Gibbs ensemble - Revision history

Carl McBride: /* Entropy */

2007-11-21T14:45:48Z

Entropy

← Older revision		Revision as of 16:45, 21 November 2007
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	:<math>\langle \psi \rangle = \overline \psi</math>		:<math>\langle \psi \rangle = \overline \psi</math>

	===Entropy===		===[[Entropy]]===
	Ref. 1 Eq. 2.70		Ref. 1 Eq. 2.70

Carl McBride: /* Ergodic theory */

2007-11-21T14:45:25Z

Ergodic theory

← Older revision		Revision as of 16:45, 21 November 2007
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	</math>		</math>

	===Ergodic theory===		===[[Ergodic hypothesis \|Ergodic theory]]===
	Ref. 1 Eq. 2.6		Ref. 1 Eq. 2.6

Carl McBride at 13:57, 10 July 2007

2007-07-10T13:57:35Z

← Older revision		Revision as of 15:57, 10 July 2007
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	(Ref. 1 Eq. 2.2)		(Ref. 1 Eq. 2.2)

	:<math>\mathcal{G}_{(N)} (X_{(N)},t)= \frac{\Gamma_{(N)}^{(0)}}{\mathcal{N}} \frac{{\rm d}\mathcal{N}}{{\rm d}\Gamma_{(N)}}</math>		:<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		where <math>\Gamma_{(N)}^{(0)}</math> is a normalized constant with the dimensions
	of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>.		of the [[phase space]] <math>\left. \Gamma_{(N)} \right.</math>.

	:<math>\~~left. X_~~{(N)} ~~\right.~~= \{ ~~r_1~~ , ..., ~~r_N~~ ; ~~p_1~~ , ..., ~~p_N~~ \}</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):		Normalization condition (Ref. 1 Eq. 2.3):
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	Macroscopic mean values are given by (Ref. 1 Eq. 2.5)		Macroscopic mean values are given by (Ref. 1 Eq. 2.5)

	:<math>\langle \psi (r,t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}}		:<math>\langle \psi ({\mathbf r},t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}}
	\int_{\Gamma_{(N)}} \psi (X_{(N)}) \mathcal{G}_{(N)} (X_{(N)},t) {\rm d}\Gamma_{(N)}</math>		\int_{\Gamma_{(N)}} \psi ({\mathbf X}_{(N)}) \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t) {\rm d}\Gamma_{(N)}
			</math>

	===Ergodic theory===		===Ergodic theory===
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	where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12)		where <math>\Omega</math> is the ''N''-particle [[thermal potential]] (Ref. 1 Eq. 2.12)

	:<math>\Omega_{(N)} (X_{(N)},t)= \ln \mathcal{G}_{(N)} (X_{(N)},t)</math>		:<math>\Omega_{(N)} ({\mathbf X}_{(N)},t)= \ln \mathcal{G}_{(N)} ({\mathbf X}_{(N)},t)</math>

	==References==		==References==
	# G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)		# G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
	[[category: statistical mechanics]]		[[category: statistical mechanics]]

Carl McBride at 09:39, 31 May 2007

2007-05-31T09:39:59Z

← Older revision	Revision as of 11:39, 31 May 2007
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		Here we have the ''N-particle distribution function''
		(Ref. 1 Eq. 2.2)

		:<math>\mathcal{G}_{(N)} (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>\left. X_{(N)} \right.= \{ r_1 , ..., r_N ; p_1 , ..., 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 (r,t)\rangle= \frac{1}{\Gamma_{(N)}^{(0)}}
		\int_{\Gamma_{(N)}} \psi (X_{(N)}) \mathcal{G}_{(N)} (X_{(N)},t) {\rm d}\Gamma_{(N)}</math>

		===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)} (X_{(N)},t)= \ln \mathcal{G}_{(N)} (X_{(N)},t)</math>

		==References==
		# G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
		[[category: statistical mechanics]]

Juan: Removing all content from page

2007-05-31T09:24:14Z

Removing all content from page

← Older revision		Revision as of 11:24, 31 May 2007
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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 employ simulations in the semi-grand canonical ensemble, histogram reweighting, or characterization of the spinodal curve from the study of computed collective scattering function. <p>
	The Gibbs ensemble Monte Carlo method has been specificly designed to characterize phase transitions. It was mainly developed by Panagiotopoulos [1] 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.

	~~==References==~~
	~~[1] A. Z. Panagiotopoulos, Mol Phys. 61, 813 (1987).~~

Juan at 09:21, 31 May 2007

2007-05-31T09:21:53Z

← Older revision		Revision as of 11:21, 31 May 2007
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	==References==		==References==
	[1]A. Z. Panagiotopoulos, Mol Phys. 61, 813 (1987).		[1] A. Z. Panagiotopoulos, Mol Phys. 61, 813 (1987).

Juan at 09:21, 31 May 2007

2007-05-31T09:21:39Z

← Older revision		Revision as of 11:21, 31 May 2007
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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>		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 employ simulations in the semi-grand canonical ensemble, histogram reweighting, or characterization of the spinodal curve from the study of computed collective scattering function. <p>
	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.		The Gibbs ensemble Monte Carlo method has been specificly designed to characterize phase transitions. It was mainly developed by Panagiotopoulos [1] 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.

			==References==
			[1]A. Z. Panagiotopoulos, Mol Phys. 61, 813 (1987).

Juan at 09:15, 31 May 2007

2007-05-31T09:15:33Z

← Older revision		Revision as of 11:15, 31 May 2007
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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.		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>
	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.		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.

Juan: New page: 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 t...

2007-05-31T09:10:46Z

New page: 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 t...

New page

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.
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.