\section{Quantum fluctuations\\
\footnotesize \textit{The philosophy in athermal transitions}}

\textsc{Onsager}'s model \cite{onsager1949effects} of athermal transition through rival entropies consists of rigid rods (referred to as \textsc{Onsager}'s rods hereafter) of infinite length-diametre ratio $\epsilon^{-1} \coloneqq l/d$. The system shall be \textit{isotropic} (with $[1+s]$ different axes along which the rods tend to align) and \textit{nematic} (all rods align uniaxially) respectively at two extremum concentrations \begin{math}
  c \coloneqq C/V
 
\end{math},\footnote{
I shall stick to the notational conventions in apps.~\ref{app:BLtheo} and \ref{app:statprefmaxientr&minifreeener}, in which some elements of statistical physics are `popularised' for interested audiences.
} or equivalently, packing fractions \cite{parisi2020theory} \begin{math}
  \varphi \coloneqq 2dl^2 c \sim
 
  \epsilon^{\pm 1}
\end{math}.\footnote{
Imagine two perpendicular rods excluding each other's geometrical centres from occupying a certain volume, the average \emph{excluded volume} produced by a single rod $\sim 2dl^2$; \textit{cf.}\ fig.~1.a of \cite[\textbf{b}]{frenkel2000&2015}. Other examples of excluded volumes could be
(i) \begin{math}
  \mlr{4 \pi l^3}{3}
\end{math} for a radius-$l$ sphere
(ii) \begin{math}
  2 \pi d^2 l
\end{math} for a cylinder of length $2l$ and diametre $2d$, with packing fractions \begin{math}
  \sim \mlr{\qty(5\pm 1)\epsilon^{\pm 1}}{4\pi} 
\end{math} respectively.
}
To locate the transition point $\varphi^*$ between the least and the most concentrated phases, one may refer to the energy/entropy difference\footnote{
\textit{Cf.}\ app.\!~\ref{app:statprefmaxientr&minifreeener}; app.\!~\ref{app:misccalc} shall elaborate somewhat the sketch of the model here.
} \cite{frenkel2000&2015}
\begin{eqnarray}
  &&\ub{\textstyle
    \ln\qty{
      V - 2 \pi d^2l C
    } + \ln{1}
  }_{
    S_{\varphi_{\text{nem}}}
    = S_{\varphi_{\text{nem}}}^{\text{translational}} + S_{\varphi_{\text{nem}}}^{\text{rotational}}
  } - \ub{\textstyle\bigl(
    \ln\qty{
      V - 2 dl^2 C
    } + \ln\qty{1+s}
  \bigr)}_{
    S_{\varphi_{\text{iso}}}
    = S_{\varphi_{\text{iso}}}^{\text{tra}} + S_{\varphi_{\text{iso}}}^{\text{rot}}
  } \eqqcolon \notag\\&&\delta{S}_\varphi = \ln{\frac{
    1 - \epsilon \pi \varphi
  }{
    \qty(1 - \varphi)(1+s)
  }} \xRightarrow[
    \epsilon \rightarrow 0
  ]{
    0 \equiv \delta{S}_{\varphi = \varphi^*}
  } \varphi^* = \frac{s}{1+s} \begin{cases}
    \xrightarrow{s\rightarrow 0} 0,
    \\\xle{s = 1} 1/2,
    \\\xrightarrow{s\rightarrow \infty} 1,
  \end{cases}
\label{eq:del{S}}\end{eqnarray}
where
(i) \begin{math}\qty(
  V - 2\pi d^2 l C
)\end{math} is the explorable space for rods in a nematic single phase, and \begin{math}\qty(
  V - 2 d l^2 C
)\end{math} the one in an isotropic case
(ii) $(1+s)$ is interpreted as number of component species of a self-assembly in \cite[\textbf{b}]{frenkel2000&2015}
(iii) we shall call a $\varphi \sim \varphi^*$ \textit{semidilute} \cite{doi1988theory}, and the transition depends trivially on the exact value of $\qty[\mlr{s}{(1+s)}]$ if it remains some finite constant.

