\Part{stat}
    \desc{Statistichal Mechanics}{}{}
    \desc[german]{Statistische Mechanik}{}{}


\begin{ttext}
    \eng{
        \textbf{Extensive quantities:} Additive for subsystems (system size dependent): $S(\lambda E, \lambda V, \lambda N)  = \lambda S(E, V, N)$\\
        \textbf{Intensive quantities:} Independent of system size, ratio of two extensive quantities
    }
    \ger{
        \textbf{Extensive Größen:} Additiv für Subsysteme (Systemgrößenabhänig): $S(\lambda E, \lambda V, \lambda N)  = \lambda S(E, V, N)$\\
        \textbf{Intensive Größen:} Unabhängig der Systemgröße, Verhältnis zweier extensiver Größen
    }
\end{ttext}

\begin{formula}{liouville}
    \desc{Liouville equation}{}{$\{\}$ poisson bracket}
    \desc[german]{Liouville-Gleichung}{}{$\{\}$ Poisson-Klammer}
    \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\}}
\end{formula}

\Section{entropy}
    \desc{Entropy}{}{}
    \desc[german]{Entropie}{}{}

    \begin{formula}{properties}
        \desc{Positive-definite and additive}{}{}
        \desc[german]{Positiv Definit und Additiv}{}{}
        \eq{
            S &\ge 0 \\
            S(E_1, E_2) &= S_1 + S_2
        }
    \end{formula}

    \begin{formula}{von_neumann}
        \desc{Von-Neumann}{}{$\rho$ density matrix}
        \desc[german]{Von-Neumann}{}{$\rho$ Dichtematrix}
        \eq{S = - \kB \braket{\log \rho} = - \kB \tr(\rho \log\rho)}
    \end{formula}

    \begin{formula}{gibbs}
        \desc{Gibbs}{}{$p_n$ probability for micro state $n$}
        \desc[german]{Gibbs}{}{$p_n$ Wahrscheinlichkeit für Mikrozustand $n$}
        \eq{S = - \kB \sum_n p_n \log p_n}
    \end{formula}

    \begin{formula}{boltzmann}
        \desc{Boltzmann}{}{$\Omega$ \#micro states}
        \desc[german]{Boltzmann}{}{$\Omega$ \#Mikrozustände}
        \eq{S = \kB \log\Omega}
    \end{formula}

    \begin{formula}{temp}
        \desc{Temperature}{}{}
        \desc[german]{Temperatur}{}{}
        \eq{\frac{1}{T} \coloneq \pdv{S}{E}_V}
    \end{formula}

    \begin{formula}{pressure}
        \desc{Pressure}{}{}
        \desc[german]{Druck}{}{}
        \eq{p = T \pdv{S}{V}_E}
    \end{formula}

\Part{td}
    \desc{Thermodynamics}{}{}
    \desc[german]{Thermodynamik}{}{}

\begin{formula}{therm_wavelength}
    \desc{Thermal wavelength}{}{}
    \desc[german]{Thermische Wellenlänge}{}{}
    \eq{\lambda = \frac{\hbar}{\sqrt{2\pi m \kB T}}}
\end{formula}

\Section{process}
    \desc{Processes}{}{}
    \desc[german]{Prozesse}{}{}
    \begin{ttext}
        \eng{
            \begin{itemize}
                \item \textbf{isobaric}: constant pressure $p = \const$
                \item \textbf{isochoric}: constant volume $V = \const$
                \item \textbf{isothermal}: constant temperature $T = \const$
                \item \textbf{isentropic}: constant entropy $S = \const$
                \item \textbf{isenthalpic}: constant entalphy $H = \const$
                \item \textbf{adiabatic}: no heat transfer $\Delta Q=0$
                \item \textbf{quasistatic}: happens so slow, the system always stays in td. equilibrium
                \item \textbf{reversivle}: reversible processes are always quasistatic and no entropie is created $\Delta S = 0$
            \end{itemize}
        }
        \ger{
            \begin{itemize}
                \item \textbf{isobar}: konstanter Druck $p = \const$
                \item \textbf{isochor}: konstantes Volumen $V = \const$
                \item \textbf{isotherm}: konstante Temperatur $T = \const$
                \item \textbf{isentrop}: konstante Entropie $S = \const$
                \item \textbf{isenthalp}: konstante Entalphie $H = \const$
                \item \textbf{adiabatisch}: kein Wärmeübertrag $\Delta Q=0$
                \item \textbf{quasistatsch}: läuft so langsam ab, dass das System durchgehend im t.d Equilibrium bleibt
                \item \textbf{reversibel}: reversible Prozesse sind immer quasistatisch und es wird keine Entropie erzeugt $Delta S = 0$
            \end{itemize}
        }
    \end{ttext}

