Master equation: Difference between revisions
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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> | |||
==Equations of evolution== | |||
The equations of evolution for the distribution function <math>\rho</math> for the diagonal fragments(Ref. 1 Eq. 3-1) | |||
:<math>\psi_{ \{k\}\{k\}}(z) = \sum_{n=2}^\infty (-\lambda)^n \langle \{k\} \vert \delta L \left[ \frac{1}{L_0-z} \delta L \right]^n \vert \{k\} \rangle </math> | |||
for the creation fragments (Ref. 1 Eq. 3-2) | |||
:<math>\tilde{C}_{ \{k\}\{k'\}}(z) = \sum_{n=1}^\infty (-\lambda)^n \langle \{k\} \vert \left[ \frac{1}{L_0-z} \delta L \right]^n \vert \{k'\} \rangle </math> | |||
and for the destruction regions (Ref. 1 Eq. 3-3) | |||
:<math>\mathcal{D}_{ \{k'\}\{k''\}}(z) = \sum_{n=1}^\infty (-\lambda)^n \langle \{k'\} \vert \left[ \delta L \frac{1}{L_0-z} \right]^n \vert \{k''\} \rangle </math> | |||
==References== | ==References== | ||
#[http://dx.doi.org/10.1016/0031-8914(61)90008-8 I. | #[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]] | ||
Latest revision as of 11:07, 1 July 2008
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The master equation describes the exact behavior of the velocity distribution for any time (Ref. 1 Eq. 3-11)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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'}
where the time dependent functional of the initial conditions is given by (Ref. 1 Eq. 3-9)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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) }
and the diagonal fragment is given by (Ref. 1 Eq. 3-10)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{00}(\tau) = \frac{1}{2\pi i} \oint_c \exp (-iz \tau) \psi^+_{00} (z)~ {\mathrm d}z }
Equations of evolution[edit]
The equations of evolution for the distribution function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} for the diagonal fragments(Ref. 1 Eq. 3-1)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{ \{k\}\{k\}}(z) = \sum_{n=2}^\infty (-\lambda)^n \langle \{k\} \vert \delta L \left[ \frac{1}{L_0-z} \delta L \right]^n \vert \{k\} \rangle }
for the creation fragments (Ref. 1 Eq. 3-2)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{C}_{ \{k\}\{k'\}}(z) = \sum_{n=1}^\infty (-\lambda)^n \langle \{k\} \vert \left[ \frac{1}{L_0-z} \delta L \right]^n \vert \{k'\} \rangle }
and for the destruction regions (Ref. 1 Eq. 3-3)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{D}_{ \{k'\}\{k''\}}(z) = \sum_{n=1}^\infty (-\lambda)^n \langle \{k'\} \vert \left[ \delta L \frac{1}{L_0-z} \right]^n \vert \{k''\} \rangle }