Fixing the alignment of table with equations and images

Question

My desired features for the table alignment are:

• The cell sizes should be auto-adjustable according to the contents.
• Every table cell content should be vertically and horizontally center-aligned.
• The first column's contents (except its header) should be vertically center-aligned but horizontally left-aligned.
• A fair amount of padding should appear between images and table cell borders.
• The text-color and background-color of the header row should be different.

\documentclass{article}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage[export]{adjustbox}
\tabcolsep10pt\fboxrule0pt\fboxsep\tabcolsep
\usepackage{geometry}
\usepackage{array}
\renewcommand{\arraystretch}{1.2}
\usepackage{tabularx}
\newcolumntype{L}[1]{>{\flushleft\arraybackslash}m{#1}}
\renewcommand{\tabularxcolumn}[1]{>{\centering\arraybackslash}m{#1}}

\begin{document}
\begin{center}
\begin{tabularx}{\linewidth}{|L{.15\linewidth}|X|X|X|} \hline
          \multicolumn{1}{|c|}{Property} & Microcanonical & Canonical & Grand-canonical  \\ \hline
          Contact with environment 
          & Isolated system, so the energy and number of particles are constant.
          & System are in thermal contact, such that it can exchange energy but no particles. 
          &  System can exchange both energy and particles. \\ \hline
          Probability density, $\rho(E)$
          &  {$$\begin{cases} \text{C,} & E \text{ to } (E+dE)\\ 0, & \text{otherwise} \end{cases}$$}
          here, $C=$ constant
          & $Ae^{(F-E)/KT}$
          & $Ae^{(\Omega + \mu N - E)/KT}$ \\ \hline
          Diagram
          & \includegraphics[width=\linewidth, valign=c]{example-image}
          & \includegraphics[width=\linewidth, valign=c]{example-image}
          & \includegraphics[width=\linewidth, valign=c]{example-image} \\ \hline
\end{tabularx}
\end{center}

\end{document}

Issues that needed to be fixed:

• The first column's contents are not vertically center-aligned
• The second column's cell with the displayed equation is not properly aligned at the center.
• The padding around images is not appearing perfectly.

Answer 1

Use \(...\) and not $$...$$; \raggedright and not \flushleft. Also valign=C, plus a trick to get some padding at the top.

\documentclass{article}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage[export]{adjustbox}
\usepackage{geometry}
\usepackage{array}
\usepackage{tabularx}

\newcolumntype{L}[1]{>{\raggedright\arraybackslash}m{#1}}
\renewcommand{\tabularxcolumn}[1]{>{\centering\arraybackslash}m{#1}}

\begin{document}

\begin{center}
\begin{tabularx}{\linewidth}{|L{.15\linewidth}|X|X|X|} \hline
  \multicolumn{1}{|c|}{Property} & Microcanonical & Canonical & Grand-canonical \\
  \hline
  Contact with environment 
    & Isolated system, so the energy and number of particles are constant.
    & System are in thermal contact, such that it can exchange energy but no particles. 
    & System can exchange both energy and particles. \\
  \hline
  Probability density, $\rho(E)$
    & \(\begin{cases} \text{C,} & E \text{ to } (E+dE)\\ 0, & \text{otherwise} \end{cases}\)
      \vspace{1ex}\newline here, $C$ is a constant
    & $Ae^{(F-E)/KT}$
    & $Ae^{(\Omega + \mu N - E)/KT}$ \\
  \hline
  Diagram
    & \includegraphics[width=\linewidth, valign=C,margin=0pt 0pt 0pt 4pt]{example-image}
    & \includegraphics[width=\linewidth, valign=C,margin=0pt 0pt 0pt 4pt]{example-image}
    & \includegraphics[width=\linewidth, valign=C,margin=0pt 0pt 0pt 4pt]{example-image} \\
  \hline
\end{tabularx}
\end{center}

\end{document}

Answer 2

With tabularray table and use of the \adjincludegraphics macro from the adjustbox package:

\documentclass{article}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage[export]{adjustbox}
\usepackage{tabularray}

\begin{document}
    \begin{center}
    \adjustboxset{height=0.15\textwidth, % had to be defined
                  keepaspectratio,
                  valign=m}
\begin{tblr}{hlines, vlines,
             colspec={X[0.6,l,m] *{3}{X[1,c,m]}},
             rowsep=4pt,
             }
\SetCell{c} Property
    & Microcanonical
        & Canonical
            & Grand-canonical                                       \\
Contact with environment
    &   Isolated system, so the energy and number of particles are constant.
        &   System are in thermal contact, such that it can exchange energy but no particles.
            &   System can exchange both energy and particles.      \\
Probability density, $\rho(E)$
    & \parbox{\linewidth}{$\begin{cases}
        \text{C,} & E \text{ to } (E+dE)\\
               0, & \text{otherwise}
        \end{cases}$

    \medskip
      here, $C$ is a constant}
    & $Ae^{(F-E)/KT}$
        & $Ae^{(\Omega + \mu N - E)/KT}$ \\
Diagram
    &   \adjincludegraphics{example-image-duck}
        &   \adjincludegraphics{example-image-duck} 
            &   \adjincludegraphics{example-image-duck} \\
\end{tblr}
    \end{center}
\end{document}

Edit: The weak point of the above solution is determining the size of the images. To adjust them correctly to the width of the columns, it is necessary to guess the allowed their heights. Determining their widths by column widths unfortunately now is inserted (due to bug in package) a large amount of vertical space above/below them.

With the new version of the package, this bug will be removed and the correct image size can will be possible to determine simply by the width of the columns with the following setting:

    \adjustboxset{width=\linewidth,   % <---
                  keepaspectratio,
                  valign=m}

Resulted table will be the same as before. More information you can find in this answer.

Answer 3

With {NiceTabular} of nicematrix.

\documentclass{article}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage[export]{adjustbox}
\usepackage{nicematrix}


\begin{document}

\begin{center}
\begin{NiceTabular}{m[l]{.15\linewidth}X[m,c]X[m,c]X[m,c]}[hvlines,cell-space-top-limit=3pt,colortbl-like]
  \RowStyle[color=blue]{\centering}\rowcolor{lightgray}
  Property & Microcanonical & Canonical & Grand-canonical \\
  Contact with environment 
    & Isolated system, so the energy and number of particles are constant.
    & System are in thermal contact, such that it can exchange energy but no particles. 
    & System can exchange both energy and particles. \\
  Probability density, $\rho(E)$
    & \(\begin{cases} \text{C,} & E \text{ to } (E+dE)\\ 0, & \text{otherwise} \end{cases}\)
      \vspace{1ex}\newline here, $C$ is a constant
    & $Ae^{(F-E)/KT}$
    & $Ae^{(\Omega + \mu N - E)/KT}$ \\
  Diagram
    & \includegraphics[width=\linewidth, valign=C]{example-image}
    & \includegraphics[width=\linewidth, valign=C]{example-image}
    & \includegraphics[width=\linewidth, valign=C]{example-image} \\
\end{NiceTabular}
\end{center}

\end{document}