    \Subsection{gay}
        \desc{Irreversible gas expansion (Gay-Lussac experiment)}{}{}
        \desc[german]{Irreversible Gasexpansion (Gay-Lussac-Versuch)}{}{}

        \begin{bigformula}{experiment}
            \desc{Gay-Lussac experiment}{}{}
            \desc[german]{Gay-Lussac-Versuch}{}{}
            \begin{minipage}{0.6\textwidth}
                \vfill
                \begin{ttext}
                    \eng{
                        A classical gas in a system with volume $V_1$ is separated from another system with volume $V_2$. 
                        In the Gay-Lussac experiment, the separation is removed and the gas flows into $V_2$.
                    }
                    \ger{
                        Ein klassisches Gas in einem System mit Volumen $V_1$ ist getrennt von einem zweiten System mit Volumen $V_2$.
                        Beim Gay-Lussac Versuch wird die Trennwand entfern und das Gas fließt in das Volumen $V_2$.
                    }
                \end{ttext}
                \vfill
            \end{minipage}
            \hfill
            \begin{minipage}{0.3\textwidth}
                \begin{figure}[H]
                    \centering
                    \includegraphics[width=\textwidth]{img/td_gay_lussac.pdf}
                \end{figure}
            \end{minipage}
        \end{bigformula}


        \begin{formula}{entropy}
            \desc{Entropy change}{}{}
            \desc[german]{Entropieänderung}{}{}
            \eq{\Delta S=N\kB \ln \left(\frac{V_1 + V_2}{V_1}\right) > 0}
        \end{formula}

        \TODO{Reversible}

        \TODO{Quasistatischer T-Ausgleich}

        \TODO{Joule-Thompson Prozess}


    \Section{phases}
        \desc{Phase transitions}{}{}
        \desc[german]{Phasenübergänge}{}{}

        \begin{ttext}
            \eng{
                A phase transition is a discontinuity in the free energy $F$ or Gibbs energy $G$ or in one of their derivatives.
                The degree of the phase transition is the degree of the derivative which exhibits the discontinuity.
            }
            \ger{
                Ein Phasenübergang ist eine Unstetigkeit in the Freien Energie $F$ oder in der Gibbs-Energie $G$ oder in ihrer Ableitungen.
                Die Ordnung des Phasenübergangs ist die Ordnung der Ableitung, in welcher die Unstetigkeit auftritt.
            }
        \end{ttext}

        \begin{formula}{latent_heat}
            \desc{Latent heat}{Heat required to bring substance from phase 1 to phase 2}{$\Delta S$ entropy change of the phase transition}
            \desc[german]{Latente Wärme}{Für den Phasenübergang von Phase 1 nach Phase 2 benötigte Wärme}{$\Delta S$ Entropieänderung des Phasenübergangs}
            \eq{Q_\text{L} = T \Delta S}
        \end{formula}
        \begin{formula}{clausis_clapeyron}
            \desc{Clausius-Clapyeron equation}{Slope of the coexistence curve}{$\Delta V$ Volume change of the phase transition}
            \desc[german]{Clausius-Clapeyron Gleichung}{Steigung der Phasengrenzlinie}{$\Delta V$ Volumenänderung des Phasenübergangs}
            \eq{\odv{p}{T} = \frac{Q_\text{L}}{T\Delta V}}
        \end{formula}
        \begin{formula}{coexistence}
            \desc{Phase transition}{At the coexistence curve}{}
            \desc[german]{Phasenübergang}{Im Koexistenzbereich}{}
            \eq{G_1 = G_2 \\ \shortintertext{\GT{and_therefore}} \mu_1 = \mu_2}
        \end{formula}

        \begin{formula}{gibbs_phase_rule}
            \desc{Gibbs rule / Phase rule}{}{$c$ \#components, $f$ \#degrees of freedom, $p$ \#phases}
            \desc[german]{Gibbsche Phasenregel}{}{$c$ \#Komponenten, $f$ \#Freiheitsgrade, $p$ \#Phasen}
            \eq{f = c - p + 2}
        \end{formula}

