839 lines
37 KiB
TeX
839 lines
37 KiB
TeX
\Part[
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\eng{Statistichal Mechanics}
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\ger{Statistische Mechanik}
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]{stat}
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\begin{ttext}
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\eng{
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\textbf{Extensive quantities:} Additive for subsystems (system size dependent): $S(\lambda E, \lambda V, \lambda N) = \lambda S(E, V, N)$\\
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\textbf{Intensive quantities:} Independent of system size, ratio of two extensive quantities
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}
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\ger{
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\textbf{Extensive Größen:} Additiv für Subsysteme (Systemgrößenabhänig): $S(\lambda E, \lambda V, \lambda N) = \lambda S(E, V, N)$\\
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\textbf{Intensive Größen:} Unabhängig der Systemgröße, Verhältnis zweier extensiver Größen
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}
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\end{ttext}
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\begin{formula}{liouville}
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\desc{Liouville equation}{}{$\{\}$ poisson bracket}
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\desc[german]{Liouville-Gleichung}{}{$\{\}$ Poisson-Klammer}
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\eq{\pdv{\rho}{t} = - \sum_{i=1}^{N} \left(\pdv{\rho}{q_i} \pdv{H}{p_i} - \pdv{\rho}{p_i} \pdv{H}{q_i} \right) = \{H, \rho\}}
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\end{formula}
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\Section[
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\eng{Entropy}
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\ger{Entropie}
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]{entropy}
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\begin{formula}{properties}
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\desc{Positive-definite and additive}{}{}
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\desc[german]{Positiv Definit und Additiv}{}{}
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\eq{
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S &\ge 0 \\
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S(E_1, E_2) &= S_1 + S_2
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}
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\end{formula}
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\begin{formula}{von_neumann}
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\desc{Von-Neumann}{}{$\rho$ density matrix}
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\desc[german]{Von-Neumann}{}{$\rho$ Dichtematrix}
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\eq{S = - \kB \braket{\log \rho} = - \kB \tr(\rho \log\rho)}
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\end{formula}
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\begin{formula}{gibbs}
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\desc{Gibbs}{}{$p_n$ probability for micro state $n$}
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\desc[german]{Gibbs}{}{$p_n$ Wahrscheinlichkeit für Mikrozustand $n$}
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\eq{S = - \kB \sum_n p_n \log p_n}
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\end{formula}
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\begin{formula}{boltzmann}
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\desc{Boltzmann}{}{$\Omega$ \#micro states}
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\desc[german]{Boltzmann}{}{$\Omega$ \#Mikrozustände}
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\eq{S = \kB \log\Omega}
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\end{formula}
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\begin{formula}{temp}
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\desc{Temperature}{}{}
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\desc[german]{Temperatur}{}{}
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\eq{\frac{1}{T} \coloneq \pdv{S}{E}_V}
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\end{formula}
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\begin{formula}{pressure}
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\desc{Pressure}{}{}
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\desc[german]{Druck}{}{}
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\eq{p = T \pdv{S}{V}_E}
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\end{formula}
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\Part[
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\eng{Thermodynamics}
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\ger{Thermodynamik}
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]{td}
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\begin{formula}{therm_wavelength}
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\desc{Thermal wavelength}{}{}
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\desc[german]{Thermische Wellenlänge}{}{}
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\eq{\lambda = \frac{\hbar}{\sqrt{2\pi m \kB T}}}
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\end{formula}
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\Section[
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\eng{Processes}
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\ger{Prozesse}
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]{process}
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\begin{ttext}
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\eng{
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\begin{itemize}
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\item \textbf{isobaric}: constant pressure $p = \const$
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\item \textbf{isochoric}: constant volume $V = \const$
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\item \textbf{isothermal}: constant temperature $T = \const$
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\item \textbf{isentropic}: constant entropy $S = \const$
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\item \textbf{isenthalpic}: constant entalphy $H = \const$
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\item \textbf{adiabatic}: no heat transfer $\Delta Q=0$
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\item \textbf{quasistatic}: happens so slow, the system always stays in td. equilibrium
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\item \textbf{reversivle}: reversible processes are always quasistatic and no entropie is created $\Delta S = 0$
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\end{itemize}
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}
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\ger{
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\begin{itemize}
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\item \textbf{isobar}: konstanter Druck $p = \const$
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\item \textbf{isochor}: konstantes Volumen $V = \const$
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\item \textbf{isotherm}: konstante Temperatur $T = \const$
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\item \textbf{isentrop}: konstante Entropie $S = \const$
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\item \textbf{isenthalp}: konstante Entalphie $H = \const$
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\item \textbf{adiabatisch}: kein Wärmeübertrag $\Delta Q=0$
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\item \textbf{quasistatsch}: läuft so langsam ab, dass das System durchgehend im t.d Equilibrium bleibt
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\item \textbf{reversibel}: reversible Prozesse sind immer quasistatisch und es wird keine Entropie erzeugt $Delta S = 0$
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\end{itemize}
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}
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\end{ttext}
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\Subsection[
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\eng{Irreversible gas expansion (Gay-Lussac experiment)}
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\ger{Irreversible Gasexpansion (Gay-Lussac-Versuch)}
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]{gay}
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\begin{bigformula}{experiment}
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\desc{Gay-Lussac experiment}{}{}
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\desc[german]{Gay-Lussac-Versuch}{}{}
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\begin{minipage}{0.6\textwidth}
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\vfill
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\begin{ttext}
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\eng{
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A classical gas in a system with volume $V_1$ is separated from another system with volume $V_2$.
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In the Gay-Lussac experiment, the separation is removed and the gas flows into $V_2$.
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}
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\ger{
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Ein klassisches Gas in einem System mit Volumen $V_1$ ist getrennt von einem zweiten System mit Volumen $V_2$.
