TikZ-PGF Circles – How to Place Text and Arrows on a TikZ Circle

Question

I have the figure below from the following MWE:

\documentclass{article}
\usepackage{tikz}
\usepackage{etoolbox}
\usetikzlibrary{arrows.meta}
\usetikzlibrary{bending}
\usetikzlibrary{decorations.text}

\begin{document}

\begin{center}

\begin{tikzpicture}[scale=5.3,cap=round,>=latex]
\def\Radius{.5cm}

\draw (0cm,0cm) circle[radius=\Radius];

\begin{scope}[
    -{Stealth[round, length=8pt, width=8pt, bend]},
    shorten >=4pt,
    very thin,
  ]
    \draw (\Radius, 0) arc(-3:3:\Radius);
    \draw (-\Radius, 0) arc(180+3:180-3:\Radius);
  \end{scope}

  % draw the points 
  \fill[radius=.7pt]
    (90:\Radius) circle[] node[above left] {$\tau =$ 0 fm}
    (10:\Radius) circle[] node[above right] {$\tau \sim$ 0.6 fm}
    (-43:\Radius) circle[] node[below right] {$\tau \sim$ 1 fm}
    (170:\Radius) circle[] node[below left] {$\tau \sim$ 2 fm}
  ;

  \def\Item#1#2(#3:#4){%
    \path[
      decoration={
        text along path,
        text={text1},
        text={some text},
        text={some other text},
        text align=center,
      },
      decorate,
    ]
      (#3:\Radius-#2) arc(#3:#4:\Radius-#2)
    ;
  }
  \Item A 1pt (90:10)
  \Item B 1pt (10:-43)
  \Item C 1pt (-43:-170)
  %\Item D 2pt (60:0)
\end{tikzpicture}
    
\end{center}
\end{document}

which gives me:
But I want to have 2 circles with some text in the centre of the circle, and only one arrow in the 90-180 deg quadrant and lastly, with different texts in place of "text3" with all of them positioned along the arcs and towards the south of the paper. Any help is greatly appreciated.

Answer 1

Your question is not entirely clear (see my comment below question). So far, I guess that you are after something like this:

\documentclass[border=3.141592mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,
                bending,
                decorations.markings, decorations.text}

\newlength{\R}\setlength{\R}{2cm}

\begin{document}
\begin{tikzpicture}[
  C/.style = {circle, draw, 
              double,double distance=1mm, % <--- ??
              minimum size=#1, align=center},
decoration = {text effects along path, reverse path,
              text={\i},
              text align=center,
              raise=3pt,
             },
dot/.style = {circle, fill, minimum size =1mm,
              label=#1, node contents={}},
->-/.style = {decoration={markings,
              mark=at position 0.8 with {\arrow{Stealth[angle=60:5pt 3]}},
              raise=0pt},
              postaction={decorate}},
                    ]
\foreach \i/\sa/\ea [count=\j] in {{}/170/90, alpha/10/90, beta/-43/10, gamma/170/317}
{
\path   [thin, draw=blue, text=orange,    % <--- new
         postaction={decorate},           % <--- changed
         text effects={text along path}]  % <--- new
        (\sa:\R) arc (\sa:\ea:\R);
}
\path   [->-]  
        (170:\R) arc (170:317:\R);
\path   ( 90:\R) node[dot={$\tau=0$} fm]
        ( 10:\R) node[dot=right:$\tau \sim 0.6$ fm]
        (-43:\R) node[dot=right:$\tau \sim 1$ fm]
        (170:\R) node[dot= left:$\tau \sim 2$ fm];
% circle in center
\node[C=21mm]   {where\\should I\\be???};
\end{tikzpicture}
\end{document}