        \Subsubsection{osmosis}
            \desc{Osmosis}{}{}
            \desc[german]{Osmose}{}{}
            \begin{ttext}
                \eng{
                    Osmosis is the spontaneous net movement or diffusion of solvent molecules 
                    through a selectively-permeable membrane, which allows through the solvent molecules, but not the solute molecules.
                    The direction of the diffusion is from a region of high water potential 
                    (region of lower solute concentration) to a region of low water potential 
                    (region of higher solute concentration), in the direction that tends to equalize the solute concentrations on the two sides.
                }
                \ger{
                    Osmosis ist die spontane Passage oder Diffusion Lösungsmittelmolekülen durch eine semi-permeable Membran
                    die für das Lösungsmittel, jedoch nicht die darin gelösten Stoffe durchlässig ist.
                    Die Richtung der Diffusion ist vom Gebiet mit hohem chemischen Potential (niedrigere Konzentration des gelösten Stoffes)
                    in das mit niedrigem chemischem Potential (höherere Konzentraion des gelösten Stoffes), sodass die Konzentration des gelösten Stoffes ausgeglichen wird.
                }
            \end{ttext}
        \begin{formula}{osmosis}
            \desc{Osmotic pressure}{}{$N_c$ \#dissolved particles}
            \desc[german]{Osmotischer Druck / Van-\'t-hoffsches Gesetz}{}{$N_c$ \#gelöster Teilchen}
            \eq{p_\text{osm} = \kB T \frac{N_c}{V}}
        \end{formula}   


    \Subsection{props}
        \desc{Material properties}{}{}
        \desc[german]{Materialeigenschaften}{}{}
        \begin{formula}{heat_capacity}
            \desc{Heat capacity}{}{\QtyRef{heat}}
            \desc[german]{Wärmekapazität}{}{}
            \quantity{c}{\joule\per\kelvin}{}
            \eq{c = \frac{Q}{\Delta T}}
        \end{formula}
        \begin{formula}{heat_capacity_V}
            \desc{Isochoric heat capacity}{}{\QtyRef{heat}, \QtyRef{internal_energy} \QtyRef{temperature}, \QtyRef{volume}}
            \desc[german]{Isochore Wärmekapazität}{}{}
            \eq{c_v = \pdv{Q}{T}_V = \pdv{U}{T}_V}
        \end{formula}

        \begin{formula}{heat_capacity_p}
            \desc{Isobaric heat capacity}{}{\QtyRef{heat}, \QtyRef{enthalpy} \QtyRef{temperature}, \QtyRef{pressure}}
            \desc[german]{Isobare Wärmekapazität}{}{}
            \eq{c_p = \pdv{Q}{T}_p = \pdv{H}{T}_p}
        \end{formula}

        \begin{formula}{bulk_modules}
            \desc{Bulk modules}{}{\QtyRef{pressure}, $V$ initial \qtyRef{volume}}
            \desc[german]{Kompressionsmodul}{}{\QtyRef{pressure}, $V$ Anfangsvolumen}
            \eq{K = -V \odv{p}{V} }
        \end{formula}

        \begin{formula}{compressibility}
            \desc{Compressibility}{}{}
            \desc[german]{Kompressibilität}{}{}
            \eq{\kappa = -\frac{1}{V} \pdv{V}{p} }
        \end{formula}

        \begin{formula}{compressibility_T}
            \desc{Isothermal compressibility}{}{}
            \desc[german]{Isotherme Kompressibilität}{}{}
            \eq{\kappa_T = -\frac{1}{V} \pdv{V}{p}_{T} = \frac{1}{K}}
        \end{formula}

        \begin{formula}{compressibility_S}
            \desc{Adiabatic compressibility}{}{}
            \desc[german]{Adiabatische Kompressibilität}{}{}
            \eq{\kappa_S = -\frac{1}{V} \pdv{V}{p}_{S}}
        \end{formula}

        \begin{formula}{therm_expansion}
            \desc{Thermal expansion coefficient}{}{}
            \desc[german]{Thermaler Ausdehnungskoeffizient}{}{}
            \eq{\alpha = \frac{1}{V} \pdv{V}{T}_{p,N}}
        \end{formula}