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Beim Gay-Lussac Versuch wird die Trennwand entfern und das Gas fließt in das Volumen $V_2$.
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}
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\end{ttext}
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\vfill
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\end{minipage}
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\hfill
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\begin{minipage}{0.3\textwidth}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{img/td_gay_lussac.pdf}
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\end{figure}
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\end{minipage}
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\end{bigformula}
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\begin{formula}{entropy}
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\desc{Entropy change}{}{}
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\desc[german]{Entropieänderung}{}{}
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\eq{\Delta S=N\kB \ln \left(\frac{V_1 + V_2}{V_1}\right) > 0}
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\end{formula}
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\TODO{Reversible}
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\TODO{Quasistatischer T-Ausgleich}
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\TODO{Joule-Thompson Prozess}
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\Section[
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\eng{Phase transitions}
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\ger{Phasenübergänge}
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]{phases}
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\begin{ttext}
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\eng{
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A phase transition is a discontinuity in the free energy $F$ or Gibbs energy $G$ or in one of their derivatives.
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The degree of the phase transition is the degree of the derivative which exhibits the discontinuity.
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}
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\ger{
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Ein Phasenübergang ist eine Unstetigkeit in the Freien Energie $F$ oder in der Gibbs-Energie $G$ oder in ihrer Ableitungen.
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Die Ordnung des Phasenübergangs ist die Ordnung der Ableitung, in welcher die Unstetigkeit auftritt.
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}
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\end{ttext}
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\begin{formula}{latent_heat}
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\desc{Latent heat}{Heat required to bring substance from phase 1 to phase 2}{$\Delta S$ entropy change of the phase transition}
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\desc[german]{Latente Wärme}{Für den Phasenübergang von Phase 1 nach Phase 2 benötigte Wärme}{$\Delta S$ Entropieänderung des Phasenübergangs}
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\eq{Q_\text{L} = T \Delta S}
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\end{formula}
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\begin{formula}{clausis_clapeyron}
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\desc{Clausius-Clapyeron equation}{Slope of the coexistence curve}{$\Delta V$ Volume change of the phase transition}
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\desc[german]{Clausius-Clapeyron Gleichung}{Steigung der Phasengrenzlinie}{$\Delta V$ Volumenänderung des Phasenübergangs}
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\eq{\odv{p}{T} = \frac{Q_\text{L}}{T\Delta V}}
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\end{formula}
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\begin{formula}{coexistence}
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\desc{Phase transition}{At the coexistence curve}{}
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\desc[german]{Phasenübergang}{Im Koexistenzbereich}{}
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\eq{G_1 = G_2 \\ \shortintertext{\GT{and_therefore}} \mu_1 = \mu_2}
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\end{formula}
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\begin{formula}{gibbs_phase_rule}
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\desc{Gibbs rule / Phase rule}{}{$c$ \#components, $f$ \#degrees of freedom, $p$ \#phases}
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\desc[german]{Gibbsche Phasenregel}{}{$c$ \#Komponenten, $f$ \#Freiheitsgrade, $p$ \#Phasen}
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\eq{f = c - p + 2}
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\end{formula}
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\Subsubsection[
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\eng{Osmosis}
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\ger{Osmose}
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]{osmosis}
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\begin{ttext}
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\eng{
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Osmosis is the spontaneous net movement or diffusion of solvent molecules
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through a selectively-permeable membrane, which allows through the solvent molecules, but not the solute molecules.
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The direction of the diffusion is from a region of high water potential
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(region of lower solute concentration) to a region of low water potential
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(region of higher solute concentration), in the direction that tends to equalize the solute concentrations on the two sides.
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}
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\ger{
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Osmosis ist die spontane Passage oder Diffusion Lösungsmittelmolekülen durch eine semi-permeable Membran
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die für das Lösungsmittel, jedoch nicht die darin gelösten Stoffe durchlässig ist.
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Die Richtung der Diffusion ist vom Gebiet mit hohem chemischen Potential (niedrigere Konzentration des gelösten Stoffes)
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in das mit niedrigem chemischem Potential (höherere Konzentraion des gelösten Stoffes), sodass die Konzentration des gelösten Stoffes ausgeglichen wird.