Summaries of the essence and pedagogical practices have been abundant \cite{frenkel2000&2015}--\cite{vroege1992phase} since \textsc{Onsager}'s first insight. Three basic facts about the model are
\begin{description}[wide=0pt]
\item[Athermality]
as \textsc{Mayer}'s function used to polynomial/\textsc{Virial} expand a partition function depends trivially on temperatures (footnote~\ref{foot:Mayefct}); this shall be rephrased in the language of quantum fluctuations in a moment
\item[Discontinuous phase transition]
whose many links to the ferroic hystereses\footnote{
  Below some critical/\textit{order disruption} temperatures of \textsc{Curie}'s, \textsc{N\'eel}'s \textit{etc.} \cite{aharoni2000introduction}.
}
are in evidence
\item[Two-well potential]
as higher-than-second-order \textsc{Virial}'s terms are greatly suppressed by $\epsilon$, the free energy $f[\varphi]$ shall have a two-well landscape by \textsc{Landau}'s mean-field (m.f.) idea in the close vicinity of the semidilute transition point; if the isotropic minima\footnote{
Strictly speaking, of some \begin{math}
  \delta{f}_{\varphi}\qty[\text{order parameter }\sigma]
\end{math} now for different m.f.\ solutions $\varphi$; but the m.f.\ constraint/saddle-point equation sets some relations between $\sigma$ and its conjugate force $\varphi$, so everything could eventually be expressed as a function of $\varphi$ (or $\sigma$). \textit{Cf.}\ app.\!~\ref{app:misccalc}.
}
are pinned, then any \textit{errors} in a desired micro-state pattern can only concern the nematic ones.
\end{description}
The essay's goal is to encapsulate ultimately \textsc{Onsager}'s spirit into ferroics, with the trial prototype here \textsc{Stoner-Wohlfarth}'s (S-W's) macro-spin \cite{tannous2008stoner,carrey2011simple}. \textit{Viz.}, to fashion a ferroic switch triggered by a variable field with the usual outcome of polarisation reversals, but the internal mechanism governed by \textsc{Onsager}'s entropy competitions---hence athermal, and a reliable two-state toggling free from pitfalls of some muddled energy landscapes.

Before continuing the main thread of narration, I shall give a few more legitimacy checks on the aforementioned model features and some philosophies involved.
\begin{description}[wide=0pt]
\item[Quantum phase transition]{
stands out typically when phasing out the `internal energy-entropy' tradeoff in eq~\eqref{def:S} at zero temperature, with frozen particles and proliferating quantum fluctuations by \textsc{Heisenberg}'s uncertainty principle.\footnote{
Some \textsc{Schwarz}'s/triangle inequality, \cite{weinberg2015quantum} such that a vanishing root-mean-square deviation $\Delta_{\Psi} \vbx$ of the position operator $\vbx$ evaluated at a given system state $\Psi$
\begin{math}
\xRightarrow[\text{uncertainty relation}]{
  \Delta_{\Psi} \vbx . \Delta_{\Psi} \vbp \geq \text{constant}
}\end{math} momentum root-mean-square \begin{math}
  \Delta_{\Psi} \vbp \rightarrow \infty
\end{math}.
} In \textsc{Onsager}'s model, the rod exploration space or positional uncertainty is abruptly reduced when $\varphi$ drops across $\varphi^*$, leading to multiplication of the quantised rotational degrees of freedom,\footnote{
Sending centres of all rods to a fixed $\bbR^3$-origin, $\bbR^3$-rotations of the directional unit vectors $\vbn$ are then typified by group elements $\msR \in$ SO$_3$. Note some $\bbC^2$-`spin rotations' in quantum mechanics can be represented by
\begin{math}
  \msU_{\theta} = \me^{\mlr{
    \theta \vbn . \vv{\sigma}
  }{2\mi}}
  \in 
\end{math} SU$_2$ generated by \textsc{Pauli}'s matrices \begin{math}
  \sigma_{x,y,z}
\end{math}, such that SU$_2$($\simeq$ a unit $3$-sphere S$^3$, topologically) acts locally like SO$_3$($\simeq$ quotient group \begin{math}
  \mlr{{\text{S}}^3}{\qty{\pm 1}}
\end{math}).
}
and thus some athermal `quantum phase transition' like those of quantum \textsc{Ising}/rotor models \textit{etc.}\ in \cite{sachdev2006&2011}.
}\item[Mean-field theory\label{sec:meanfieltheo}]{\!\!\footnote{
\textit{Cf.}\ chaps. 1--2 of \cite{cardy1996scaling}, and \cite{parisi2020theory} \textit{etc.}
}
(i) coarse-grains fractions of a statistical system into self-equilibrial `molecules'
\begin{equation*}
  m^i \coloneqq \expval{\textstyle
    \sigma^i = \delta{\sigma^i} + \qty(
      m \equiv \sum_{j=1}^C \mlr{m^j}{C}
    ) 
  }_{\mcZ},
\end{equation*}
with $\sigma^i$ a variable (named \textit{spin}) representative of fraction $i$
(ii) approximates the partition functions/statistic $\mcZ$ by a series of the local means. Such m.f.\ predictions may break down by \textsc{Ginzburg}'s criterion \cite{cardy1996scaling} for a low-dimensional system, with finite-range interactions, at a \textit{continuous}\footnote{
In 1st derivatives of the free energy with respect to applied forces.
}
phase transition where the correlation length $\xi$ (characteristic size of fluctuations \begin{math}
  \expval{\delta{\sigma^i}.\delta{\sigma^j}}
\end{math}) explodes.\footnote{
\textit{I.e.}, nonnegligible fluctuations over all distances force the entire system to form a unique critical phase; the properly compatible ideas of scaling and renormalisation are then needed for quantitative descriptions.
}
As for a \emph{discontinuous} transition like \textsc{Onsager}'s athermal one, an instability of the continuation from one coexisting phase to another shall yield some \textit{hysteresis}/\textit{memory effect}. \textit{I.e.}, the system needs time to readjust between single phases, with $\xi$ finite at such transitions in general. Note the expected transitions in the toy model to be proposed are \textit{abrupt}. As we are not concerned anyway with the asymptotic exponents of critical behaviours during the abrupt periods, a first qualitatively m.f.\ treatment seems adequate.