\Section{laws}
    \desc{Laws of thermodynamics}{}{}
    \desc[german]{Hauptsätze der Thermodynamik}{}{}

    \Subsection{law0}
        \desc{Zeroeth law}{}{}
        \desc[german]{Nullter Hauptsatz}{}{}
        \begin{ttext}
            \eng{If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.}
            \ger{Wenn sich zwei Siesteme jeweils im thermischen Gleichgewicht mit einem dritten befinden, befinden sie sich auch untereinander im thermischen Gleichgewicht.}
        \end{ttext}

        \Eng[teq]{th. eq.}
        \Ger[teq]{th. GGW.}
        \def\ggwarrow{\overset{\GT{teq}}{\leftrightarrow}}
        \begin{equation}
            \label{eq:\fqname}
            A \ggwarrow C \quad\wedge\quad B \ggwarrow C \quad\Rightarrow\quad A \ggwarrow B
        \end{equation}

    \Subsection{law1}
        \desc{First law}{}{}
        \desc[german]{Erster Hauptsatz}{}{}
        \begin{ttext}
            \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.}
            \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 $W$.}
        \end{ttext}

        \begin{formula}{internal_energy}
            \desc{Internal energy change}{}{}
            \desc[german]{Änderung der inneren Energie}{}{}
            \eq{
                \Delta U &= \delta Q - \delta W \\
                \d U &= T \d S - p \d V
            }
        \end{formula}
        

    \Subsection{law2}
        \desc{Second law}{}{}
        \desc[german]{Zweiter Hauptsatz}{}{}
        \begin{ttext}
            \eng{
                \textbf{Clausius}: Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.\\
                \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.
            }
            \ger{
                \textbf{Clausius}: Es gibt keine Zustansänderung, deren einziges Ergebnis die Übertragung von Wärme von einem Körper niederer Temperatur auf einen Körper höherer Temperatur ist.\\
                \textbf{Kelvin}: Es ist unmöglich, eine periodisch arbeitende Maschine zu konstruieren, die weiter nichts bewirkt als Hebung einer Last und Abkühlung eines Wärmereservoirs.
            }
        \end{ttext}
    \Subsection{law3}
        \desc{Third law}{}{}
        \desc[german]{Dritter Hauptsatz}{}{}
        \begin{ttext}
            \eng{It is impussible to cool a system to absolute zero.}
            \ger{Es ist unmöglich, ein System bis zum absoluten Nullpunkt abzukühlen.}
        \end{ttext}

        \begin{formula}{3d_law}
            \desc{Entropy density}{}{$s = \frac{S}{N}$}
            \desc[german]{Entropiedichte}{}{$s = \frac{S}{N}$}
            \eq{
                \lim_{T\to 0} s(T) &= 0 \\
                \shortintertext{\GT{and_therefore_also}}
                \lim_{T\to 0} c_V &= 0
            }
        \end{formula}   

\Section{ensembles}
    \desc{Ensembles}{}{}
    \desc[german]{Ensembles}{}{}
    \Eng[const_variables]{Constant variables}
    \Ger[const_variables]{Konstante Variablen}

    \begin{bigformula}{nve} \absLabel[mc_ensemble]
        \desc{Microcanonical ensemble}{}{}
        \desc[german]{Mikrokanonisches Ensemble}{}{}
        \begin{minipagetable}{nve}
            \entry{const_variables} {$E$, $V,$ $N$           }
            \entry{partition_sum}   {$\Omega = \sum_n 1$     }
            \entry{probability}     {$p_n = \frac{1}{\Omega}$}
            \entry{td_pot}          {$S = \kB\ln\Omega$      }
            \rentry{\qtyRef{pressure}}        {$p = T \pdv{S}{V}_{E,N}$}
            \rentry{\qtyRef{entropy}}         {$S = \kB = \ln\Omega$   }
        \end{minipagetable}
    \end{bigformula}