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}
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\end{ttext}
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\begin{formula}{osmosis}
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\desc{Osmotic pressure}{}{$N_c$ \#dissolved particles}
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\desc[german]{Osmotischer Druck / Van-\'t-hoffsches Gesetz}{}{$N_c$ \#gelöster Teilchen}
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\eq{p_\text{osm} = \kB T \frac{N_c}{V}}
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\end{formula}
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\Subsection[
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\eng{Material properties}
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\ger{Materialeigenschaften}
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]{props}
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\begin{formula}{heat_cap}
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\desc{Heat capacity}{}{$Q$ heat}
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\desc[german]{Wärmekapazität}{}{$Q$ Wärme}
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\eq{c = \frac{Q}{\Delta T}}
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\end{formula}
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\begin{formula}{heat_cap_V}
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\desc{Isochoric heat capacity}{}{$U$ internal energy}
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\desc[german]{Isochore Wärmekapazität}{}{$U$ innere Energie}
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\eq{c_v = \pdv{Q}{T}_V = \pdv{U}{T}_V}
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\end{formula}
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\begin{formula}{heat_cap_p}
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\desc{Isobaric heat capacity}{}{$H$ enthalpy}
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\desc[german]{Isobare Wärmekapazität}{}{$H$ Enthalpie}
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\eq{c_p = \pdv{Q}{T}_P = \pdv{H}{T}_P}
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\end{formula}
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\begin{formula}{bulk_modules}
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\desc{Bulk modules}{}{$p$ pressure, $V$ initial volume}
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\desc[german]{Kompressionsmodul}{}{$p$ Druck, $V$ Anfangsvolumen}
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\eq{K = -V \odv{p}{V} }
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\end{formula}
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\begin{formula}{compressibility}
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\desc{Compressibility}{}{}
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\desc[german]{Kompressibilität}{}{}
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\eq{\kappa = -\frac{1}{V} \pdv{V}{p} }
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\end{formula}
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\begin{formula}{compressibility_T}
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\desc{Isothermal compressibility}{}{}
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\desc[german]{Isotherme Kompressibilität}{}{}
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\eq{\kappa_T = -\frac{1}{V} \pdv{V}{p}_{T} = \frac{1}{K}}
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\end{formula}
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\begin{formula}{compressibility_S}
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\desc{Adiabatic compressibility}{}{}
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\desc[german]{Adiabatische Kompressibilität}{}{}
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\eq{\kappa_S = -\frac{1}{V} \pdv{V}{p}_{S}}
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\end{formula}
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\begin{formula}{therm_expansion}
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\desc{Thermal expansion coefficient}{}{}
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\desc[german]{Thermaler Ausdehnungskoeffizient}{}{}
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\eq{\alpha = \frac{1}{V} \pdv{V}{T}_{p,N}}
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\end{formula}
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\Section[
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\eng{Laws of thermodynamics}
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\ger{Hauptsätze der Thermodynamik}
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]{laws}
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\Subsection[
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\eng{Zeroeth law}
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\ger{Nullter Hauptsatz}
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]{law0}
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\begin{ttext}
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\eng{If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.}
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\ger{Wenn sich zwei Siesteme jeweils im thermischen Gleichgewicht mit einem dritten befinden, befinden sie sich auch untereinander im thermischen Gleichgewicht.}
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\end{ttext}
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\Eng[teq]{th. eq.}
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\Ger[teq]{th. GGW.}
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\def\ggwarrow{\overset{\GT{teq}}{\leftrightarrow}}
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\begin{equation}
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\label{eq:\fqname}
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A \ggwarrow C \quad\wedge\quad B \ggwarrow C \quad\Rightarrow\quad A \ggwarrow B
