Editing Beispiel des Großkanonischen Ensenbles
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| Latest revision | Your text | ||
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Illustration am Anhand von | Illustration am Anhand von | ||
<math>\begin{align} | |||
& {{G}_{\nu }}=\left\{ H,N \right\} \\ | & {{G}_{\nu }}=\left\{ H,N \right\} \\ | ||
& {{h}_{\alpha }}=\left\{ V \right\} \\ | & {{h}_{\alpha }}=\left\{ V \right\} \\ | ||
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<math>\begin{align} | |||
& R=\frac{1}{Z}{{e}^{-\sum\limits_{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}} \\ | & R=\frac{1}{Z}{{e}^{-\sum\limits_{\nu }{{{\lambda }_{\nu }}{{G}_{\nu }}}}} \\ | ||
& {{R}_{gk}}=\frac{1}{{{Z}_{gk}}}{{e}^{-{{\lambda }_{1}}H-{{\lambda }_{2}}N}} | & {{R}_{gk}}=\frac{1}{{{Z}_{gk}}}{{e}^{-{{\lambda }_{1}}H-{{\lambda }_{2}}N}} | ||
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oftmals <math>{{\lambda }_{1}}=\beta ,\quad {{\lambda }_{2}}=-\beta \mu </math> | oftmals <math>{{\lambda }_{1}}=\beta ,\quad {{\lambda }_{2}}=-\beta \mu </math> | ||
<math>\left( {{\lambda }_{1}},{{\lambda }_{2}} \right)\to \left( \beta ,\mu \right)</math> | |||
wir zeigen: | wir zeigen: | ||
<math>\beta =\frac{1}{kT}</math> Temperatur taucht auf muss gezeigt werden | |||
<math>\mu</math> = Chemisches Potential ist die Energie die man braucht um 1 Teilchen hinzu zufügen | |||
<math>{{R}_{gk}}=\frac{1}{Z}{{e}^{-\beta \left( H-\mu N \right)}}</math> | |||
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braucht man um Zustandsgleichung festzulegen | braucht man um Zustandsgleichung festzulegen | ||
<math>S=S\left( \left\langle {{G}_{\nu }} \right\rangle ,{{h}_{\alpha }} \right)</math> | |||
<math>\Rightarrow {{S}_{gk}}={{S}_{gk}}\left( \left\langle H \right\rangle ,\left\langle N \right\rangle ,V \right)</math> | |||
<math>{{S}_{gk}}\left( E,\overline{N},V \right)=k\beta E-k\beta \mu \overline{N}+k\ln {{Z}_{gk}}\left( \beta \mu V \right)</math> | |||
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Beziehungen der partiellen Ableitungen aus Gibbsgleichung | Beziehungen der partiellen Ableitungen aus Gibbsgleichung | ||
<math>k{{\lambda }_{\nu }}={{\partial }_{\left\langle {{G}_{\nu }} \right\rangle }}S;\quad k\sum\limits_{\nu }{{{\lambda }_{\nu }}{{M}_{\nu ,\alpha }}={{\partial }_{{{h}_{\alpha }}}}S}</math> | |||
für <math>\nu=1</math> | für <math>\nu=1</math> | ||
<math>k{{\lambda }_{\nu }}={{\partial }_{\left\langle {{G}_{\nu }} \right\rangle }}S\Rightarrow k\beta ={{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}}};\quad k\sum\limits_{\nu }{{{\lambda }_{\nu }}{{M}_{\nu ,\alpha }}={{\partial }_{{{h}_{\alpha }}}}S}\Rightarrow {{\left( \frac{\partial S}{\partial N} \right)}_{E,\bar{N}}}=-k\beta \operatorname{Tr}\left( \frac{\partial H}{\partial V}R \right)</math> | |||
<math>\begin{align} | |||