One more thing to stress is that a uniform \begin{math}
  m \equiv m^{\forall i}
\end{math} is \textit{not} a necessity for optimising a m.f.\ free energy. The implicit homogeneity from a m.f.\ idea is explicitly lost in a disordered system. \textsc{Parisi}'s remedy \cite{parisi2020theory} is to
(i) \textit{replicate} the statistics $\mcZ$ $s$ times and allow `$\sigma^i_a$-$\sigma^j_b$' interactions, but every ($1+s$) replicas $\sigma_{a = 1,2,\dots,1+s}^{\forall i \in \qty{1,\dots,C}}$ form a `molecule' bound by a strong-localisation constraint,\footnote{
Reminiscent of the $s$ extra rotational degrees of freedom for each \textsc{Onsager}'s rod in an isotropic phase than in a nematic one.
} with $
  \qty{\sigma_{a=1}^i
  \equiv \sigma^i}_{i=1}^C
$ the reference \textit{quenched} disorder; such molecular liquid shall recover some translational and rotational invariance, and one could use again methods like \textsc{Virial}'s expansion \textit{etc.}\ generalised from theories of conventional atomic/molecular liquids
(ii) \textit{conflate} dissenting statistics into a full average/integral $
  \mcZ^{1+s} \sim \prod_{a=1}^{1+s} \mcZ_a
$, and the final desired \textit{glass free energy} is an analytic continuation of \begin{equation*}
  - \mlr{\qty(
      \partial_s \mcZ^{1+s}
    )}{\mcZ C}
  \xrightarrow[]{s\rightarrow 0} \mlr{\qty(
    - \ln{\mcZ}
  )}{C}
  = f_{\text{gla}}.
\end{equation*}
}
\end{description}