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\end{equation}
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\Subsection[
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\eng{First law}
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\ger{Erster Hauptsatz}
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]{law1}
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\begin{ttext}
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\eng{In a process without transfer of matter, the change in internal energy, $\Delta U$, of a thermodynamic system is equal to the energy gained as heat, $Q$, less the thermodynamic work, W, done by the system on its surroundings.}
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\ger{In einem abgeschlossenem System ist die Änderung der inneren Energie $U$ gleich der gewonnenen Wärme $Q$ minus der vom System an der Umgebung verrichteten Arbeit $W$.}
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\end{ttext}
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\begin{formula}{internal_energy}
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\desc{Internal energy change}{}{}
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\desc[german]{Änderung der inneren Energie}{}{}
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\eq{
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\Delta U &= \delta Q - \delta W \\
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\d U &= T \d S - p \d V
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}
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\end{formula}
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\Subsection[
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\eng{Second law}
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\ger{Zweiter Hauptsatz}
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]{law2}
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\begin{ttext}
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\eng{
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\textbf{Clausius}: Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.\\
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\textbf{Kelvin}: It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.
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}
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\ger{
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\textbf{Clausius}: Es gibt keine Zustansänderung, deren einziges Ergebnis die Übertragung von Wärme von einem Körper niederer Temperatur auf einen Körper höherer Temperatur ist.\\
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\textbf{Kelvin}: Es ist unmöglich, eine periodisch arbeitende Maschine zu konstruieren, die weiter nichts bewirkt als Hebung einer Last und Abkühlung eines Wärmereservoirs.
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}
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\end{ttext}
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\Subsection[
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\eng{Third law}
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\ger{Dritter Hauptsatz}
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]{law3}
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\begin{ttext}
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\eng{It is impussible to cool a system to absolute zero.}
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\ger{Es ist unmöglich, ein System bis zum absoluten Nullpunkt abzukühlen.}
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\end{ttext}
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\begin{formula}{3d_law}
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\desc{Entropy density}{}{$s = \frac{S}{N}$}
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\desc[german]{Entropiedichte}{}{$s = \frac{S}{N}$}
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\eq{
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\lim_{T\to 0} s(T) &= 0 \\
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\shortintertext{\GT{and_therefore_also}}
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\lim_{T\to 0} c_V &= 0
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}
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\end{formula}
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\Section[
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\eng{Ensembles}
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\ger{Ensembles}
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]{ensembles}
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\Eng[const_variables]{Constant variables}
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\Ger[const_variables]{Konstante Variablen}
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\begin{bigformula}{nve} \absLabel[mc_ensemble]
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\desc{Microcanonical ensemble}{}{}
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\desc[german]{Mikrokanonisches Ensemble}{}{}
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\begin{minipagetable}{nve}
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\entry{const_variables} {$E$, $V,$ $N$ }
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\entry{partition_sum} {$\Omega = \sum_n 1$ }
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\entry{probability} {$p_n = \frac{1}{\Omega}$}
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\entry{td_pot} {$S = \kB\ln\Omega$ }
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\entry{pressure} {$p = T \pdv{S}{V}_{E,N}$}
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\entry{entropy} {$S = \kB = \ln\Omega$ }
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\end{minipagetable}
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\end{bigformula}
