Inverse temperature - Revision history

Nice and Tidy: Slight tidy.

2010-03-05T13:39:23Z

Slight tidy.

Carl McBride: /* References */ Added ISBN to book reference.

2010-03-04T11:49:38Z

References: Added ISBN to book reference.

← Older revision		Revision as of 13:49, 4 March 2010
Line 36:		Line 36:

	==References==		==References==
	#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987)		#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1
	[[category: Classical thermodynamics]]		[[category: Classical thermodynamics]]
	[[category: statistical mechanics]]		[[category: statistical mechanics]]
	[[category: Non-equilibrium thermodynamics]]		[[category: Non-equilibrium thermodynamics]]

Dduque: cleaned a bit, added reference to quantum stat. mech.

2010-03-04T11:15:36Z

cleaned a bit, added reference to quantum stat. mech.

← Older revision		Revision as of 13:15, 4 March 2010
Line 5:		Line 5:
	This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.		This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.

	Indeed, it shown in Ref. 1 that this is the way it enters. The task is to maximize number of ways $N$ particles may be asigned to $K$ space-momentum cells, such that one has a set of occupation numbers <math>n_i</math>. Introducing the [[partition function]]:		Indeed, it shown in Ref. 1 (pp. 79-85) that this is the way it enters. The task is to maximize number of ways $N$ particles may be asigned to $K$ space-momentum cells, such that one has a set of occupation numbers <math>n_i</math>. Introducing the [[partition function]]:

	:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>		:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>
Line 19:		Line 19:
	An additional constraint, which applies only to dilute gases, is:		An additional constraint, which applies only to dilute gases, is:

	:<math>\sum_i n_i e_i=E</math>,		:<math>\sum_i n_i e_i=E, </math>

	where <math>E</math> is the total energy and <math>e_i=p_i^2/2m</math> is the energy of cell <math>i</math>.		where <math>E</math> is the total energy and <math>e_i=p_i^2/2m</math> is the energy of cell <math>i</math>.

	The method of [[Lagrange multipliers]] ~~entail~~ finding the extremum of the function		The method of [[Lagrange multipliers]] entails finding the extremum of the function

	~~:<math>L=\log\Omega - \alpha (\sum_i n_i - N ) - \beta ( \sum_i n_i e_i - E )</math>,~~

			:<math>L=\log\Omega - \alpha (\sum_i n_i - N ) - \beta ( \sum_i n_i e_i - E ),</math>
	where the two Lagrange multipliers enforce the two conditions and permit the treatment of		where the two Lagrange multipliers enforce the two conditions and permit the treatment of
	the occupations as independent variables. The minimization leads to		the occupations as independent variables. The minimization leads to
Line 35:		Line 33:
	:<math>\beta := \frac{1}{k_BT} .</math>		:<math>\beta := \frac{1}{k_BT} .</math>

			Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).

	==References==		==References==
	#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition~~, pp. 79-85~~ (1987)		#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987)
	[[category: Classical thermodynamics]]		[[category: Classical thermodynamics]]
	[[category: statistical mechanics]]		[[category: statistical mechanics]]
	[[category: Non-equilibrium thermodynamics]]		[[category: Non-equilibrium thermodynamics]]

Dduque: Eqn fixed

2010-03-04T11:07:18Z

Eqn fixed

← Older revision		Revision as of 13:07, 4 March 2010
Line 30:		Line 30:
	the occupations as independent variables. The minimization leads to		the occupations as independent variables. The minimization leads to

	:<math>n_i=Ce^{-\beta e_i), </math>		:<math>n_i=C e^{-\beta e_i}, </math>

	and an application to the case of an ideal gas reveals the connection with the temperature,		and an application to the case of an ideal gas reveals the connection with the temperature,