    \begin{bigformula}{nvt} \absLabel[c_ensemble]
        \desc{Canonical ensemble}{}{}
        \desc[german]{Kanonisches Ensemble}{}{}
        \begin{minipagetable}{nvt}
            \entry{const_variables} {$T$, $V$, $N$                    }
            \entry{partition_sum}   {$Z = \sum_n \e^{-\beta E_n}$     }
            \entry{probability}     {$p_n = \frac{\e^{-\beta E_n}}{Z}$}
            \entry{td_pot}          {$F = - \kB T \ln Z$              }
            \rentry{\qtyRef{pressure}}        {$p = -\pdv{F}{V}_{T,N}$          }
            \rentry{\qtyRef{entropy}}         {$S = -\pdv{F}{T}_{V,N}$          }
        \end{minipagetable}
    \end{bigformula}

    \begin{bigformula}{mvt} \absLabel[gc_ensemble]
        \desc{Grand canonical ensemble}{}{}
        \desc[german]{Grosskanonisches Ensemble}{}{}
        \begin{minipagetable}{mvt}
            \entry{const_variables} {$T$, $V$, $\mu$                                      }
            \entry{partition_sum}   {$Z_\text{g} = \sum_{n} \e^{-\beta(E_n - \mu N_n)}$   }
            \entry{probability}     {$p_n = \frac{\e^{-\beta (E_n - \mu N_n)}}{Z_\text{g}}$}
            \entry{td_pot}          {$ \Phi = - \kB T \ln Z$                              }
            \rentry{\qtyRef{pressure}}        {$p = -\pdv{\Phi}{V}_{T,\mu} = -\frac{\Phi}{V}$       }
            \rentry{\qtyRef{entropy}}         {$S = -\pdv{\Phi}{T}_{V,\mu}$                         }
        \end{minipagetable}
    \end{bigformula}

    \begin{bigformula}{npt}
        \desc{Isobaric-isothermal}{Gibbs ensemble}{}
        % \desc[german]{Kanonisches Ensemble}{}{}
        \begin{minipagetable}{npt}
            \entry{const_variables}   {$N$, $p$, $T$}
            \entry{partition_sum}     {$Z = \sum_{n}\e^{-\beta(E_n+pV)}$}
            \entry{probability}       {$p_n = \frac{\e^{-\beta(E_n + pV_n-TS)}}{Z}$}
            \entry{td_pot}            {$G = -\kB \ln Z$}
            \rentry{\qtyRef{volume}}  {$V = \pdv{G}{V}_{T,N} $}
            \rentry{\qtyRef{entropy}} {$S = - \pdv{G}{T}_{p,N}$}
        \end{minipagetable}
    \end{bigformula}

    \begin{bigformula}{nph}
        \desc{Isoenthalpic-isobaric ensemble}{Enthalpy ensemble}{}
        % \desc[german]{Kanonisches Ensemble}{}{}
        \begin{minipagetable}{nph}
            \entry{const_variables}    {$N$, $p$, $H$}
            % \entry{partition_sum}      {$ $}
            % \entry{probability}        {$ $}
            \entry{td_pot}             {$H$}
            % \rentry{\qtyRef{pressure}} {$ $}
            % \rentry{\qtyRef{entropy}}  {$ $}
        \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{pots}
        \desc{Potentials}{}{}
        \desc[german]{Potentiale}{}{}
        \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}
            \hiddenQuantity{U}{\joule}{s}
        \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}
            \hiddenQuantity{F}{\joule}{s}
        \end{formula}
        \begin{formula}{enthalpy}
            \desc{Enthalpy}{}{}
            \desc[german]{Enthalpie}{früher "Wärmeinhalt"}{}
            \eq{\d H(S,p,N) = T\d S +V\d p + \mu\d N}
            \hiddenQuantity{H}{\joule}{s}
        \end{formula}
        \begin{formula}{free_enthalpy}
            \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}
            \hiddenQuantity{G}{\joule}{s}
        \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}
            \hiddenQuantity{\Phi}{\joule}{s}
        \end{formula}

