Semi-grand ensembles - Revision history

Carl McBride: /* References */ Added a recent publication

2012-01-20T12:05:12Z

References: Added a recent publication

← Older revision		Revision as of 14:05, 20 January 2012
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	==References==		==References==
			<references/>
			;Related reading
			*[http://dx.doi.org/10.1063/1.3677193 Yiping Tang "A new method of semigrand canonical ensemble to calculate first-order phase transitions for binary mixtures", Journal of Chemical Physics '''136''' 034505 (2012)]
	[[category: Statistical mechanics]]		[[category: Statistical mechanics]]

Nice and Tidy: Slight tidy

2008-02-28T08:59:25Z

Slight tidy

← Older revision		Revision as of 10:59, 28 February 2008
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	~~== General features ==~~		'''Semi-grand ensembles''' are used in [[Monte Carlo]] simulation of [[mixtures]]. In these ensembles the total number of molecules is fixed, but the composition can change.
	Semi-grand ensembles are used in Monte Carlo simulation of mixtures. In these ensembles the total number of molecules is fixed, but the composition can change.

	== Canonical ensemble: fixed volume, temperature and number(s) of molecules ==		== Canonical ensemble: fixed volume, temperature and number(s) of molecules ==

	We shall consider a system consisting of ''c'' components;.		We shall consider a system consisting of ''c'' components;.
	In the [[Canonical ensemble\|canonical ensemble]], the differential		In the [[Canonical ensemble\|canonical ensemble]], the differential
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	*<math> A </math> is the [[Helmholtz energy function]]		*<math> A </math> is the [[Helmholtz energy function]]
	*<math> \beta ~~\equiv~~ 1/k_B T </math>		*<math> \beta := 1/k_B T </math>
	*<math> k_B</math> is the [[Boltzmann constant]]		*<math> k_B</math> is the [[Boltzmann constant]]
	*<math> T </math> is the absolute [[temperature]]		*<math> T </math> is the absolute [[temperature]]
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	where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>.		where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>.
	* Now considering the ~~thermodynamical~~ potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>		* Now considering the thermodynamic potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

	:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N -		:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N -
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	:<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>,		:<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>,

	where the ''new'' ~~thermodynamical Potential~~ <math> \beta \Phi </math> is given by:		where the ''new'' thermodynamic potential <math> \beta \Phi </math> is given by:

	:<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N		:<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N

Carl McBride: /* Canonical ensemble: fixed volume, temperature and number(s) of molecules */

2008-02-12T15:14:04Z

Canonical ensemble: fixed volume, temperature and number(s) of molecules

← Older revision		Revision as of 17:14, 12 February 2008
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	*<math> \beta \equiv 1/k_B T </math>		*<math> \beta \equiv 1/k_B T </math>
	*<math> k_B</math> is the [[Boltzmann constant]]		*<math> k_B</math> is the [[Boltzmann constant]]
	*<math> T </math> is the [[~~absolute~~ temperature]]		*<math> T </math> is the absolute [[temperature]]
	*<math> E </math> is the [[internal energy]]		*<math> E </math> is the [[internal energy]]
	*<math> p </math> is the [[pressure]]		*<math> p </math> is the [[pressure]]

Nice and Tidy at 09:53, 7 September 2007

2007-09-07T09:53:24Z

← Older revision		Revision as of 11:53, 7 September 2007
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	\int \left( \prod_{i=1}^c d (R_i^)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^)^{3N_1} , \cdots \right) \right].		\int \left( \prod_{i=1}^c d (R_i^)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^)^{3N_1} , \cdots \right) \right].
	</math>		</math>

	~~... to be continued ...~~

	==References==		==References==
	[[category: Statistical mechanics]]		[[category: Statistical mechanics]]

Noe: /* Fixed pressure and temperature: Semi-grand ensemble: partition function */

2007-09-07T09:16:56Z

Fixed pressure and temperature: Semi-grand ensemble: partition function

← Older revision		Revision as of 11:16, 7 September 2007
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	\int \left( \prod_{i=1}^c d (R_i^)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^)^{3N_1} , \cdots \right) \right].		\int \left( \prod_{i=1}^c d (R_i^)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^)^{3N_1} , \cdots \right) \right].
	</math>		</math>

			... to be continued ...

	==References==		==References==
	[[category: Statistical mechanics]]		[[category: Statistical mechanics]]

Noe: /* Semi-grand ensemble at fixed volume and temperature */

2007-09-07T09:15:37Z

Semi-grand ensemble at fixed volume and temperature

← Older revision		Revision as of 11:15, 7 September 2007
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	* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>		* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

	:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - i		:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N -
	\sum_{i=2}^c N_i d \left( \beta \mu_{i1} \right).		\sum_{i=2}^c N_i d \left( \beta \mu_{i1} \right).
	</math>		</math>

Noe: still correcting errors

2007-09-07T09:14:56Z

still correcting errors

← Older revision		Revision as of 11:14, 7 September 2007
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	* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>		* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

	:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - ~~N_2~~ d \left( \beta \mu_{21} \right).		:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - i
			\sum_{i=2}^c N_i d \left( \beta \mu_{i1} \right).
	</math>		</math>

Noe: corrections

2007-09-07T09:07:03Z

corrections

← Older revision		Revision as of 11:07, 7 September 2007
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	: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \~~mu_2~~ d ~~N_2~~; </math>		: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_i d N_i; </math>


	: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\~~mu_2~~-\~~mu_i~~) d N_i; </math>		: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_i-\mu_1) d N_i; </math>

	or,		or,

Carl McBride at 14:10, 28 May 2007

2007-05-28T14:10:51Z

← Older revision		Revision as of 16:10, 28 May 2007
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	\int \left( \prod_{i=1}^c d (R_i^)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^)^{3N_1} , \cdots \right) \right].		\int \left( \prod_{i=1}^c d (R_i^)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^)^{3N_1} , \cdots \right) \right].
	</math>		</math>
			==References==
			[[category: Statistical mechanics]]

Carl McBride: /* Fixed pressure and temperature */

2007-03-21T11:59:38Z

Fixed pressure and temperature

← Older revision		Revision as of 13:59, 21 March 2007
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	== Fixed pressure and temperature ==		== Fixed pressure and temperature ==

	In the [[~~Isothermal~~-isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> one can write:		In the [[isothermal-isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> one can write:

	:<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>		:<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>