\section{Phenomenologies for hystereses}
\subsection{A thought experiment}
The task is simply to well encapsulate $C$ magnetised \textsc{Onsager}'s rods in an elastic cuboid\footnote{
Or any similar geometry like a cylinder of length $L$ and diametre $D$, for negligible boundary effects.
}
of height $L \sim l$ and volume $D^2 L \sim l^3$. Small variations of the cuboid volume/occupied volume of the rods allow variations of packing fractions \begin{math}
  \varphi
\end{math} around the critical \begin{math}
  \varphi^*
\end{math} ($\equiv \mlr{1}{2}$ for simplicity) in eq~\eqref{eq:del{S}}.\footnote{
The average nematic excluded volume for rods in the cramped box is now a cylinder $\pi d^2 l$ of length $l$ (rather than $2l$) and diametre $2d$, but this will not change the final \begin{math}
  \eval{\varphi^* \textstyle}_{\epsilon \rightarrow 0}
\end{math} anyway.
}
Label the three axes through the centres of the opposite cuboid faces as $x$, $y$ and $z$; let the vertical axis be $z$, so that the principal fa\c{c}ade and the aerial view are respectively in the $x$-$z$ and $x$-$y$ planes.
\subsubsection{\label{sec:initconf}
Initial configuration
\\\textit{Introduction of a mechanism for protecting rods' impenetrability}}
\begin{enumerate}[wide=0pt,label=(\roman*)]
\item all north poles ${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$ are perfectly up in a strong applied force $\vbB \equiv \vbB^z$ (table~\ref{tab:iden}), inducing naturally a unique \textit{nematic axis} as the $z$-axis
\item the horizontal `${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}\hspace{-3pt}\leftrightarrow\hspace{-3pt}{\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$' and `${\protect\substack{\text{\footnotesize\color{Blue}\faMinusCircle}}}\hspace{-3pt}\leftrightarrow\hspace{-3pt}{\protect\substack{\text{\footnotesize\color{Blue}\faMinusCircle}}}$' repulsions keep the confined rods from collapsing together but aligning at equal spacings \begin{math}
  a \lesssim 2 \sqrt{\epsilon} l \ll D
\end{math}, as the preset \begin{itemize*}[label=\textcolor{gray}{$\diamond$}]
\item \begin{math}
  L \equiv l
\end{math}
\item $\qty[\mlr{D}{l}]$ finite
\item \begin{math}
  \varphi = \qty[\mlr{2dl^2\qty(\mlr{D}{a})^2}{D^2 L}]
  \gtrsim \varphi^*
\end{math} for now; note that \begin{math}
  a \gg d = \epsilon l
\end{math}, `\begin{math}
  \uspin \!\!\cdots\!\! \uspin
\end{math}' is hence morphologically nothing more than a bundle of fibres/lines of induction
\end{itemize*}
\item dipolar interactions (\begin{math}
  \sim \mlr{1}{\qty|\vbr^{i,j}|^3}
\end{math} for parallel dipoles $i$ and $j$ at a distance $\vbr^{i,j}$)\footnote{
\textit{Cf.}\ prob.\!~6.21 in \cite{griffiths2017introduction} for a general potential between angled dipoles.
} are assumed to be so weak and well \textit{screened} (\cite{altland2010condensed}, say the characteristic length of damping is \begin{math}
  \sim a
\end{math}) that they compare minorly with the prescribed two-well potential in the system free energy $\mcF$; \textit{cf.}\ discussions of the non-local forces \textit{etc.}\ in some \textsc{Langevin}'s equation \begin{math}
  \partial_t \phi = \qty(- \fdv*{\mcF}{\phi})
\end{math} of \cite{jaglajagla2005hysteresis}, with $\phi(\vbx,t)$ a scalar variable forming the phenomenological $\mcF[\phi]$.
\end{enumerate}
\begin{table}[b]
\caption{\label{tab:iden}
Ferroic dipoles $\vbM$ subject to their conjugate force $\vec{B}$. The varying $\vbB$ is along a fixed vertical axis, on which the projection of the sum of dipoles is represented by a macro-spin
(\textcircled{1}--\textcircled{3}) $\Uspin$\;$\simeq$ \begin{math}
  \uspin \!\!\cdots\!\! \uspin
\end{math}
(\textcircled{5}--\textcircled{7}) $\Dspin$\;\begin{math}
  \simeq \dspin \!\!\cdots\!\! \dspin
\end{math}
(\textcircled{4}, \textcircled{8}) ${\protect \substack{\text{${\protect \substack{\text{{\footnotesize \faCircle}}}}$}}}$\;\begin{math}
  \simeq \mlr{\qty(
    \Uspin + \Dspin
  )}{2}
\end{math}.
}
\begin{ruledtabular}
\begin{tabular}{lcccccccc}
&\textcircled{1}&\textcircled{2}&\textcircled{3}&\textcircled{4}&\textcircled{5}&\textcircled{6}&\textcircled{7}&\textcircled{8}\\\colrule
&\blue{-}&0&\red{+}&\red{+}&\red{+}&0&\blue{-}&\blue{-}\\\bottomrule
$\vbM^{\text{total}}$
& \multicolumn{3}{|@{}c@{}|}{
  $\Uspin$
}&${\protect \substack{\text{${\protect \substack{\text{{\footnotesize \faCircle}}}}$}}}$&
\multicolumn{3}{|@{}c@{}|}{
  $\Dspin$
}&${\protect \substack{\text{${\protect \substack{\text{{\footnotesize \faCircle}}}}$}}}$\\\toprule
