Xxencoding - Revision history

en>FrescoBot: Bot: link syntax

2014-07-17T01:06:56Z

← Older revision		Revision as of 03:06, 17 July 2014
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	~~{{Statistical mechanics\|cTopic=[[Thermodynamic potential\|Potentials]]}}~~		Wilber Berryhill is the name his mothers and fathers gave him and he completely digs that name. Invoicing is my profession. Playing badminton is a factor that he is totally addicted to. Her family lives in Alaska but her spouse desires them to transfer.<br><br>Visit my page; online psychics ([http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 visit this site])
	~~The '''grand potential''' is a quantity used in [[statistical mechanics]], especially for [[irreversible process]]es in [[Open system (systems theory)\|open system]]s.~~
	~~The grand potential is the characteristic state function for the [[grand canonical ensemble]].~~

	~~==Definition==~~

	~~Grand potential is defined by~~
	~~:<math>~~
	~~\Phi_{G} \ \stackrel{\mathrm{def}}{=}\ U - T S - \mu N~~
	~~</math>~~
	~~where U is the [[internal energy]], ''T'' is the [[temperature]] of the system, ''S'' is the [[entropy]], μ~~ is the ~~[[chemical potential]],~~ and ~~''N'' is the number of particles in the system.~~

	~~The change in the grand potential is given by~~
	~~:<math>\begin{align}~~
	~~d\Phi_{G} & = dU - TdS - SdT - \mu dN - Nd\mu \\~~
	~~& = - P dV - S dT - N d\mu~~
	~~\end{align}</math>~~
	~~where ''P'' is [[pressure]]~~ and ~~''V'' is [[volume]], using the [[fundamental thermodynamic relation]] (combined [[first law of thermodynamics\|first]] and [[first law of thermodynamics\|second]] [[thermodynamic laws]]);~~

	~~:<math>dU = TdS - PdV + \mu dN</math>~~

	~~When the system is in [[thermodynamic equilibrium]], Φ<sub>''G''</sub> is a minimum~~. ~~This can be seen by considering that dΦ<sub>''G''</sub>~~ is ~~zero if the volume is fixed and the temperature and chemical potential have stopped evolving~~.

	~~===Landau free energy===~~

	Some authors refer to the ''Landau free energy'' or '''Landau potential''' as:<ref>{{citebook\|author=Lee, J. Chang\|year=2002\|title=Thermal Physics - Entropy and Free Energies\|chapter=5\|publisher=New Jersey: World Scientific}}</ref><ref>Reference on "Landau potential" is ~~found in the book: {{citebook\|author=D. Goodstein\|year=\|title=States of Matter\|page=19\|publisher=}}</ref>~~

	~~:<math>~~
	~~\Omega \ \stackrel{\mathrm{def}}{=}\ F - \mu N = U - T S - \mu N~~
	~~</math>~~

	~~named after Russian physicist [[Lev Landau]], which may be~~ a ~~synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains <math> \Omega = -PV \,\;</math>~~

	~~==Grand potential for homogeneous systems (vs. inhomogeneous systems)==~~
	In the case of a scale-invariant type of system (where a system of volume <math>\lambda V</math> has exactly the same set of microstates as <math>\lambda</math> systems of volume <math>V</math>), then when we grow the system new particles and energy will flow in from the reservoir to fill the new volume with a homogeneous extension of the original system.
	The pressure then must be constant with respect to changes in volume: <math>(\partial \langle P \rangle/\partial V)_{\mu,T} = 0</math>, and the particle and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g., <math>(\partial \langle N \rangle/\partial V)_{\mu,T} = N/V</math>.
	~~In this case we have simply <math> \Phi_{G} = - \langle P\rangle V</math>, as well as the familiar relationship <math>G = \langle N \rangle \mu</math> for the [[Gibbs free energy]].~~
	The value of <math> \Phi_{G} </math> can be understood as the work we can extract from the system by shrinking it down to nothing (putting all the particles and energy back into the reservoir). The fact that ~~<math> \Phi_{G} = - \langle P\rangle V</math>~~ is ~~negative implies that it takes energy~~ to ~~perform this ''extraction''~~.

