Monte Carlo in the microcanonical ensemble - Revision history

Carl McBride: Reverted edits by 70.158.67.162 (Talk); changed back to last version by Noe

2007-07-04T09:20:35Z

Reverted edits by 70.158.67.162 (Talk); changed back to last version by Noe

← Older revision		Revision as of 11:20, 4 July 2007
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	: <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} U \left( X^{3N} \right) </math>; (Eq.1)		: <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right) </math>; (Eq.1)

	where:		where:

70.158.67.162 at 01:46, 4 July 2007

2007-07-04T01:46:44Z

← Older revision		Revision as of 03:46, 4 July 2007
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	: <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right) </math>; (Eq.1)		: <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} U \left( X^{3N} \right) </math>; (Eq.1)

	where:		where:

Noe at 09:55, 2 March 2007

2007-03-02T09:55:03Z

← Older revision		Revision as of 11:55, 2 March 2007
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	Consider a system of <math> \left. N \right. </math> identical particles, with total energy <math> \left. H \right. </math> given by:		Consider a system of <math> \left. N \right. </math> identical particles, with total energy <math> \left. H \right. </math> given by:

	~~: <math> H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right), </math>~~

			: <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right) </math>; (Eq.1)

	where:		where:
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			The first term on the right hand side of (Eq. 1) is the [[Kinetic energy \|kinetic energy]], whereas the second term is




	The first term on the right hand side is the [[Kinetic energy \|kinetic energy]], whereas the second ~~one~~ is
	the [[Potential energy \| potential energy]] (a function of the positional coordinates).		the [[Potential energy \| potential energy]] (a function of the positional coordinates).

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	\int d P^{3N} \delta \left[ K(P^{3N})		\int d P^{3N} \delta \left[ K(P^{3N})
	- \Delta E \right]		- \Delta E \right]
	</math> ; (Eq. 1)		</math> ; (Eq. 2)
	where <math> \left. ~~P^{3N}~~ \right. </math> ~~stands for~~ the ~~<math>3N</math> momenta, and~~		where:
			* <math> \left. \delta(x) \right. </math> is the [[Dirac delta distribution\|Dirac's delta function]]

	: <math> \Delta E = E - U\left(X^{3N}\right) </math>.		* <math> \Delta E = E - U\left(X^{3N}\right) </math>.

	The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radius		The Integral in the right hand side of (Eq. 2) corresponds to the surface of a 3N-dimensional (<math> p_i; i=1,2,3,\cdots 3N </math>) hyper-sphere of radius
	<math> r = \left. \sqrt{ 2 m \Delta E } \right. </math> ;		<math> r = \left. \sqrt{ 2 m \Delta E } \right. </math> ;
	therefore:		therefore:

Noe at 09:44, 2 March 2007

2007-03-02T09:44:50Z

← Older revision		Revision as of 11:44, 2 March 2007
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	~~where the~~ first term on the right hand side is the [[Kinetic energy \|kinetic energy]], whereas the second one is		The first term on the right hand side is the [[Kinetic energy \|kinetic energy]], whereas the second one is
	the [[Potential energy \| potential energy]] (a function of the positional coordinates).		the [[Potential energy \| potential energy]] (a function of the positional coordinates).

Noe at 09:44, 2 March 2007

2007-03-02T09:44:33Z

← Older revision		Revision as of 11:44, 2 March 2007
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	: <math> H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right), </math>		: <math> H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right), </math>


			where:

			* <math> \left. X^{3N} \right. </math> represents the 3N Cartesian position coordinates of the particles
			* <math> \left. P^{3N} \right. </math> stands for the the 3N momenta.







	where the first term on the right hand side is the [[Kinetic energy \|kinetic energy]], whereas the second one is		where the first term on the right hand side is the [[Kinetic energy \|kinetic energy]], whereas the second one is

Noe: /* Integration of the kinetic degrees of freedom */

2007-02-28T16:19:56Z

Integration of the kinetic degrees of freedom

← Older revision		Revision as of 18:19, 28 February 2007
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	Consider a system of <math> \left. N \right. </math> identical particles, with total energy <math> \left. H \right. </math> given by:		Consider a system of <math> \left. N \right. </math> identical particles, with total energy <math> \left. H \right. </math> given by:

	: <math> H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right). </math>		: <math> H = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U \left( X^{3N} \right), </math>

	where the first term on the right hand side is the [[Kinetic energy \|kinetic energy]], whereas the second one is		where the first term on the right hand side is the [[Kinetic energy \|kinetic energy]], whereas the second one is
	the [[Potential energy \| potential energy]] (a function of the positional coordinates)		the [[Potential energy \| potential energy]] (a function of the positional coordinates).

	Now, let us consider the system in a [[Microcanonical ensemble \|microcanonical ensemble]];		Now, let us consider the system in a [[Microcanonical ensemble \|microcanonical ensemble]];
	~~Let~~ <math> \left. E \right. </math> be the total energy of the system (constrained in this ensemble)		let <math> \left. E \right. </math> be the total energy of the system (constrained in this ensemble).

	The probability, <math> \left. \Pi \right. </math> of a given position configuration <math> \left. X^{3N} \right. </math>, with potential energy		The probability, <math> \left. \Pi \right. </math> of a given position configuration <math> \left. X^{3N} \right. </math>, with potential energy
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	where <math> \left. P^{3N} \right. </math> stands for the <math>3N</math> momenta, and		where <math> \left. P^{3N} \right. </math> stands for the <math>3N</math> momenta, and

	: <math> \Delta E = E - U\left(X^{3N}\right) </math>		: <math> \Delta E = E - U\left(X^{3N}\right) </math>.

	The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radius		The Integral in the right hand side of Eq. 1 corresponds to the surface of a 3N-dimensional hyper-sphere of radius

Nice and Tidy: /* References */

2007-02-28T16:16:11Z

References

← Older revision		Revision as of 18:16, 28 February 2007
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	#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" ~~Phys. Rev.~~ E 64, 042501 (2001) (4 pages) ]		#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Physical Review E 64, 042501 (2001) (4 pages) ]

Nice and Tidy: /* References */

2007-02-28T16:15:43Z

References

← Older revision		Revision as of 18:15, 28 February 2007
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	#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Phys. Rev. E 64, 042501 (2001) [4 pages] ]		#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Phys. Rev. E 64, 042501 (2001) (4 pages) ]

Nice and Tidy: /* Integration of the kinetic degrees of freedom */

2007-02-28T16:03:56Z

Integration of the kinetic degrees of freedom

← Older revision		Revision as of 18:03, 28 February 2007
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	</math>.		</math>.

	See Ref 1. for an application of Monte Carlo simulation using this ensemble.		See Ref. 1 for an application of Monte Carlo simulation using this ensemble.

	[[Category: Monte Carlo]]		[[Category: Monte Carlo]]

Nice and Tidy: /* Integration of the kinetic degrees of freedom */

2007-02-28T16:03:44Z

Integration of the kinetic degrees of freedom

← Older revision		Revision as of 18:03, 28 February 2007
Line 29:		Line 29:
	</math>.		</math>.

	See Ref 1 for an application of Monte Carlo simulation using this ensemble.		See Ref 1. for an application of Monte Carlo simulation using this ensemble.

	[[Category: Monte Carlo]]		[[Category: Monte Carlo]]