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\begin{bigformula}{nvt} \absLabel[c_ensemble]
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\desc{Canonical ensemble}{}{}
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\desc[german]{Kanonisches Ensemble}{}{}
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\begin{minipagetable}{nvt}
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\entry{const_variables} {$T$, $V$, $N$ }
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\entry{partition_sum} {$Z = \sum_n \e^{-\beta E_n}$ }
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\entry{probability} {$p_n = \frac{\e^{-\beta E_n}}{Z}$}
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\entry{td_pot} {$F = - \kB T \ln Z$ }
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\entry{pressure} {$p = -\pdv{F}{V}_{T,N}$ }
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\entry{entropy} {$S = -\pdv{F}{T}_{V,N}$ }
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\end{minipagetable}
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\end{bigformula}
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\begin{bigformula}{mvt} \absLabel[gc_ensemble]
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\desc{Grand canonical ensemble}{}{}
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\desc[german]{Grosskanonisches Ensemble}{}{}
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\begin{minipagetable}{mvt}
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\entry{const_variables} {$T$, $V$, $\mu$ }
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\entry{partition_sum} {$Z_\text{g} = \sum_{n} \e^{-\beta(E_n - \mu N_n)}$ }
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\entry{probability} {$p_n = \frac{\e^{-\beta (E_n - \mu N_n}}{Z_\text{g}}$}
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\entry{td_pot} {$ \Phi = - \kB T \ln Z$ }
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\entry{pressure} {$p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$ }
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\entry{entropy} {$S = -\pdv{\Phi}{T}_{V,\mu}$ }
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\end{minipagetable}
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\end{bigformula}
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\begin{bigformula}{npt}
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\desc{Isobaric-isothermal}{Gibbs ensemble}{}
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% \desc[german]{Kanonisches Ensemble}{}{}
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\begin{minipagetable}{npt}
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\entry{const_variables} {$N$, $p$, $T$}
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\entry{partition_sum} {}
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\entry{probability} {$p_n ? \frac{\e^{-\beta(E_n + pV_n)}}{Z}$}
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\entry{td_pot} {}
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\entry{pressure} {}
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\entry{entropy} {}
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\end{minipagetable}
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\end{bigformula}
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\begin{bigformula}{nph}
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\desc{Isonthalpic-isobaric ensemble}{Enthalpy ensemble}{}
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% \desc[german]{Kanonisches Ensemble}{}{}
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\begin{minipagetable}{nph}
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\entry{const_variables} {}
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\entry{partition_sum} {}
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\entry{probability} {}
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\entry{td_pot} {}
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|
\entry{pressure} {}
|
|
\entry{entropy} {}
|
|
\end{minipagetable}
|
|
\end{bigformula}
|
|
|
|
\TODO{complete, link potentials}
|
|
|
|
|
|
\begin{formula}{ergodic_hypo}
|
|
\desc{Ergodic hypothesis}{Over a long periode of time, all accessible microstates in the phase space are equiprobable}{$A$ Observable}
|
|
\desc[german]{Ergodenhypothese}{Innerhalb einer langen Zeitspanne sind alle energetisch erreichbaren Mikrozustände im Phasenraum gleich wahrscheinlich}{$A$ Messgröße}
|
|
\eq{\braket{A}_\text{\GT{time}} = \braket{A}_\text{\GT{ensemble}}}
|
|
\end{formula}
|
|
|
|
|
|
\Subsection[
|
|
\eng{Potentials}
|
|
\ger{Potentiale}
|
|
]{pots}
|
|
\begin{formula}{internal_energy}
|
|
\desc{Internal energy}{}{}
|
|
\desc[german]{Innere Energie}{}{}
|
|
\eq{\d U(S,V,N) = T\d S -p\d V + \mu\d N}
|
|
\end{formula}
|
|
\begin{formula}{free_energy}
|
|
\desc{Free energy / Helmholtz energy }{}{}
|
|
\desc[german]{Freie Energie / Helmholtz Energie}{}{}
|
|
\eq{\d F(T,V,N) = -S\d T -p\d V + \mu\d N}
|
|
\end{formula}
|
|
\begin{formula}{enthalpy}
|
|
\desc{Enthalpy}{}{}
|
|
\desc[german]{Enthalpie}{}{}
|
|
\eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N}
|
|
\end{formula}
|
|
\begin{formula}{gibbs_energy}
|
|
\desc{Free enthalpy / Gibbs energy}{}{}
|
|
\desc[german]{Freie Entahlpie / Gibbs-Energie}{}{}
|
|
\eq{\d G(T,p,N) = -S\d T + V\d p + \mu\d N}
|
|
\end{formula}
|
|
\begin{formula}{grand_canon_pot}
|
|
\desc{Grand canonical potential}{}{}
|
|
\desc[german]{Großkanonisches Potential}{}{}
|
|
\eq{\d \Phi(T,V,\mu) = -S\d T -p\d V - N\d\mu}
|
|
\end{formula}
|
|
|
|
\TODO{Maxwell Relationen, TD Quadrat}
|
|
\begin{formula}{td-square}
|
|
\desc{Thermodynamic squre}{}{}
|
|
\desc[german]{Themodynamisches Quadrat}{Guggenheim Quadrat}{}
|
|
\begin{minipage}{0.3\textwidth}
|
|
\begin{tikzpicture}
|
|
\draw[thick] (0,0) grid (3,3);
|
|
\node at (0.5, 2.5) {$-S$};
|
|
\node at (1.5, 2.5) {\color{blue}$U$};
|
|
\node at (2.5, 2.5) {$V$};