	~~Such homogeneous scaling does not exist~~ in many systems. For example, when analyzing the ensemble of electrons in a single molecule or even a piece of metal floating in space, doubling the volume of the space does double the number of electrons in the material.<~~ref~~>~~{{cite doi\|10.1063/1.1740312}}~~<~~/ref~~>
	~~The problem here is that, although electrons and energy are exchanged with a reservoir, the material host is not allowed to change.~~
	~~Generally in small systems, or systems with long range interactions~~ (~~those outside the~~ [~~[thermodynamic limit]]), <math>\Phi_{G} \neq - \langle P\rangle V</math>.<ref>{{cite isbn\|9780486495095}}</ref>~~

	~~==Ideal gas==~~

	~~For an ideal gas,~~
	:~~<math>~~
	~~\Phi_{G} = - k_{B} T \ln(\Xi) = - k_{B} T Z_{1} e^{\beta \mu}~~
	</~~math>~~
	~~where Ξ is the [[grand partition function]], ''k<sub>B<~~/~~sub>'' is [[Boltzmann constant]], ''Z''<sub>1</sub> is the [[partition function (statistical mechanics)\|partition function]] for 1 particle and β = 1/k<sub>B</sub>T is the [[inverse temperature]]~~. ~~The factor ''e''<sup>βμ<~~/~~sup> is the [[Boltzmann factor]]~~.

	==~~References~~==
	~~{{Reflist}}~~

	~~== See also ==~~
	* [[Gibbs free energy\|Gibbs energy]]
	* [[Helmholtz free energy\|Helmholtz energy]]

	~~== External links ==~~
	* [http://theory.ph.man.ac.uk/~judith/stat_therm/node88.html Grand Potential (Manchester University)]

	~~[[Category:Thermodynamics]]~~

en>Attila.lendvai: /* See also */ add link to Binary-to-text encoding instead of the local partial list of binary to text encodings

2012-11-27T12:19:09Z

See also: add link to Binary-to-text encoding instead of the local partial list of binary to text encodings

← Older revision		Revision as of 14:19, 27 November 2012
Line 1:		Line 1:
	~~Alyson~~ is the ~~title individuals use to contact me~~ and ~~I believe it sounds quite great when you say it~~. ~~Some time~~ in the ~~past~~ ~~free psychic~~ ([~~https~~://~~www~~.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 machlitim~~.~~org~~.il]) ~~he chose~~ to ~~reside in North Carolina and he doesn~~'~~t strategy on altering it~~. ~~Distributing production is where her main earnings arrives from~~. ~~I am truly fond~~ of to ~~go to karaoke but I've been using on new things recently~~.<br><br>~~my web page~~ :~~: psychics~~ [~~http:~~//~~isaworld~~.pe.~~kr/?document_srl~~=~~392088 online psychic~~], [http://~~www~~.~~khuplaza~~.~~com~~/~~dent~~/~~14869889 simply click the up coming site~~],		{{Statistical mechanics\|cTopic=[[Thermodynamic potential\|Potentials]]}}
			The '''grand potential''' is a quantity used in [[statistical mechanics]], especially for [[irreversible process]]es in [[Open system (systems theory)\|open system]]s.
			The grand potential is the characteristic state function for the [[grand canonical ensemble]].

			==Definition==

			Grand potential is defined by
			:<math>
			\Phi_{G} \ \stackrel{\mathrm{def}}{=}\ U - T S - \mu N
			</math>
			where U is the [[internal energy]], ''T'' is the [[temperature]] of the system, ''S'' is the [[entropy]], μ is the [[chemical potential]], and ''N'' is the number of particles in the system.

