Maxwell Boltzmann Distribution

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maxwell-boltzmann-distribution.tex (36 lines)

\documentclass{standalone}

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

\def\kB{1.38e-23} % Boltzmann constant
\def\mu{1.66e-27} % unified atomic mass unit/Dalton (symbols: m_u or Da)

\def\maxwellboltzmann#1{4 * pi * (\mu / (2 * pi * \kB * #1))^(3/2) * x^2 * exp(-\mu * x^2 / (2 * \kB * #1))}

\begin{document}

\begin{tikzpicture}

  \begin{axis}[
      domain = 0:8000,
      xlabel = {$v$ [\si{\metre\per\second}]},
      ylabel = $P(v)$,
      smooth, thick,
      axis lines = left,
      every tick/.style = {thick},
    ]

    \addplot[color=red,samples=100]{\maxwellboltzmann{100}};

    \addplot[color=yellow]{\maxwellboltzmann{300}};

    \addplot[color=blue]{\maxwellboltzmann{1000}};

    \legend{\SI{100}{\kelvin}, \SI{300}{\kelvin}, \SI{1000}{\kelvin}}

  \end{axis}
\end{tikzpicture}

\end{document}

The Mexican hat potential exhibits spontaneous symmetry breaking (SSB), a process by which a physical system starting in a symmetric state spontaneously enters and remains in an asymmetric state. This usually applies to systems whose equations of motion obey a set of symmetries while the lowest-energy state(s) do(es) not. When the system assumes one of its ground states, its symmetry is broken even though the Lagrangian as a whole retains it. See https://janosh.github.io/tikz/higgs-potential for a very similar image.

In this image, the system starts out in the naive $O(N)$-invariant vacuum (blue dot) but quantum fluctuations quickly push it into the real vacuum (red dot) where $O(N)$ is broken down to $O(N-1)$.

Maxwell Boltzmann Distribution

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maxwell-boltzmann-distribution.tex (36 lines)