|
|
\node at (0.5, 1.5) {\color{blue}$H$};
|
|
\node at (2.5, 1.5) {\color{blue}$F$};
|
|
\node at (0.5, 0.5) {$-p$};
|
|
\node at (1.5, 0.5) {\color{blue}$G$};
|
|
\node at (2.5, 0.5) {$T$};
|
|
\end{tikzpicture}
|
|
\end{minipage}
|
|
\begin{ttext}
|
|
\eng{The corners opposite from the potential are the coefficients and each coefficients differential is opposite to it.}
|
|
\ger{Die Ecken gegenüber des Potentials sind die Koeffizienten, das Differential eines Koeffizienten ist in der Ecke gegenüber.}
|
|
\end{ttext}
|
|
\end{formula}
|
|
|
|
\Section[
|
|
\eng{Ideal gas}
|
|
\ger{Ideales Gas}
|
|
]{id_gas}
|
|
\begin{ttext}
|
|
\eng{The ideal gas consists of non-interacting, undifferentiable particles.}
|
|
\ger{Das ideale Gas besteht aus nicht-wechselwirkenden, ununterscheidbaren Teilchen.}
|
|
\end{ttext}
|
|
|
|
\begin{formula}{phase_space_vol}
|
|
\desc{Phase space volume}{$3N$ sphere}{$N$ \#particles, $h^{3N}$ volume of a microstate, $N!$ particles are undifferentiable}
|
|
\desc[german]{}{$3N$ Kugel}{$N$ \#Teilchen, $h^{3N}$ Volumen eines Mikrozustandes, $N!$ Teilchen sind ununterscheidbar}
|
|
\eq{
|
|
\Omega(E) &= \int_V\d^3q_1 \sdots \int_V\d^3q_N \int \d^3p_1 \sdots \int\d^3p_N \frac{1}{N!\,h^{3N}} \Theta\left(E - \sum_{i} \frac{\vec{p_i}^2}{2m}\right) \\
|
|
&= \left(\frac{V}{N}\right)^N \left(\frac{4\pi m E}{3 h^2 N}\right)^{\frac{3N}{2}} \e^\frac{5N}{2}
|
|
}
|
|
\end{formula}
|
|
\begin{formula}{entropy}
|
|
\desc{Entropy}{}{}
|
|
\desc[german]{Entropie}{}{}
|
|
\eq{
|
|
S = \frac{5}{2} N\kB + N\kB\ln\left(\frac{V}{N}\left(\frac{2\pi m E}{3 h^2 N}\right)^{\frac{3}{2}}\right)
|
|
}
|
|
\end{formula}
|
|
|
|
\begin{formula}{id_gas_eq}
|
|
\desc{Ideal gas equation}{}{}
|
|
\desc[german]{Ideale Gasgleichung}{Thermische Zustandsgleichung idealer Gase}{}
|
|
\eq{pV &= nRT \\ &= N\kB T}
|
|
\end{formula}
|
|
|
|
\begin{formula}{equation_of_state}
|
|
\desc{Equation of state}{}{}
|
|
\desc[german]{Kalorische Zustangsgleichung}{}{}
|
|
\eq{U = \frac{3}{2} N\kB T}
|
|
\end{formula}
|
|
|
|
% \Subsubsection[
|
|
% \eng{Equipartitiontheorem}
|
|
% \ger{Äquipartitionstheorem}
|
|
% ]{equipart}
|
|
\begin{formula}{equipart}
|
|
\desc{Equipartitiontheorem}{Each degree of freedom contributes $U_\text{D}$ (for classical particle systems)}{}
|
|
\desc[german]{Äquipartitionstheorem}{Jedem Freiheitsgrad steht die Energie $U_\text{D}$ zur Verfügung}{}
|
|
\eq{U_\text{D} = \frac{1}{2} \kB T}
|
|
\end{formula}
|
|
|
|
|
|
\begin{formula}{maxwell_velocity}
|
|
\desc{Maxwell velocity distribution}{See \absRef{maxwell-boltzmann_distribution}}{}
|
|
\desc[german]{Maxwellsche Geschwindigkeitsverteilung}{Siehe auch \absRef{maxwell-boltzmann_distribution}}{}
|
|
\eq{w(v) \d v = 4\pi \left(\frac{\beta m}{2\pi}\right)^\frac{3}{2} v^2 \e^{-\frac{\beta m v^2}{2}} \d v}
|
|
\end{formula}
|
|
|
|
\begin{formula}{avg_velocity}
|
|
\desc{Average quadratic velocity}{per particle in a 3D gas}{}
|
|
\desc[german]{Mittlere quadratosche Geschwindigkeit}{pro Teilchen im 3D-Gas}{}
|
|
\eq{\braket{v^2} = \int_0^\infty \d v\,v^2 w(v) = \frac{3\kB T}{m}}
|
|
\end{formula}
|
|
|
|
\Subsubsection[
|
|
\eng{Molecule gas}
|
|
\ger{Molekülgas}
|
|
]{molecule_gas}
|
|
|
|
\begin{formula}{desc}
|
|
\desc{Molecule gas}{2 particles of mass $M$ connected by a ``spring'' with distance $L$}{}
|
|
\desc[german]{Molekülgas}{2 Teilchen der Masse $M$ sind verbunden durch eine ``Feder'' mit Länge $L$}{}
|
|
% \begin{figure}[h]
|
|
\centering
|
|
\tikzstyle{spring}=[thick,decorate,decoration={coil,aspect=0.8,amplitude=5,pre length=0.1cm,post length=0.1cm,segment length=10}]
|
|
\begin{tikzpicture}
|
|
\def\radius{0.5}
|
|
\coordinate (left) at (-3, 0);
|
|
\coordinate (right) at (3, 0);
|
|
\draw (left) circle (\radius);
|
|
\draw[spring] ($(left) + (\radius,0)$) -- ($(right) - (\radius,0)$);
|
|
\draw (right) circle (\radius);
|
|
\end{tikzpicture}
|
|
% \end{figure}
|
|
\end{formula}
|
|
|
|
\begin{formula}{translation}
|
|
\desc{Translation}{}{$n_i \in \N_0$, $i=x,\,y,\,z$}
|
|
\desc[german]{Translation}{}{$n_i \in \N_0$, $i=x,\,y,\,z$}
|
|
\eq{p_i &= \frac{2\pi\hbar}{L}n_i \\
|
|
E_\text{kin} &= \frac{\vec{p}_r^2}{2M}
|
|
}
|
|
\end{formula}
|
|
\begin{formula}{vibration}
|
|
\desc{Vibration}{}{$n \in \N_0$}
|
|
\desc[german]{Schwingungen}{}{$n \in \N_0$}
|
|
\eq{E_\text{vib} = \hbar \omega \left(n+\frac{1}{2}\right)}
|
|
\end{formula}
|
|
\begin{formula}{rotation}
|
|
\desc{Rotation}{}{$j\in \N_0$}
|
|
\desc[german]{Rotation}{}{$j\in \N_0$}
|
|
\eq{E_\text{rot} = \frac{\hbar^2}{2I}j(j+1)}
|
|
\end{formula}
|
|
\TODO{Diagram für verschiedene Temperaturen, Weiler Skript p.83}
|
|
|
|
|
|
\Section[
|
|
\eng{Real gas}
|
|
\ger{Reales Gas}
|
|
]{real_gas}
|
|
|
|
\Subsection[
|
|
\eng{Virial expansion}
|
|
\ger{Virialentwicklung}
|
|
]{virial}
|
|
\begin{ttext}
|
|
\eng{Expansion of the pressure $p$ in a power series of the density $\rho$.}
|
|
\ger{Entwicklung desw Drucks $p$ in eine Potenzreihe der Dichte $\rho$.}
|
|
\end{ttext}
|
|
|
|
\begin{formula}{series}
|
|
\desc{Virial expansion}{The 2\ts{nd} and 3\ts{d} virial coefficient are tabelated for many substances}{$B$ and $C$ 2\ts{nd} and 3\ts{d} virial coefficient, $\rho = \frac{N}{V}$}
|
|
\desc[german]{Virialentwicklung}{Der zweite und dritte Virialkoeffizient ist für viele Substanzen tabelliert}{$B$ und $C$ 2. und 3. Virialkoeffizient, $\rho = \frac{N}{V}$}
|
|
\eq{p = \kB T \rho\,[1 + B(T) \rho + C(T) \rho^2 + \dots]}
|
|
\end{formula}
|
|
|
|
\begin{formula}{mayer_function}
|
|
\desc{Mayer function}{}{$V(i,j)$ pair potential}
|
|
\desc[german]{Mayer-Funktion}{}{$V(i,j)$ Paarpotential}
|
|
\eq{f(\vec{r}) = \e^{-\beta V(i,j)} - 1}
|
|
\end{formula}
|
|
|
|
\begin{formula}{second_coefficient}
|
|
\desc{Second virial coefficient}{Depends on pair potential between two molecules}{}
|
|
\desc[german]{Zweiter Virialkoeffizient}{Hängt vom Paarpotential zweier Moleküle ab}{}
|
|
\eq{B = -\frac{1}{2} \int_V \d^3 \vec{r} f(\vec{r})}
