Gibbs distribution - Revision history

Carl McBride: Changed Kelvin to kelvin

2008-02-14T13:01:13Z

Changed Kelvin to kelvin

← Older revision		Revision as of 15:01, 14 February 2008
Line 5:		Line 5:
	where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system		where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system
	and <math>\Theta</math> is the temperature (to convert <math>\Theta</math> into the more familiar		and <math>\Theta</math> is the temperature (to convert <math>\Theta</math> into the more familiar
	[[temperature \|~~Kelvin~~ scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>).		[[temperature \|kelvin scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>).
	The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38)		The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38)

Carl McBride at 15:15, 12 February 2008

2008-02-12T15:15:17Z

← Older revision		Revision as of 17:15, 12 February 2008
Line 5:		Line 5:
	where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system		where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system
	and <math>\Theta</math> is the temperature (to convert <math>\Theta</math> into the more familiar		and <math>\Theta</math> is the temperature (to convert <math>\Theta</math> into the more familiar
	[[Kelvin scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>).		[[temperature \|Kelvin scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>).
	The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38)		The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38)

Carl McBride: New page: Ref 1 Eq. 3.37: : $\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right)$ where $N$ is the number of particles, $H$ is the [[H...

2007-05-29T08:55:26Z

New page: Ref 1 Eq. 3.37: :<math>\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right)</math> where <math>N</math> is the number of particles, <math>H</math> is the [[H...

New page

Ref 1 Eq. 3.37:

:<math>\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right)</math>

where <math>N</math> is the number of particles, <math>H</math> is the [[Hamiltonian]] of the system
and <math>\Theta</math> is the temperature (to convert <math>\Theta</math> into the more familiar
[[Kelvin scale]] one divides by the [[Boltzmann constant]] <math>k_B</math>).
The constant <math>Z_{(N)}</math> is found from the normalization condition (Ref. 1 Eq. 3.38)

:<math>\frac{1}{\Gamma_{(N)}^{(0)}Z_{(N)}} \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1 ... {\rm d}^3r_N \int_{- \infty}^{\infty} \exp \left( - \frac{K_{(N)}}{\Theta}\right) ~{\rm d}^3p_1 ... {\rm d}^3p_N =1</math>

which leads to (Ref. 1 Eq. 3.40)

:<math>Z_{(N)}= \frac{1}{V^N} Q_{(N)}</math>

where (Ref. 1 Eq. 3.41)

:<math>Q_{(N)} = \int_V \exp \left( - \frac{U_1,..., _N}{\Theta}\right) ~{\rm d}^3r_1 ... {\rm d}^3r_N</math>

this is the [[statistical integral]]

:<math>Z \equiv \sum_n e^{-E_n/T}= {\rm tr} ~ e^{-H|T}</math>

where <math>H</math> is the [[Hamiltonian]] of the system.
==References==
# G. A. Martynov "Fundamental Theory of Liquids. Method of Distribution Functions", Adam Hilger (out of print)
[[category: statistical mechanics]]

Gibbs distribution - Revision history

Carl McBride: Changed Kelvin to kelvin

Carl McBride at 15:15, 12 February 2008

Carl McBride: New page: Ref 1 Eq. 3.37: :\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right) where N is the number of particles, H is the [[H...

Carl McBride: New page: Ref 1 Eq. 3.37: : $\mathcal{G}_{(N)} = \frac{1}{Z_{(N)}} \exp \left( - \frac{H_{(N)}}{\Theta}\right)$ where $N$ is the number of particles, $H$ is the [[H...