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Statistical Energy Distributions

physicsstatistical-mechanicsBose-EinsteinBoltzmannFermi-Diracpgfplotscetztikz

A plot comparing the identical particle distribution functions of Bose-Einstein, Boltzmann, and Fermi-Dirac statistics as a function of the reduced chemical potential β(ϵ−μ)\beta (\epsilon - \mu). This visualization highlights the differences between the three types of distribution functions, which are used to describe the behavior of particles in different statistical systems.


Statistical Energy Distributions

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  Code

  LaTeX

statistical-energy-distributions.tex (29 lines)

\documentclass{standalone}

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

\begin{document}
\begin{tikzpicture}
  \begin{axis}[
      xlabel = $\beta (\epsilon - \mu)$,
      ylabel = $\langle n\rangle$,
      ymin = 0,ymax = 1.8,
      smooth,thick,
      axis lines = center,
      every tick/.style = {thick},
      legend cell align=left,
      legend style={legend pos=north east,font=\tiny},
      width=10cm,height=5cm]

    \def\xmax{7}
    \addplot[color=blue,domain=0:\xmax]{1/(e^x - 1)};
    \addplot[color=orange,domain=-1:\xmax]{1/e^x};
    \addplot[color=red,domain = -\xmax:\xmax]{1/(e^x + 1)};

    \legend{Bose-Einstein,Boltzmann,Fermi-Dirac}

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

  Typst

statistical-energy-distributions.typ (64 lines)

#import "@preview/cetz:0.3.2": canvas, draw
#import "@preview/cetz-plot:0.1.1": plot

#set page(width: auto, height: auto, margin: 8pt)

// Distribution functions
#let bose-einstein(x) = 1 / (calc.exp(x) - 1)
#let boltzmann(x) = 1 / calc.exp(x)
#let fermi-dirac(x) = 1 / (calc.exp(x) + 1)

#canvas({
  draw.set-style(
    axes: (
      y: (mark: (end: "stealth", fill: black), label: (anchor: "north-west", offset: -0.2)),
      x: (mark: (end: "stealth", fill: black), label: (anchor: "south-east", offset: -0.2)),
    ),
  )

  plot.plot(
    size: (8, 5),
    x-label: $beta (epsilon - mu)$,
    y-label: $angle.l n angle.r$,
    x-min: -7,
    x-max: 7,
    y-min: 0,
    y-max: 1.8,
    x-tick-step: 2,
    y-tick-step: 0.5,
    axis-style: "school-book",
    x-grid: true,
    y-grid: true,
    legend: "inner-north-east",
    legend-style: (stroke: 0.5pt, scale: 80%),
    {
      // Bose-Einstein distribution
      plot.add(
        style: (stroke: blue + 1.5pt),
        domain: (0.1, 7), // Avoid x=0 since BE diverges there
        samples: 200,
        label: "Bose-Einstein",
        bose-einstein,
      )

      // Boltzmann distribution
      plot.add(
        style: (stroke: orange + 1.5pt),
        domain: (-1, 7),
        samples: 100,
        label: "Boltzmann",
        boltzmann,
      )

      // Fermi-Dirac distribution
      plot.add(
        style: (stroke: red + 1.5pt),
        domain: (-7, 7),
        samples: 100,
        label: "Fermi-Dirac",
        fermi-dirac,
      )
    },
  )
})