Master equation - Revision history

Carl McBride: Added equations of evolution

2008-07-01T10:07:47Z

Added equations of evolution

@@ Line 11: / Line 11: @@
 :<math>G_{00}(\tau)  = \frac{1}{2\pi i} \oint_c \exp (-iz \tau) \psi^+_{00} (z)~ {\mathrm d}z </math>
 ==References==
 #[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigogine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646  (1961)]
 [[category: Non-equilibrium thermodynamics]]

Carl McBride: /* References */ Corrected spelling error

2008-06-27T12:26:30Z

References: Corrected spelling error

← Older revision		Revision as of 14:26, 27 June 2008
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	:<math>G_{00}(\tau) = \frac{1}{2\pi i} \oint_c \exp (-iz \tau) \psi^+_{00} (z)~ {\mathrm d}z </math>		:<math>G_{00}(\tau) = \frac{1}{2\pi i} \oint_c \exp (-iz \tau) \psi^+_{00} (z)~ {\mathrm d}z </math>
	==References==		==References==
	#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. ~~Prigonine~~ and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646 (1961)]		#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigogine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646 (1961)]
	[[category: Non-equilibrium thermodynamics]]		[[category: Non-equilibrium thermodynamics]]

Carl McBride: Added terms

2008-06-26T16:52:10Z

Added terms

← Older revision		Revision as of 18:52, 26 June 2008
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	{{stub-general}}		{{stub-general}}
	The '''master equation''' describes the exact behavior of the velocity distribution for any time (Ref. 1 Eq. 3-11)		The '''master equation''' describes the exact behavior of the [[velocity distribution]] for any time (Ref. 1 Eq. 3-11)

	:<math>\partial_{t \rho_0} \left( \{ {\mathbf \upsilon} \},t \right) = {\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right) + \int_0^t G_{00}(t-t') \rho_0 \left( \{ {\upsilon} \},t' \right) {\mathrm d}t'</math>		:<math>\partial_{t \rho_0} \left( \{ {\mathbf \upsilon} \},t \right) = {\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right) + \int_0^t G_{00}(t-t') \rho_0 \left( \{ {\upsilon} \},t' \right) {\mathrm d}t'</math>

	where		where the time dependent functional of the initial conditions is given by (Ref. 1 Eq. 3-9)

	:<math>{\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right)</math>		:<math>{\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right) = \frac{-1}{2\pi} \oint_c \exp (-izt) \sum_{ \{k'' \} \neq 0} {\mathcal D}^+_{0 \{k'' \}} (z) \rho_{\{k'' \}} \left( \{ {\mathbf \upsilon} \},0 \right) </math>

			and the diagonal fragment is given by (Ref. 1 Eq. 3-10)

			:<math>G_{00}(\tau) = \frac{1}{2\pi i} \oint_c \exp (-iz \tau) \psi^+_{00} (z)~ {\mathrm d}z </math>
	==References==		==References==
	#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigonine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646 (1961)]		#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigonine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646 (1961)]
	[[category: Non-equilibrium thermodynamics]]		[[category: Non-equilibrium thermodynamics]]

Carl McBride: Started adding the master equation

2008-06-26T13:32:51Z

Started adding the master equation

← Older revision		Revision as of 15:32, 26 June 2008
Line 1:		Line 1:
	{{stub-general}}		{{stub-general}}
			The '''master equation''' describes the exact behavior of the velocity distribution for any time (Ref. 1 Eq. 3-11)

			:<math>\partial_{t \rho_0} \left( \{ {\mathbf \upsilon} \},t \right) = {\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right) + \int_0^t G_{00}(t-t') \rho_0 \left( \{ {\upsilon} \},t' \right) {\mathrm d}t'</math>

			where

			:<math>{\mathcal D}_0 \left(t, \rho_{ \{k'' \} } \left( \{ {\mathbf \upsilon} \},0 \right) \right)</math>
	==References==		==References==
	#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigonine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646 (1961)]		#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigonine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646 (1961)]
	[[category: Non-equilibrium thermodynamics]]		[[category: Non-equilibrium thermodynamics]]

Carl McBride: New page: {{stub-general}} ==References== #[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigonine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-64...

2008-06-26T12:42:56Z

New page: {{stub-general}} ==References== #[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigonine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-64...

New page

{{stub-general}}
==References==
#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. Prigonine and P. Résibois "On the kinetics of the approach to equilibrium", Physica '''27''' pp. 629-646 (1961)]
[[category: Non-equilibrium thermodynamics]]