& k{{\lambda }_{\nu }}={{\partial }_{\left\langle {{G}_{\nu }} \right\rangle }}S\Rightarrow k\beta ={{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}\left( \left( \text{V},\text{N sind nicht anzufassen bei der partiellen Ableitung} \right) \right)}} \\ | & k{{\lambda }_{\nu }}={{\partial }_{\left\langle {{G}_{\nu }} \right\rangle }}S\Rightarrow k\beta ={{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}\left( \left( \text{V},\text{N sind nicht anzufassen bei der partiellen Ableitung} \right) \right)}} \\ | ||
& k\sum\limits_{\nu }{{{\lambda }_{\nu }}{{M}_{\nu ,\alpha }}={{\partial }_{{{h}_{\alpha }}}}S}\Rightarrow {{\left( \frac{\partial S}{\partial N} \right)}_{E,\bar{N}}}=-k\beta \operatorname{Tr}\left( \frac{\partial H}{\partial V}R \right)\quad \left( {{\partial }_{V}}N\to 0 \right) \\ | & k\sum\limits_{\nu }{{{\lambda }_{\nu }}{{M}_{\nu ,\alpha }}={{\partial }_{{{h}_{\alpha }}}}S}\Rightarrow {{\left( \frac{\partial S}{\partial N} \right)}_{E,\bar{N}}}=-k\beta \operatorname{Tr}\left( \frac{\partial H}{\partial V}R \right)\quad \left( {{\partial }_{V}}N\to 0 \right) \\ | ||
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für <math>\nu=2</math> | für <math>\nu=2</math> | ||
<math>\begin{align} | |||
& -k\beta \mu ={{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}}} \\ | & -k\beta \mu ={{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}}} \\ | ||
& k{{\partial }_{V}}\ln {{Z}_{gk}}=k\beta p\Rightarrow p=\frac{1}{\beta }{{\partial }_{V}}\ln {{Z}_{gk}} \\ | & k{{\partial }_{V}}\ln {{Z}_{gk}}=k\beta p\Rightarrow p=\frac{1}{\beta }{{\partial }_{V}}\ln {{Z}_{gk}} \\ | ||
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<math>\begin{align} | |||
& {{T}^{-1}}={{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}}} \\ | & {{T}^{-1}}={{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}}} \\ | ||
& \mu =-T{{\left( \frac{\partial S}{\partial \bar{N}} \right)}_{V,E}} \\ | & \mu =-T{{\left( \frac{\partial S}{\partial \bar{N}} \right)}_{V,E}} \\ | ||
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es ist zu zeigen, dass die Temperaturdefinition sinnvoll ist | es ist zu zeigen, dass die Temperaturdefinition sinnvoll ist | ||
<math>{{T}^{-1}}=\left( \frac{\partial S}{\partial E} \right)</math> | |||
sonst darf man es nicht Temeratur nennen | sonst darf man es nicht Temeratur nennen | ||
dazu zeigen: | dazu zeigen: | ||
<math>{{\left( \frac{\partial S}{\partial E} \right)}_{V,\bar{N}}}</math> ist als Eigenschaft bei 2 System die in Konakt über eine Grenzfläche stehen gleich | |||
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| <math>{{{\bar{N}}}_{1}},{{V}_{1}},{{E}_{1}}</math> | | <math>{{{\bar{N}}}_{1}},{{V}_{1}},{{E}_{1}}</math> | ||
|| | || | ||
<math>{{{\bar{N}}}_{2}},{{V}_{2}},{{E}_{2}}</math> | |||
|} | |} | ||
<math>\begin{align} | |||
& E={{E}_{1}}+{{E}_{2}} \\ | & E={{E}_{1}}+{{E}_{2}} \\ | ||
& V={{V}_{1}}+{{V}_{2}} \\ | & V={{V}_{1}}+{{V}_{2}} \\ | ||
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\end{align}</math> | \end{align}</math> | ||
Zu zeugen: | Zu zeugen: | ||
<math>S\overset{!}{\mathop{=}}\,{{S}_{1}}+{{S}_{2}}</math> | |||