			The change in the grand potential is given by
			:<math>\begin{align}
			d\Phi_{G} & = dU - TdS - SdT - \mu dN - Nd\mu \\
			& = - P dV - S dT - N d\mu
			\end{align}</math>
			where ''P'' is [[pressure]] and ''V'' is [[volume]], using the [[fundamental thermodynamic relation]] (combined [[first law of thermodynamics\|first]] and [[first law of thermodynamics\|second]] [[thermodynamic laws]]);

			:<math>dU = TdS - PdV + \mu dN</math>

			When the system is in [[thermodynamic equilibrium]], Φ<sub>''G''</sub> is a minimum. This can be seen by considering that dΦ<sub>''G''</sub> is zero if the volume is fixed and the temperature and chemical potential have stopped evolving.

			===Landau free energy===

			Some authors refer to the ''Landau free energy'' or '''Landau potential''' as:<ref>{{citebook\|author=Lee, J. Chang\|year=2002\|title=Thermal Physics - Entropy and Free Energies\|chapter=5\|publisher=New Jersey: World Scientific}}</ref><ref>Reference on "Landau potential" is found in the book: {{citebook\|author=D. Goodstein\|year=\|title=States of Matter\|page=19\|publisher=}}</ref>

			:<math>
			\Omega \ \stackrel{\mathrm{def}}{=}\ F - \mu N = U - T S - \mu N
			</math>

			named after Russian physicist [[Lev Landau]], which may be a synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains <math> \Omega = -PV \,\;</math>

			==Grand potential for homogeneous systems (vs. inhomogeneous systems)==
			In the case of a scale-invariant type of system (where a system of volume <math>\lambda V</math> has exactly the same set of microstates as <math>\lambda</math> systems of volume <math>V</math>), then when we grow the system new particles and energy will flow in from the reservoir to fill the new volume with a homogeneous extension of the original system.
			The pressure then must be constant with respect to changes in volume: <math>(\partial \langle P \rangle/\partial V)_{\mu,T} = 0</math>, and the particle and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g., <math>(\partial \langle N \rangle/\partial V)_{\mu,T} = N/V</math>.
			In this case we have simply <math> \Phi_{G} = - \langle P\rangle V</math>, as well as the familiar relationship <math>G = \langle N \rangle \mu</math> for the [[Gibbs free energy]].
			The value of <math> \Phi_{G} </math> can be understood as the work we can extract from the system by shrinking it down to nothing (putting all the particles and energy back into the reservoir). The fact that <math> \Phi_{G} = - \langle P\rangle V</math> is negative implies that it takes energy to perform this ''extraction''.

			Such homogeneous scaling does not exist in many systems. For example, when analyzing the ensemble of electrons in a single molecule or even a piece of metal floating in space, doubling the volume of the space does double the number of electrons in the material.<ref>{{cite doi\|10.1063/1.1740312}}</ref>
			The problem here is that, although electrons and energy are exchanged with a reservoir, the material host is not allowed to change.
			Generally in small systems, or systems with long range interactions (those outside the [[thermodynamic limit]]), <math>\Phi_{G} \neq - \langle P\rangle V</math>.<ref>{{cite isbn\|9780486495095}}</ref>

			==Ideal gas==

			For an ideal gas,
			:<math>
			\Phi_{G} = - k_{B} T \ln(\Xi) = - k_{B} T Z_{1} e^{\beta \mu}
			</math>
			where Ξ is the [[grand partition function]], ''k<sub>B</sub>'' is [[Boltzmann constant]], ''Z''<sub>1</sub> is the [[partition function (statistical mechanics)\|partition function]] for 1 particle and β = 1/k<sub>B</sub>T is the [[inverse temperature]]. The factor ''e''<sup>βμ</sup> is the [[Boltzmann factor]].

			==References==
			{{Reflist}}

			== See also ==
			* [[Gibbs free energy\|Gibbs energy]]
			* [[Helmholtz free energy\|Helmholtz energy]]

			== External links ==
			* [http://theory.ph.man.ac.uk/~judith/stat_therm/node88.html Grand Potential (Manchester University)]

			[[Category:Thermodynamics]]

67.164.57.135: added

2012-06-11T03:46:17Z

<ref> added

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