Virial pressure - Revision history

Carl McBride: Replaced page with a redirec

2012-03-14T14:09:36Z

Replaced page with a redirec

← Older revision		Revision as of 16:09, 14 March 2012
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	~~The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to~~ [[~~molecular dynamics]], since [[Newtons laws~~#~~Newton's second law of motion \|forces]] are evaluated and readily available. For pair interactions, one has:~~		#REDIRECT[[Pressure#Virial pressure]]

	~~:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>~~

	~~where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math> is the [[Boltzmann constant]].~~
	~~In this equation one can recognize an [[Equation of State: Ideal Gas \|ideal gas]] contribution, and a second term due to the [[~~Virial theorem \|virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

	This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].

	~~If the interaction is central, the force is given by~~
	~~:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , </math>~~
	~~where <math>f(r)</math> the force corresponding to the [[Intermolecular pair potential \|intermolecular potential]] <math>\Phi(r)</math>:~~

	~~:<math>-\partial \Phi(r)/\partial r.</math>~~

	~~For example, for the [[Lennard-Jones model \| Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to~~
	~~:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>~~

	~~Notice that most [[Realistic models \|realistic potentials]] are attractive at long ranges; hence the first correction to the ideal~~ pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.
	~~==See also==~~
	*[[Test volume method]]
	~~[[category: statistical mechanics~~]]

Carl McBride: Modified an internal link

2011-05-18T14:16:56Z

Modified an internal link

← Older revision		Revision as of 16:16, 18 May 2011
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	where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math> is the [[Boltzmann constant]].		where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math> is the [[Boltzmann constant]].
	In this equation one can recognize an [[Equation of State: Ideal Gas \|ideal gas]] contribution, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.		In this equation one can recognize an [[Equation of State: Ideal Gas \|ideal gas]] contribution, and a second term due to the [[Virial theorem \|virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

	This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].		This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].

79.146.196.64 at 14:51, 4 February 2011

2011-02-04T14:51:24Z

← Older revision		Revision as of 16:51, 4 February 2011
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	:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>		:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>

	where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math>is the [[Boltzmann constant]].		where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math> is the [[Boltzmann constant]].
	In this equation one can recognize an [[Equation of State: Ideal Gas \|ideal gas]] contribution, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.		In this equation one can recognize an [[Equation of State: Ideal Gas \|ideal gas]] contribution, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

211.121.168.82: Punctuation correction

2010-10-07T08:57:58Z

Punctuation correction

← Older revision		Revision as of 10:57, 7 October 2010
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	:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>		:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>

	Notice that most [[Realistic models \|realistic potentials]] are attractive at long ranges, hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.		Notice that most [[Realistic models \|realistic potentials]] are attractive at long ranges; hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.
	==See also==		==See also==
	*[[Test volume method]]		*[[Test volume method]]
	[[category: statistical mechanics]]		[[category: statistical mechanics]]

Nice and Tidy at 14:14, 7 February 2008

2008-02-07T14:14:36Z

← Older revision		Revision as of 16:14, 7 February 2008
Line 1:		Line 1:
	The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since forces are evaluated and readily available. For pair interactions, one has:		The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since [[Newtons laws#Newton's second law of motion \|forces]] are evaluated and readily available. For pair interactions, one has:

	:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>		:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>

	where one can recognize an ideal ~~term~~, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.		where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math>is the [[Boltzmann constant]].
			In this equation one can recognize an [[Equation of State: Ideal Gas \|ideal gas]] contribution, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

	This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].		This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].

Nice and Tidy at 10:51, 7 February 2008

2008-02-07T10:51:14Z

← Older revision		Revision as of 12:51, 7 February 2008
Line 17:		Line 17:

	Notice that most [[Realistic models \|realistic potentials]] are attractive at long ranges, hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.		Notice that most [[Realistic models \|realistic potentials]] are attractive at long ranges, hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.
			==See also==
			*[[Test volume method]]
	[[category: statistical mechanics]]		[[category: statistical mechanics]]

Nice and Tidy: Changed U to Phi.

2008-02-07T09:44:43Z

Changed U to Phi.

← Older revision		Revision as of 11:44, 7 February 2008
Line 5:		Line 5:
	where one can recognize an ideal term, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.		where one can recognize an ideal term, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

	This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates" <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].		This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].

	If the interaction is central, the force is given by		If the interaction is central, the force is given by
	:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , </math>		:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , </math>
	where <math>f(r)</math> the force corresponding to the intermolecular potential <math>U(r)</math>:		where <math>f(r)</math> the force corresponding to the [[Intermolecular pair potential \|intermolecular potential]] <math>\Phi(r)</math>:

	:<math>-\partial u(r)/\partial r.</math>		:<math>-\partial \Phi(r)/\partial r.</math>

	~~E.g.~~, for the [[Lennard-Jones model \| Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to		For example, for the [[Lennard-Jones model \| Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to
	:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>		:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>

	Notice that most realistic potentials are attractive at long ranges, hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.		Notice that most [[Realistic models \|realistic potentials]] are attractive at long ranges, hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.
	[[category: statistical mechanics]]		[[category: statistical mechanics]]

Dduque: A bit more info

2008-02-06T20:26:03Z

A bit more info

← Older revision		Revision as of 22:26, 6 February 2008
Line 16:		Line 16:
	:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>		:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>

			Notice that most realistic potentials are attractive at long ranges, hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from repulsive potentials, such as [[hard sphere model \| hard spheres]], are always positive.
	[[category: ~~classical~~ mechanics]]		[[category: statistical mechanics]]

Dduque: Hopefully signs are all ok now!

2008-02-06T20:22:21Z

Hopefully signs are all ok now!

← Older revision		Revision as of 22:22, 6 February 2008
Line 1:		Line 1:
	The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since forces are evaluated and readily available. For ~~central~~ interactions, one has:		The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since forces are evaluated and readily available. For pair interactions, one has:

	:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>		:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>

	where one can recognize an ideal term, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.		where one can recognize an ideal term, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

			This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates" <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress \| stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].

			If the interaction is central, the force is given by
			:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , </math>
			where <math>f(r)</math> the force corresponding to the intermolecular potential <math>U(r)</math>:

			:<math>-\partial u(r)/\partial r.</math>

			E.g., for the [[Lennard-Jones model \| Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to
			:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>


	[[category: classical mechanics]]		[[category: classical mechanics]]

Nice and Tidy at 15:10, 6 February 2008

2008-02-06T15:10:41Z

← Older revision		Revision as of 17:10, 6 February 2008
Line 1:		Line 1:
	~~This~~ is ~~a common method~~ to obtain the [[pressure]] from a general simulation ~~(it~~ is ~~best~~ suited ~~for~~ [[molecular dynamics]] since forces are evaluated ~~in this case)~~. For central interactions, one has:		The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since forces are evaluated and readily available. For central interactions, one has:

	:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>		:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>

	where one can recognize an ideal term and ~~another one~~ due to the [[virial]]. The ~~underline~~ is an average, which would be a time average in molecular dynamics, or ~~ensamble~~ average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in ~~actual~~ world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.		where one can recognize an ideal term, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.
			[[category: classical mechanics]]