 2.6
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| :<math>\langle \psi \rangle = \overline \psi</math>
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| ===[[Entropy]]===
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| Ref. 1 Eq. 2.70
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| :<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>
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| where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12)
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| :<math>\Omega_{(N)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)</math>
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| ==References==
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| # G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
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| [[category: statistical mechanics]]
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