$\vbB$ & $\Uparrow$ & 0 & $\Downarrow$ & $\Downarrow$ & $\Downarrow$ & 0 & $\Uparrow$ & $\Uparrow$\\
&\red{+}&0&\blue{-}&\blue{-}&\blue{-}&0&\red{+}&\red{+}
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[htbp]
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}[small]
\vertex (o){0};
\vertex[left = 1.5cm of o](x1);\vertex[right = 1.5cm of o](x2){$\vbb$};\vertex[below = 1.5cm of o](y1);\vertex[above = 1.5cm of o](y2){$\vbm$};
\vertex[left = .5cm of o](x3){$\Downarrow$};\vertex[right = .5cm of o](x4){$\Uparrow$};
\vertex[left = 1cm of o](d){\textcircled{4}};\vertex[right = 1cm of o](h){\textcircled{8}};\vertex[below = 1cm of o](f){\textcircled{6}};\vertex[above = 1cm of o](b){\textcircled{2}};
\vertex[right = 1cm of b](a){\textcircled{1}};\vertex[left = 1cm of b](c){\textcircled{3}};
\vertex[left = 1cm of f](e){\textcircled{5}};\vertex[right = 1cm of f](g){\textcircled{7}};
\diagram*{
(x1)--[scalar, gray](x2),(y1)--[scalar, gray](y2),
(a)--[fermion](b)
  --[fermion](c)--[fermion](d)--[fermion](e)--[fermion](e)--[fermion](f)--[fermion](g)
  --[fermion](h)--[fermion](a),
};\end{feynman}\end{tikzpicture}
\caption{
Sketch of key points in a ferroic hysteresis loop of the thought experiment depicted in table~\ref{tab:iden}. We prefer in physics dimensionless variables \begin{math}
  \vbm \coloneq \mlr{\vbM^{\text{tot}}}{\vbM_{\text{sat}}^{\text{tot}}}
\end{math} and \begin{math}
  \vbb \coloneqq \mlr{\vbB}{\vbB_{\text{coe}}}
\end{math}, with $\vbM_{\text{sat}}^{\text{tot}}$ and $\qty(-\vbB_{\text{coe}})$ respectively the \textit{saturation} magnetisation and \textit{coercive} force at \textcircled{1} and \textcircled{4}. The (major) loop is a perfect square for S-W's macro-spin \cite{tannous2008stoner,carrey2011simple} in a varying $\vbB$ along the anisotropy axis, \textit{i.e.}, \begin{math}
  1 \equiv \vbm_{\textit{remanent}}
\end{math} at \textcircled{2}, and \begin{math}
  1 \equiv \vbm_{\text{transition}}
  = - \vbb_{\text{tra}}
\end{math} at \textcircled{3}.
\label{fig:hysteresis}}\end{figure}
A small translational deviation of a single rod in the above equi-spacing configuration shall raise $\mcF$ by
\begin{eqnarray*}
  &&\delta \mcF \propto \sum_{s= \pm} \qty{
    \sqrt{\qty[
      (x-sa)^2 + y^2
    ]^{-3}} + \sqrt{\text{`$x \leftrightarrow y$'}}
  } - 4a^{-3}
  \quad\notag\\&&\geq 0\qc
  x^2+y^2 \leq a^2 < \qty(|x| - a)^2 + \qty(|y| - a)^2
  ,
\end{eqnarray*}
and makes the system less stable. This malposition amounts to strengthening the effective couplings of adjacent (di)poles in fig.\!~\ref{fig:aeriview} (or shrinking the whole lattice a bit, keeping all dipoles parallel) on a coarse-grained level. Introducing such `pan-tilt' randomness/disorders seems interesting \cite{wiese2021theory} but not the exact task of fashioning for now a simplest m.f.\ toy model.
\begin{figure}[htbp]
\centering
\begin{equation*}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}[small]
\vertex (o){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};
\vertex[left = 1cm of o](a1);\vertex[right = 1cm of o](a2){$x$};\vertex[below = 1cm of o](b1);\vertex[above = 1cm of o](b2){$y$};
\vertex[left = .7cm of o](x1){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};\vertex[right = .7cm of o](x2){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};\vertex[below = .7cm of o](y1){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};\vertex[above = .7cm of o](y2){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};
\diagram*{
(a1)--[scalar, gray](a2),(b1)--[scalar, gray](b2),
};\end{feynman}\end{tikzpicture}
\longrightarrow \begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}[small]
\vertex (o){};
\vertex[left = 1cm of o](a1);\vertex[right = 1cm of o](a2){$x$};\vertex[below = 1cm of o](b1);\vertex[above = 1cm of o](b2){$y$};
\vertex[left = .7cm of o](x1){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};\vertex[right = .7cm of o](x2){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};\vertex[below = .7cm of o](y1){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};\vertex[above = .7cm of o](y2){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};
\vertex[right = .2cm of o](x3);\vertex[above = -.05
cm of x3](z){${\protect\substack{\text{\footnotesize\color{Red}\faPlusCircle}}}$};
\diagram*{
(a1)--[scalar, gray](a2),(b1)--[scalar, gray](b2),
};\end{feynman}\end{tikzpicture}
\end{equation*}
\caption{
Aerial view of a deviation of the central pole.
\label{fig:aeriview}}\end{figure}