<math>S\tilde{\ }\operatorname{Tr}\left( \rho \ln \rho \right)=\operatorname{Tr}\left( {{\rho }_{1}}{{\rho }_{2}}\ln \left( {{\rho }_{1}}{{\rho }_{2}} \right) \right)</math> | |||
statistischer Operator faktorisiert für '''kleine''' Grenzflächen | statistischer Operator faktorisiert für '''kleine''' Grenzflächen | ||
<math>\operatorname{Tr}\left( {{\rho }_{1}}{{\rho }_{2}}\ln \left( {{\rho }_{1}} \right) \right)+\operatorname{Tr}\left( {{\rho }_{1}}{{\rho }_{2}}\ln \left( {{\rho }_{2}} \right) \right)</math> | |||
mit | mit | ||
<math>\operatorname{Tr}\overset{\wedge}{=}\left\langle {{n}_{1}} \right|\left\langle {{n}_{2}} \right|\ldots \left| {{n}_{1}} \right\rangle \left| {{n}_{2}} \right\rangle </math> | |||
<math>\begin{align} | |||
& ={{\operatorname{Tr}}_{1}}\left( {{\rho }_{1}}\ln \left( {{\rho }_{1}} \right) \right)\underbrace{{{\operatorname{Tr}}_{2}}\left( \rho \right)}_{1}+{{\operatorname{Tr}}_{2}}\left( {{\rho }_{2}}\ln \left( {{\rho }_{2}} \right) \right)\underbrace{\operatorname{Tr}\left( {{\rho }_{1}} \right)}_{1} \\ | & ={{\operatorname{Tr}}_{1}}\left( {{\rho }_{1}}\ln \left( {{\rho }_{1}} \right) \right)\underbrace{{{\operatorname{Tr}}_{2}}\left( \rho \right)}_{1}+{{\operatorname{Tr}}_{2}}\left( {{\rho }_{2}}\ln \left( {{\rho }_{2}} \right) \right)\underbrace{\operatorname{Tr}\left( {{\rho }_{1}} \right)}_{1} \\ | ||
& \Rightarrow S={{S}_{1}}+{{S}_{2}} | & \Rightarrow S={{S}_{1}}+{{S}_{2}} | ||
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<math>\begin{align} | |||
& dE=d{{E}_{1}}+d{{E}_{2}}=0\to -d{{E}_{1}}=d{{E}_{2}} \\ | & dE=d{{E}_{1}}+d{{E}_{2}}=0\to -d{{E}_{1}}=d{{E}_{2}} \\ | ||
& dV=d{{V}_{1}}+d{{V}_{2}}=0\to -d{{V}_{1}}=d{{V}_{2}} \\ | & dV=d{{V}_{1}}+d{{V}_{2}}=0\to -d{{V}_{1}}=d{{V}_{2}} \\ | ||
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nutze bei dS: | nutze bei dS: | ||
<math>\begin{align} | |||
& d{{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}=\frac{\partial {{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial {{V}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}d{{V}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{\partial {{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial {{{\bar{N}}}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}d{{{\bar{N}}}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{\partial {{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial {{E}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}d{{E}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}} \\ | & d{{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}=\frac{\partial {{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial {{V}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}d{{V}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{\partial {{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial {{{\bar{N}}}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}d{{{\bar{N}}}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}+\frac{\partial {{S}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial {{E}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}d{{E}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}} \\ | ||