        \TODO{Maxwell Relationen}
        \begin{formula}{td-square}
            \desc{Thermodynamic square}{}{}
            \desc[german]{Themodynamisches Quadrat}{Guggenheim Quadrat}{}
            \fsplit[0.3]{
                \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}
            }{
                \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{id_gas}
    \desc{Ideal gas}{}{}
    \desc[german]{Ideales 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{
            \begin{split}
                \Omega(E) &= \int_V\d^3q_1 \sdots \int_V\d^3q_N \int \d^3p_1 \sdots \int\d^3p_N \\ 
                            &\qquad\qquad \frac{1}{N!\,h^{3N}} \Theta\left(E - \sum_{i} \frac{\vec{p_i}^2}{2m}\right)
            \end{split}\\
            \begin{split}
                    &= \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{split}
        }
    \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{molecule_gas}
        \desc{Molecule gas}{}{}
        \desc[german]{Molekülgas}{}{}

        \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{real_gas}
    \desc{Real gas}{}{}
    \desc[german]{Reales Gas}{}{}

    \Subsection{virial}
        \desc{Virial expansion}{}{}
        \desc[german]{Virialentwicklung}{}{}
        \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{vdw}
        \desc{Van der Waals equation}{}{}
        \desc[german]{Van der Waals Gleichung}{}{}
        \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{id_qgas}
    \desc{Ideal quantum gas}{}{}
    \desc[german]{Ideales Quantengas}{}{}
    \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}{}{$g_s$ \fRef{td:id_qgas:spin_degeneracy_factor}}
        \desc[german]{Zustandsdichte}{}{$g_s$ \fRef{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}{}{\fRef{td:id_qgas:dos}, \bosfer}
        \desc[german]{Besetzungszahl pro Energie}{}{\fRef{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{bos}
        \desc{Bosons}{}{}
        \desc[german]{Bosonen}{}{}
        \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{fer}
        \desc{Fermions}{}{}
        \desc[german]{Fermionen}{}{}
        \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) \\
                    p_1(\vecr_2) & p_2(\vecr_2) & \dots & p_N(\vecr_2) \\
                    \vdots       & \vdots       & \ddots & \vdots \\
                    p_1(\vecr_N) & p_2(\vecr_N) & \dots & p_N(\vecr_N) \\
                \end{vmatrix}
            }
        \end{formula}

        \begin{formula}{fermi_energy}
            \desc{Fermi energy}{}{}
            \desc[german]{Fermienergie}{}{}
            \eq{\epsilon_\text{F} \coloneq \mu(T = 0)}
        \end{formula}
        \begin{formula}{fermi_temperature}
            \desc{Fermi temperature}{}{}
            \desc[german]{Fermi Temperatur}{}{}
            \eq{T_\text{F} \coloneq \frac{\epsilon_\text{F}}{\kB}}
        \end{formula}

        \begin{formula}{fermi_impulse}
            \desc{Fermi impulse}{Radius of the \textit{Fermi sphere} in impulse space. States with $p_\text{F}$ are in the \textit{Fermi surface}}{}
            \desc[german]{Fermi-Impuls}{Radius der \textit{Fermi-Kugel} im Impulsraum. Zustände mit $P_\text{F}$ sind auf der \textit{Fermi-Fläche}}{}
            \eq{p_\text{F} = \hbar k_\text{F} = (2mE_\text{F})^\frac{1}{2}}
        \end{formula}

        \begin{formula}{specific_density}
            \desc{Specific density}{}{$f$ \fRef{td:id_qgas:generalized_zeta}, $g$ degeneracy factor, $z$ \fRef{td:id_qgas:fugacity}}
            \desc[german]{Spezifische Dichte}{}{$f$ \fRef{td:id_qgas:generalized_zeta}, $g$ Entartungsfaktor, $z$ \fRef{td:id_qgas:fugacity}}
            \eq{v = \frac{N}{V} = \frac{g}{\lambda^3}f_{3/2}(z)}
        \end{formula}

        \Subsubsection{degenerate}
            \desc{Strong degeneracy}{}{}
            \desc[german]{Starke Entartung}{}{}
            \eng[low_temps]{for low temperatures $T \ll T_\text{F}$}
            \ger[low_temps]{für geringe Temperaturen $T\ll T_\text{F}$}

            \begin{formula}{sommerfeld}
                \desc{Sommerfeld expansion}{\gt{low_temps}}{}
                \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)}
            \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_capacity}
                \desc{Heat capacity}{\gt{low_temps}}{differs from \absRef{petit-dulong}}
                \desc[german]{Wärmecapacity}{\gt{low_temps}}{weicht ab vom \absRef{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}
