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Fermi-Dirac Distribution

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physicsstatistical-mechanicsFermi-Diracdistribution-functionstemperaturepgfplots

A plot illustrating the Fermi-Dirac distribution function for different temperatures relative to the chemical potential (ϵ/μ\epsilon/\mu). The plot demonstrates how the occupation number (n(ϵ)n(\epsilon)) changes as the temperature increases from zero to higher values. The visualization also highlights the effect of thermal fluctuations, which increase with higher temperatures.


Fermi-Dirac Distribution

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fermi-dirac-vs-temp.tex (36 lines)

\documentclass{standalone}

\usepackage{pgfplots,mathtools}
\pgfplotsset{compat=newest}

\def\xmax{2.3}\def\ymax{1.2}

\begin{document}
\begin{tikzpicture}
  \begin{axis}[
      xlabel=$\epsilon/\mu$,
      ylabel=$n(\epsilon)$,
      domain=0:\xmax,ymax=\ymax,
      ytick={0.5,1},
      smooth,thick,
      axis lines=center,
      every tick/.style={thick},
      legend cell align=left]

    % Graphs
    \def\chempot{1}
    \def\n#1{1/(e^(#1*(x - \chempot)) + 1)}
    \addplot[color=red]{\n{5}};
    \addplot[color=orange,samples=100]{\n{25}};
    \addplot[const plot,color=blue] coordinates {(0,1) (\chempot,0) (\xmax,0)};

    \legend{$k_\text{B} T = \frac{1}{5} \mu$,$k_\text{B} T = \frac{1}{25} \mu$,$T = 0$}

    % Thermal fluctuations
    \draw [thin,dashed] (\chempot-0.25,1.1) -- (\chempot-0.25,0) -| (\chempot+0.25,1.1);
    \draw [thin,<->,shorten >=1,shorten <=1] (\chempot-0.25,1.05) -- (\chempot+0.25,1.05) node[midway,above] {$\propto 1/\beta$};

  \end{axis}
\end{tikzpicture}
\end{document}