Dduque at 11:04, 4 March 2010

2010-03-04T11:04:25Z

@@ Line 5: / Line 5: @@
 This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.
-Indeed, it shown in Ref. 1 that this is the way it enters. The task is to maximize number of ways $N$ particles may be asigned to $K$ space-momentum cells. Introducing the  [[partition function]]:
+Indeed, it shown in Ref. 1 that this is the way it enters. The task is to maximize number of ways $N$ particles may be asigned to $K$ space-momentum cells, such that one has a set of occupation numbers <math>n_i</math>. Introducing the  [[partition function]]:
 :<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>
@@ Line 13: / Line 13: @@
 :<math>\log \Omega \approx \log N -N - \sum_ i ( \log n_i + n_i) + \mathrm{consts} ,</math>
-where [[Stirling's approximation]] for large numbers has been used.
+where [[Stirling's approximation]] for large numbers has been used. The maximization must be performed subject to the constraint:
 ==References==

Dduque: temp save

2010-03-04T10:56:28Z

temp save

New page

It is often convenient to define a dimensionless ''inverse'' temperature, <math>\beta</math>:

:<math>\beta := \frac{1}{k_BT}</math>

This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.

Indeed, it shown in Ref. 1 that this is the way it enters. The task is to maximize number of ways $N$ particles may be asigned to $K$ space-momentum cells. Introducing the [[partition function]]:

:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>

one could maximize its logarithm (a monotonous function):

:<math>\log \Omega \approx \log N -N - \sum_ i ( \log n_i + n_i) + \mathrm{consts} ,</math>

where [[Stirling's approximation]] for large numbers has been used.

==References==
#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition, pp. 79-85 (1987)
[[category: Classical thermodynamics]]
[[category: statistical mechanics]]
[[category: Non-equilibrium thermodynamics]]

← Older revision		Revision as of 15:39, 5 March 2010
Line 1:		Line 1:
	It is often convenient to define a dimensionless ''inverse'' ~~temperature~~, <math>\beta</math>:		It is often convenient to define a dimensionless '''inverse temperature''', <math>\beta</math>:

	:<math>\beta := \frac{1}{k_BT}</math>		:<math>\beta := \frac{1}{k_BT}</math>

	This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.		This notation likely comes from its origin as a Lagrangian multiplier, for which Greek letters are customarily written.
			Indeed, it shown in Ref. 1 (pp. 79-85) that this is the way it enters. The task is to maximize number of ways <math>N</math> particles may be assigned to <math>K</math> space-momentum cells, such that one has a set of occupation numbers <math>n_i</math>. Introducing the [[partition function]]:
	Indeed, it shown in Ref. 1 (pp. 79-85) that this is the way it enters. The task is to maximize number of ways $N$ particles may be ~~asigned~~ to $K$ space-momentum cells, such that one has a set of occupation numbers <math>n_i</math>. Introducing the [[partition function]]:

	:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>		:<math>\Omega\propto\frac{N!}{n_1! n_2! \ldots n_K!} ,</math>
Line 20:		Line 19:

	:<math>\sum_i n_i e_i=E, </math>		:<math>\sum_i n_i e_i=E, </math>

	where <math>E</math> is the total energy and <math>e_i=p_i^2/2m</math> is the energy of cell <math>i</math>.		where <math>E</math> is the total energy and <math>e_i=p_i^2/2m</math> is the energy of cell <math>i</math>.

	The method of [[Lagrange multipliers]] entails finding the extremum of the function		The method of [[Lagrange multipliers]] entails finding the extremum of the function

	:<math>L=\log\Omega - \alpha (\sum_i n_i - N ) - \beta ( \sum_i n_i e_i - E ),</math>		:<math>L=\log\Omega - \alpha (\sum_i n_i - N ) - \beta ( \sum_i n_i e_i - E ),</math>

	where the two Lagrange multipliers enforce the two conditions and permit the treatment of		where the two Lagrange multipliers enforce the two conditions and permit the treatment of
	the occupations as independent variables. The minimization leads to		the occupations as independent variables. The minimization leads to
Line 31:		Line 31:

	and an application to the case of an ideal gas reveals the connection with the temperature,		and an application to the case of an ideal gas reveals the connection with the temperature,

	:<math>\beta := \frac{1}{k_BT} .</math>		:<math>\beta := \frac{1}{k_BT} .</math>

	Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).		Similar methods are used for [[quantum statistics]] of dilute gases (Ref. 1, pp. 179-185).

	==References==		==References==
	#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1		#Kerson Huang, "Statistical Physics" John Wiley and Sons, second edition (1987) ISBN 978-0-471-81518-1