\subsubsection{Interpreting the hysteresis in Onsager's spirit
\\\textit{Forging variations of concentration by tuning the force applied}}
Assuming that the above protection for impenetrability remains effective,

$\vbb_{\textcircled{1}}$$\rightarrow$$\vbb_{\textcircled{2}}$\textbf{:}
\textsc{Zeeman}'s energy, \begin{math}
  - \vbb. \vbm \xle[
    \qty|\vbb| \eqqcolon b \equiv b_z
  ]{
    \qty|\vbm| \eqqcolon m \equiv m_z
  } - bm
\end{math}, is reduced from $1$ to $0$. \textit{Viz.}, the system need less local anisotropy energy (\textit{i.e.}, the prescribed two-well potential) or dipolar interaction to compensate the \textsc{Zeeman} term. Following the same line of reasoning about fig.~\!\ref{fig:aeriview}, the system shall expand (keeping all dipoles parallel), forging a reduction in $\varphi$, while $1 \equiv m$ as the number of rods and the strength \begin{math}
  \qty|\vbM \equiv \vbM^z|
\end{math} of a single dipole both remain identical by construction.

$\vbb_{\textcircled{2}}$$\rightarrow$$\vbb_{\textcircled{3}}$$\xrightarrow{\vbb_{\textcircled{4}}}$$\vbb_{\textcircled{5}}$\textbf{:}
the same reduction continues untill a breakdown at \textcircled{3}, where $\varphi$ touches the transition $\varphi^*$, and all rods flip in \textsc{Onsager}'s spirit at once to the configuration \textcircled{5} with $\vbm$'s direction the same as $\vbb$'s again. The configuration \textcircled{4} during the flip could be a superposition
(i) \begin{math}
  \simeq \mlr{\qty(
    \Uspin + \Dspin
  )}{2}
\end{math}, of swapping spins, or
(ii) \begin{math}
  \simeq \mlr{\qty(
    \Lspin + \Rspin
  )}{2}
\end{math}, of rotating spins, but is not much of our concern as mentioned in the m.f.\ discussions in sec.\!~\ref{sec:meanfieltheo}.

The next half `\textcircled{5}$\rightarrow$\textcircled{1}' of the loop in fig.\!~\ref{fig:hysteresis} shall be symmetric with the first half. Note the number of \textsc{Onsager}'s order transitions in a loop is four, twice the one of ferroic reversals, which is reminiscent of SU$_2$'s \textit{double covering} of SO$_3$ in a simple group theory.

