Microcanonical ensemble - Revision history

Carl McBride: Added a recent publication

2015-11-13T12:27:31Z

Added a recent publication

← Older revision		Revision as of 14:27, 13 November 2015
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			'''Microcanonical ensemble'''
	== Ensemble variables ==		== Ensemble variables ==
	(One component system, 3-dimensional system, ... ):		(One component system, 3-dimensional system, ... ):
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	== References ==		== References ==
	# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press		<references/>
			;Related reading
			* D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press
			* [http://dx.doi.org/10.1063/1.4931484 Philipp Schierz, Johannes Zierenberg and Wolfhard Janke "Molecular Dynamics and Monte Carlo simulations in the microcanonical ensemble: Quantitative comparison and reweighting techniques", Journal of Chemical Physics '''143''' 134114 (2015)]

	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

2007-10-11T16:27:36Z

Partition function

← Older revision		Revision as of 18:27, 11 October 2007
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	== Partition function ==		== Partition function ==

	:<math> Q_{NVE} = \frac{1}{h^{3N} N!} \~~int \int~~ d (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E).		:<math> Q_{NVE} = \frac{1}{h^{3N} N!} \iint d (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E).
	</math>		</math>

2007-03-05T16:52:21Z

Ensemble variables

← Older revision		Revision as of 18:52, 5 March 2007
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	* <math> \left. V \right. </math>: is the volume		* <math> \left. V \right. </math>: is the volume

	* <math> \left. E \right. </math>: is the internal energy (kinetic + potential)		* <math> \left. E \right. </math>: is the [[internal energy]] (kinetic + potential)

	== Partition function ==		== Partition function ==

2007-02-28T14:49:20Z

Partition function

← Older revision		Revision as of 16:49, 28 February 2007
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	*<math> \left( p \right)^{3N} </math> represents the 3N momenta.		*<math> \left( p \right)^{3N} </math> represents the 3N momenta.

	* <math> H \left(p,q\right) </math> ~~represent~~ the [[Hamiltonian]], i.e. the total energy of the system as a function of coordinates and momenta.		* <math> H \left(p,q\right) </math> represents the [[Hamiltonian]], i.e. the total energy of the system as a function of coordinates and momenta.

	*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]]		*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]]

2007-02-28T14:48:50Z

Ensemble variables

← Older revision		Revision as of 16:48, 28 February 2007
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	(One component system, 3-dimensional system, ... ):		(One component system, 3-dimensional system, ... ):

	* <math> \left. N \right. </math>: ~~Number~~ of ~~Particles~~		* <math> \left. N \right. </math>: number of particles

	* <math> \left. V \right. </math>: ~~Volume~~		* <math> \left. V \right. </math>: is the volume

	* <math> \left. E \right. </math>: ~~Internal~~ energy (kinetic + potential)		* <math> \left. E \right. </math>: is the internal energy (kinetic + potential)

	== Partition function ==		== Partition function ==

2007-02-28T13:59:12Z

Partition function: N instead of n

← Older revision		Revision as of 15:59, 28 February 2007
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	*<math> \left. h \right. </math> is the [[Planck constant]]		*<math> \left. h \right. </math> is the [[Planck constant]]

	*<math> \left( q \right)^{3n} </math> represents the 3N Cartesian position coordinates.		*<math> \left( q \right)^{3N} </math> represents the 3N Cartesian position coordinates.

	*<math> \left( p \right)^{3n} </math> represents the 3N momenta.		*<math> \left( p \right)^{3N} </math> represents the 3N momenta.

	* <math> H \left(p,q\right) </math> represent the [[Hamiltonian]], i.e. the total energy of the system as a function of coordinates and momenta.		* <math> H \left(p,q\right) </math> represent the [[Hamiltonian]], i.e. the total energy of the system as a function of coordinates and momenta.

2007-02-27T10:52:03Z

← Older revision		Revision as of 12:52, 27 February 2007
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	*<math> \left( p \right)^{3n} </math> represents the 3N momenta.		*<math> \left( p \right)^{3n} </math> represents the 3N momenta.

	* <math> H \left(p,q\right) </math> represent the Hamiltonian, i.e. the total energy of the system as a function of coordinates and momenta.		* <math> H \left(p,q\right) </math> represent the [[Hamiltonian]], i.e. the total energy of the system as a function of coordinates and momenta.

	*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]]		*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]]
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	where:		where:

	*<math> \left. S \right. </math> is the [[Entropy\|entropy]]·		*<math> \left. S \right. </math> is the [[Entropy\|entropy]].

	*<math> \left. k_B \right. </math> is the [[Boltzmann constant]]		*<math> \left. k_B \right. </math> is the [[Boltzmann constant]]

2007-02-27T10:51:23Z

← Older revision		Revision as of 12:51, 27 February 2007
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	where:		where:

	*<math> \left. S \right. </math> is the entropy		*<math> \left. S \right. </math> is the [[Entropy\|entropy]]·

	*<math> \left. k_B \right. </math> is the [[Boltzmann constant]]		*<math> \left. k_B \right. </math> is the [[Boltzmann constant]]

2007-02-27T10:50:41Z

← Older revision		Revision as of 12:50, 27 February 2007
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	~~Microcanonical Ensemble (Clasical statistics):~~

	== Ensemble variables ==		== Ensemble variables ==
	(One component system, 3-dimensional system, ... ):		(One component system, 3-dimensional system, ... ):
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	== Partition function ==		== Partition function ==

	<math> Q_{NVE} = \frac{1}{h^{3N} N!} \int \int d (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E).		:<math> Q_{NVE} = \frac{1}{h^{3N} N!} \int \int d (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E).
	</math>		</math>

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	* <math> H \left(p,q\right) </math> represent the Hamiltonian, i.e. the total energy of the system as a function of coordinates and momenta.		* <math> H \left(p,q\right) </math> represent the Hamiltonian, i.e. the total energy of the system as a function of coordinates and momenta.

	*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution~~\|Dirac delta function~~]]		*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]]

	== Thermodynamics ==		== Thermodynamics ==

	: <math> \left. S = k_B \log Q_{NVE} \right. </math>		:<math> \left. S = k_B \log Q_{NVE} \right. </math>

	where:		where:
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	== References ==		== References ==
	# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press		# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press
			[[Category:Statistical mechanics]]

2007-02-27T10:32:18Z

Thermodynamics

← Older revision		Revision as of 12:32, 27 February 2007
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	where:		where:

	*<math> \left. S \right. </math> is the [[~~entropy~~]]		*<math> \left. S \right. </math> is the entropy

			*<math> \left. k_B \right. </math> is the [[Boltzmann constant]]

	== References ==		== References ==
	# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press		# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press