|
|
% b - \frac{a}{\kB T}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{lennard_jones}
|
|
\absLabel
|
|
\desc{Lennard-Jones potential}{Potential between two molecules. Attractive for $r > \sigma$, repulsive for $r < \sigma$.\\ In condensed matter: Attraction due to Landau Dispersion \TODO{verify} and repulsion due to Pauli exclusion principle.}{}
|
|
\desc[german]{Lennard-Jones-Potential}{Potential zwischen zwei Molekülen. Attraktiv für $r > \sigma$, repulsiv für $r < \sigma$.\\ In Festkörpern: Anziehung durch Landau-Dispersion und Abstoßung durch Pauli-Prinzip.}{}
|
|
\fig{img/potential_lennard_jones.pdf}
|
|
\eq{V(r) = 4\epsilon \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]}
|
|
\end{formula}
|
|
|
|
\Subsection[
|
|
\eng{Van der Waals equation}
|
|
\ger{Van der Waals Gleichung}
|
|
]{vdw}
|
|
\begin{ttext}
|
|
\eng{Assumes a hard-core potential with a weak attraction.}
|
|
\ger{Annahme eines Harte-Kugeln Potentials mit einer schwachen Anziehung}
|
|
\end{ttext}
|
|
\begin{formula}{partition_sum}
|
|
\desc{Partition sum}{}{$a$ internal pressure}
|
|
\desc[german]{Zustandssumme}{}{$a$ Kohäsionsdruck}
|
|
\eq{Z_N = \frac{(V-V_0)^N}{\lambda^{3N}N!} \e^{\frac{\beta N^2 a}{V}}}
|
|
\end{formula}
|
|
\begin{formula}{equation}
|
|
\desc{Van der Waals equation}{}{$b$ co-volume?}
|
|
\desc[german]{Van der Waals-Gleichung}{}{$b$ Kovolumen}
|
|
\eq{p = \frac{N \kB T}{V-b} - \frac{N^2 a}{V^2}}
|
|
\end{formula}
|
|
\TODO{sometimes N is included in a, b}
|
|
|
|
|
|
\Section[
|
|
\eng{Ideal quantum gas}
|
|
\ger{Ideales Quantengas}
|
|
]{id_qgas}
|
|
\def\bosfer{$\pm$: {$\text{bos} \atop \text{fer}$}}
|
|
|
|
\begin{formula}{fugacity}
|
|
\desc{Fugacity}{}{}
|
|
\desc[german]{Fugazität}{}{}
|
|
\eq{z = \e^{\mu\beta} = \e^\frac{\mu}{\kB T}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{occupation}
|
|
\desc{Occupation number}{}{$r$ states}
|
|
\desc[german]{Besetzungszahl}{}{$r$ Zustände}
|
|
\eq{\sum_{r} n_r = N}
|
|
\end{formula}
|
|
\begin{formula}{undiff_particles}
|
|
\desc{Undifferentiable particles}{}{$p_i$ state}
|
|
\desc[german]{Ununterscheidbare Teilchen}{}{$p_i$ Zustand}
|
|
\eq{\ket{p_1,p_2,\dots,p_N} = \ket{p_1}\ket{p_2}\dots \ket{p_N}}
|
|
\end{formula}
|
|
\begin{formula}{parity}
|
|
\desc{Applying the parity operator}{yields a \textit{symmetric} (Bosons) and a \textit{antisymmetic} (Fermions) solution}{$\hat{P}_{12}$ parity operator swaps $1$ and $2$, \bosfer}
|
|
\desc[german]{Anwenden des Paritätsoperators}{gibt eine \textit{symmetrische} (Bosonen) und eine \textit{antisymmetrische} (Fermionen) Lösung}{$\hat{O}_{12}$ Paritätsoperator tauscht $1$ und $2$, \bosfer}
|
|
\eq{\hat{P}_{12} \psi(p_i(\vec{r}_1),\,p_j(\vec{r}_2)) = \pm \psi(p_i(\vec{r}_1),\,p_j(\vec{r}_2))}
|
|
\end{formula}
|
|
|
|
\begin{formula}{spin_degeneracy_factor}
|
|
\desc{Spin degeneracy factor}{}{$s$ spin}
|
|
\desc[german]{Spinentartungsfaktor}{}{$s$ Spin}
|
|
\eq{g_s = 2s+1}
|
|
\end{formula}
|
|
\begin{formula}{dos}
|
|
\desc{Density of states}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
|
|
\desc[german]{Zustandsdichte}{}{$g_s$ \fqEqRef{td:id_qgas:spin_degeneracy_factor}}
|
|
\eq{g(\epsilon) = g_s \frac{V}{4\pi} \left(\frac{2m}{\hbar^2}\right)^\frac{3}{2} \sqrt{\epsilon}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{occupation_number_per_e}
|
|
\desc{Occupation number per energy}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
|
|
\desc[german]{Besetzungszahl pro Energie}{}{\fqEqRef{td:id_qgas:dos}, \bosfer}
|
|
\eq{n(\epsilon)\, \d\epsilon &= \frac{g(\epsilon)}{\e^{\beta(\epsilon - \mu)} \mp 1}\,\d\epsilon}
|
|
\end{formula}
|
|
|
|
\begin{formula}{occupation_number}
|
|
\desc{Occupation number}{}{\bosfer}
|
|
\desc[german]{Besetzungszahl}{}{\bosfer}
|
|
\fig{img/td_id_qgas_distributions.pdf}
|
|
\eq{
|
|
\braket{n(\epsilon)} &= \frac{1}{\e^{\beta(\epsilon - \mu)} \mp 1} \\
|
|
\shortintertext{\GT{for} $\epsilon - \mu \gg \kB T$}
|
|
&= \frac{1}{\e^{\beta(\epsilon - \mu)}}
|
|
}
|
|
|
|
\end{formula}
|
|
|
|
|
|
\begin{formula}{particle_number}
|
|
\desc{Number of particles}{}{}
|
|
\desc[german]{Teilchenzahl}{}{}
|
|
\eq{\braket{N} = \int_0^\infty n(\epsilon) \d\epsilon}
|
|
\end{formula}
|
|
|
|
\begin{formula}{energy}
|
|
\desc{Energy}{Equal to the classical ideal gas}{}
|
|
\desc[german]{Energie}{Gleich wie beim klassischen idealen Gas}{}
|
|
\eq{\braket{E} = \int_0^\infty \epsilon n(\epsilon)\,\d\epsilon = \frac{3}{2} pV}
|
|
\end{formula}
|
|
|
|
|
|
\begin{formula}{equation_of_state}
|
|
\desc{Equation of state}{Bosons: decreased pressure, they like to cluster\\Fermions: increased pressure because of the Pauli principle}{\bosfer, $v = \frac{V}{N}$ specific volume}
|
|
\desc[german]{Zustandsgleichung}{Bosonen: verringerter Druck da sie clustern\\Fermionen: erhöhter Druck durch das Pauli-Prinzip}{\bosfer, $v = \frac{V}{N}$ spezifisches Volumen}
|
|
\eq{
|
|
pV &= \kB T \ln Z_g \\
|
|
\shortintertext{\GT{after} \GT{td:real_gas:virial}}
|
|
&= N \kB T \left[1 \mp \frac{\lambda^3}{2^{5/2} g v} + \Order{\left(\frac{\lambda^3}{v}\right)^2}\right]
|
|
}
|
|
\end{formula}
|
|
\begin{formula}{relevance}
|
|
\desc{Relevance of qm. corrections}{Corrections become relevant when the particle distance is in the order of the thermal wavelength}{}
|
|
\desc[german]{Relevanz der qm. Korrekturen}{Korrekturen werden relevant, wenn der Teilchenabstand in der Größenordnung der thermischen Wellenlänge ist}{}
|
|
\eq{\left(\frac{V}{N}\right)^\frac{1}{3} \sim \frac{\lambda}{g_s^\frac{1}{3}}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{generalized_zeta}
|
|
\desc{Generalized zeta function}{}{}
|
|
\desc[german]{Verallgemeinerte Zeta-Funktion}{}{}
|
|
\eq{\left. \begin{array}{l}g_\nu(z)\\f_\nu(z)\end{array}\right\} \coloneq \frac{1}{\Gamma(\nu)} \int_0^\infty \d x\, \frac{x^{\nu-1}}{\e^x z^{-1} \mp 1}}
|
|
\end{formula}
|
|
|
|
\Subsection[
|
|
\eng{Bosons}
|
|
\ger{Bosonen}
|
|
]{bos}
|
|