\subsubsection{An effective single-domain ferroic
\\\textit{Duality to Stoner–Wohlfarth's model in a static case}}
The hysteresis-loop squareness and the rod-bundle morphology are immediate reminiscences of S-W's model for single-domain magnets \cite{tannous2008stoner,carrey2011simple}, with S-W's
(i) macro-spin $\vv{\sigma}$ identified here as projection (at the first configuration) of the sum $\vbm_{\text{sat}}$ of dipolar rods on the anisotropy (indicated by a unit vector $\vba$) axis
(ii) `anisotropy-\textsc{Zeeman}' energy ($\mcE$$_{\text{ani, \textsc{Zee}}} \propto \qty[
  \vba \wedge \mlr{\vv{\sigma}}{\qty|\vv{\sigma}|}
]^2$, $
  - \vv{\sigma} . \vbb
$) competition a mutant of the `nematic-isotropic' entropy competition. Note the above transition properties are implicitly quasistatic, which is relevant at \textit{zero temperature} or \textit{infinite frequency} of $\vbb$, \textit{viz.}, increasing/decreasing the temperature/frequency may \textit{soften} the \textit{hardness} of a ferroic material of strong anisotropy (\textit{cf.}\ the numerical results in fig.\!~3 of \cite{carrey2011simple})

\subsection{Recapture of boundary conditions \& some final remarks}
The ideal case is the cuboid immersed in a r\'eservoir of $\vbb$ absorption, with uniform $\vbb \equiv \vbb^z$ inside and $\equiv \vec{0}$ outside the cuboid. To explain \textit{any} real \cite{speliotis1965hard} and numerical \cite{kuriabova2010linear,jaglajagla2005hysteresis} experiments on strongly anisotropic materials, we shall consider some more realistic conditions. \textit{E.g.},
\begin{enumerate}[label=(\roman*),wide=0pt]
\item rather than the effectively infinite rods through the entire thickness of my model, \cite{kuriabova2010linear} simulated much shorter \textsc{Onsager}'s rods, and defined some power-law two-body potential (eq~(1) of \cite{kuriabova2010linear}) to \begin{enumerate*}
\item soften the rods' hardness or
\item provide bondings between neighbouring ends from different rods, leading to a high-density \textit{columnar} crystal state (an enhanced nematic state; \textit{cf.}\ fig.\!~2 \textit{etc.}\ of \cite{kuriabova2010linear})
\end{enumerate*}
\item \cite{jaglajagla2005hysteresis} models the $\vbm$ of \textit{thin films with easy perpendicular ($z$) axis anisotropy upon change of $\vbb$ along $z$}, with the widths of domain walls much smaller than those of the uniform domains. The system may jump abruptly between $\vbm$ of opposite signs in the hysteresis loops for weak enough exchange/dipolar interactions and Gaussian disorders (introduced to the anisotropy; \textit{cf.}\ figs.\!~2.(9) and 5 \textit{etc.}\ of \cite{jaglajagla2005hysteresis}).
\end{enumerate}
The toy model can thus be seen as a sample (with a finite `height $L$-width $D$' ratio, and $\vbb$'s effective applied area $\sim D^2$) cropped from a uniform domain (say of which the zero-force configuration \begin{math}
  \sim \uspin \!\!\cdots\!\! \uspin
\end{math}) of such a film (whose $\mlr{L}{D_{\text{fil}}} \rightarrow 0$), where the (more or less fixed) boundary conditions at \textcircled{1} and \textcircled{5} might yield some asymmetry `\begin{math}
  \uspin \Uspin \uspin
\end{math} and \begin{math}
  \uspin \Dspin \uspin
\end{math}', but a constant \textit{kink} \cite{chaikin1995principles} energy difference shall not affect significantly the transition physics \textit{etc.}

A last word to mention is that the nonperiodic ferroic model here seems different from the conventional ones of lengthless \textsc{Ising}'s spins (separated by a few crystalline spacings). \textit{Viz.}, some multi-ferroics may be yielded if one could mathematically impose some abstract `infinitely long' ferroic verticals, between which the horizontal impenetrabilities are well protected, compatibly on other periodic ferroic models.

\begin{acknowledgments}
I want to thank (i) the \textsc{sLab} in \textsc{Dongguan} for the hospitality during my gap year in search of a \textit{Ph.D.}
(ii) my lovely classmates in \textsc{Canton} and \textsc{Paris} among others, for occasionally group-studying with me online and for their encouragement to a very scattered soul.
\end{acknowledgments}