& d{{S}_{1}}=-d{{S}_{2}} | & d{{S}_{1}}=-d{{S}_{2}} | ||
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<math>\frac{\partial {{S}_{1}}}{\partial {{V}_{1}}}d{{V}_{1}}+\frac{\partial {{S}_{1}}}{\partial {{{\bar{N}}}_{1}}}d{{{\bar{N}}}_{1}}+\frac{\partial {{S}_{1}}}{\partial {{E}_{1}}}d{{E}_{1}}=-\frac{\partial {{S}_{2}}}{\partial {{V}_{2}}}d{{V}_{2}}-\frac{\partial {{S}_{2}}}{\partial {{{\bar{N}}}_{2}}}d{{{\bar{N}}}_{2\;}}-\frac{\partial {{S}_{2}}}{\partial {{E}_{2}}}d{{E}_{2}}</math> | |||
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<math>d{{E}_{1}}=-d{{E}_{2}},-d{{{\bar{N}}}_{1}}=d{{{\bar{N}}}_{2}},-d{{V}_{1}}=d{{V}_{2}}</math> | |||
<math>\begin{align} | |||
& \left( \frac{\partial {{S}_{1}}}{\partial {{E}_{1}}}-\frac{\partial {{S}_{2}}}{\partial {{E}_{2}}} \right)d{{E}_{2}}=0 \\ | & \left( \frac{\partial {{S}_{1}}}{\partial {{E}_{1}}}-\frac{\partial {{S}_{2}}}{\partial {{E}_{2}}} \right)d{{E}_{2}}=0 \\ | ||
& \left( \frac{\partial {{S}_{1}}}{\partial {{{\bar{N}}}_{1}}}-\frac{\partial {{S}_{2}}}{\partial {{{\bar{N}}}_{2}}} \right)d{{{\bar{N}}}_{2}}=0 \\ | & \left( \frac{\partial {{S}_{1}}}{\partial {{{\bar{N}}}_{1}}}-\frac{\partial {{S}_{2}}}{\partial {{{\bar{N}}}_{2}}} \right)d{{{\bar{N}}}_{2}}=0 \\ | ||
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weil N,V,E unabhängig variiert werden können gilt für alle | weil N,V,E unabhängig variiert werden können gilt für alle | ||
<math>d{{E}_{2}}</math>,<math>d{{{\bar{N}}}_{2}}</math>, | |||
<math>d{{V}_{2}}</math> | |||
→ folgende Eigenschaften des Systems im Kontakt sind gleich: | → folgende Eigenschaften des Systems im Kontakt sind gleich: | ||
<math>\begin{align} | |||
& {{\left( \frac{\partial {{S}_{1}}}{\partial {{E}_{1}}} \right)}_{{{V}_{1}},{{{\bar{N}}}_{1}}}}={{\left( \frac{\partial {{S}_{2}}}{\partial {{E}_{2}}} \right)}_{{{V}_{2}},{{{\bar{N}}}_{2}}}} \\ | & {{\left( \frac{\partial {{S}_{1}}}{\partial {{E}_{1}}} \right)}_{{{V}_{1}},{{{\bar{N}}}_{1}}}}={{\left( \frac{\partial {{S}_{2}}}{\partial {{E}_{2}}} \right)}_{{{V}_{2}},{{{\bar{N}}}_{2}}}} \\ | ||
& {{\left( \frac{\partial {{S}_{1}}}{\partial {{{\bar{N}}}_{1}}} \right)}_{{{V}_{1}},{{E}_{1}}}}={{\left( \frac{\partial {{S}_{2}}}{\partial {{{\bar{N}}}_{2}}} \right)}_{{{V}_{2}},{{E}_{2}}}} \\ | & {{\left( \frac{\partial {{S}_{1}}}{\partial {{{\bar{N}}}_{1}}} \right)}_{{{V}_{1}},{{E}_{1}}}}={{\left( \frac{\partial {{S}_{2}}}{\partial {{{\bar{N}}}_{2}}} \right)}_{{{V}_{2}},{{E}_{2}}}} \\ | ||
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<math>\beta =\frac{1}{kT}</math> | |||
beides muss am Experiment verifiziert werden | beides muss am Experiment verifiziert werden | ||
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===Dichtematrixdynamik und Zustandsgleichung=== | ===Dichtematrixdynamik und Zustandsgleichung=== | ||
Dichtematrixdynamik für 2Niveausystem: 1 Teilchen = | Dichtematrixdynamik für 2Niveausystem: 1 Teilchen = | ||
<math>{\bar{N}}</math> | |||
Besetzungszahldarstellung | Besetzungszahldarstellung | ||