\begin{formula}{partition-sum}
|
|
\desc{Partition sum}{}{$p \in\N_0$}
|
|
\desc[german]{Zustandssumme}{}{$p \in\N_0$}
|
|
\eq{Z_\text{g} = \prod_{p} \frac{1}{1-\e^{-\beta(\epsilon_p - \mu)}}}
|
|
\end{formula}
|
|
\begin{formula}{occupation}
|
|
\desc{Occupation number}{Bose-Einstein distribution}{}
|
|
\desc[german]{Besetzungszahl}{Bose-Einstein Verteilung}{}
|
|
\eq{\braket{n_p} = \frac{1}{\e^{\beta(\epsilon-\mu)}-1}}
|
|
\end{formula}
|
|
|
|
|
|
\Subsection[
|
|
\eng{Fermions}
|
|
\ger{Fermionen}
|
|
]{fer}
|
|
\begin{formula}{partition_sum}
|
|
\desc{Partition sum}{}{$p = 0,\,1$}
|
|
\desc[german]{Zustandssumme}{}{$p = 0,\,1$}
|
|
\eq{Z_\text{g} = \prod_{p} \left(1+\e^{-\beta(\epsilon_p - \mu)}\right)}
|
|
\end{formula}
|
|
\begin{formula}{occupation}
|
|
\desc{Occupation number}{Fermi-Dirac distribution. At $T=0$ \textit{Fermi edge} at $\epsilon=\mu$}{}
|
|
\desc[german]{Besetzungszahl}{Fermi-Dirac Verteilung}{Bei $T=0$ \textit{Fermi-Kante} bei $\epsilon=\mu$}
|
|
\fig{img/td_fermi_occupation.pdf}
|
|
\eq{\braket{n_p} = \frac{1}{\e^{\beta(\epsilon-\mu)}+1}}
|
|
\end{formula}
|
|
|
|
\begin{formula}{slater_determinant}
|
|
\desc{Slater determinant}{}{}
|
|
\desc[german]{Slater-Determinante}{}{}
|
|
\eq{
|
|
\psi(\vecr_1,\vecr_2,\dots,\vecr_N) = \frac{1}{\sqrt{N!}}
|
|
\begin{vmatrix}
|
|
p_1(\vecr_1) & p_2(\vecr_1) & \dots & p_N(\vecr_1) \\
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p_1(\vecr_2) & p_2(\vecr_2) & \dots & p_N(\vecr_2) \\
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\vdots & \vdots & \ddots & \vdots \\
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p_1(\vecr_N) & p_2(\vecr_N) & \dots & p_N(\vecr_N) \\
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\end{vmatrix}
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}
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\end{formula}
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\begin{formula}{fermi_energy}
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\desc{Fermi energy}{}{}
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\desc[german]{Fermienergie}{}{}
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\eq{\epsilon_\text{F} \coloneq \mu(T = 0)}
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\end{formula}
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\begin{formula}{fermi_temperature}
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\desc{Fermi temperature}{}{}
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\desc[german]{Fermi Temperatur}{}{}
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\eq{T_\text{F} \coloneq \frac{\epsilon_\text{F}}{\kB}}
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\end{formula}
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\begin{formula}{fermi_impulse}
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\desc{Fermi impulse}{Radius of the \textit{Fermi sphere} in impulse space. States with $p_\text{F}$ are in the \textit{Fermi surface}}{}
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\desc[german]{Fermi-Impuls}{Radius der \textit{Fermi-Kugel} im Impulsraum. Zustände mit $P_\text{F}$ sind auf der \textit{Fermi-Fläche}}{}
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\eq{p_\text{F} = \hbar k_\text{F} = (2mE_\text{F})^\frac{1}{2}}
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\end{formula}
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|
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\begin{formula}{specific_density}
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\desc{Specific density}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ degeneracy factor, $z$ \fqEqRef{td:id_qgas:fugacity}}
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\desc[german]{Spezifische Dichte}{}{$f$ \fqEqRef{td:id_qgas:generalized_zeta}, $g$ Entartungsfaktor, $z$ \fqEqRef{td:id_qgas:fugacity}}
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\eq{v = \frac{N}{V} = \frac{g}{\lambda^3}f_{3/2}(z)}
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\end{formula}
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|
|
|
\Subsubsection[
|
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\eng{Strong degeneracy}
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\ger{Starke Entartung}
|
|
]{degenerate}
|
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\eng[low_temps]{for low temperatures $T \ll T_\text{F}$}
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\ger[low_temps]{für geringe Temperaturen $T\ll T_\text{F}$}
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|
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\begin{formula}{sommerfeld}
|
|
\desc{Sommerfeld expansion}{\gt{low_temps}}{}
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\desc[german]{Sommerfeld-Entwicklung}{\gt{low_temps}}{}
|
|
\eq{f_\nu(z) = \frac{(\ln z)^\nu}{\Gamma(\nu+1)} \left(1+\frac{\pi^6}{6}\frac{\nu(\nu-1)}{(\ln z)^2} + \dots\right)}
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|
\end{formula}
|
|
|
|
\begin{formula}{energy_density}
|
|
\desc{Energy density}{}{}
|
|
\desc[german]{Energiedichte}{}{}
|
|
\eq{
|
|
\frac{E}{V} &= \frac{3}{2}\frac{g}{\lambda^3} \kB T f_{5/2}(z) \\
|
|
\shortintertext{\GT{td:id_qgas:fer:degenerate:sommerfeld}}
|
|
&\approx \frac{3}{5} \frac{N}{V} E_\text{F} \left(1+\frac{5\pi^2}{12}\left(\frac{\kB T}{E_\text{F}}\right)^2 \right)
|
|
}
|
|
\end{formula}
|
|
|
|
\begin{formula}{heat_cap}
|
|
\desc{Heat capacity}{\gt{low_temps}}{differs from \fqEqRef{td:TODO:petit_dulong}}
|
|
\desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \fqEqRef{td:TODO:petit_dulong}}
|
|
\fig{img/td_fermi_heat_capacity.pdf}
|
|
\eq{C_V = \pdv{E}{T}_V = N\kB \frac{\pi}{2} \left(\frac{T}{T_\text{F}}\right)}
|
|
\end{formula}
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|
|
|
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|
\TODO{Entartung und Sommerfeld}
|
|
\TODO{DULONG-PETIT Gesetz}
|
|
|