= Finite-volume angular source selection
{title2=$p_x(\theta)=e^{x\cos\theta}/\int_{-\pi}^{\pi}e^{x\cos\alpha}\,d\alpha,\quad x=Nh/(k_BT)$}
Freeze the magnitude of an <order parameter> and retain just one common orientation $\theta$ for $N$ rigidly aligned cells. Give this collective coordinate <energy> $E(\theta)=-Nh\cos\theta$ at positive <temperature> $T$. The <Boltzmann distribution> is the displayed normalized <probability density function>, where $k_B$ is the <Boltzmann constant>. Here $h$ is <energy> per cell, a different normalization from the source density in <continuous vacuum selection by an external source>. This is a deliberately restricted model of selection among already ordered orientations: it does not derive alignment or an ordering <critical temperature> from a spatial interaction.
Its mean orientation is an elementary ratio of integrals:
$$
m_N(h)=\langle\cos\theta\rangle=\frac{\int_{-\pi}^{\pi}\cos\theta\,e^{x\cos\theta}\,d\theta}{\int_{-\pi}^{\pi}e^{x\cos\theta}\,d\theta}.
$$
At $h=0$ every angle has equal <probability>, and the numerator vanishes exactly. For small $x$, expansion of the exponential gives $m_N(h)=x/2+O(x^3)$ because the angular mean of $\cos^2\theta$ is $1/2$. For $x=1$ and $5$ the means are approximately $0.446390$ and $0.893383$. At fixed $h>0$, increasing $N$ concentrates the <probability density function> near $\theta=0$; expanding $\cos\theta=1-\theta^2/2+O(\theta^4)$ gives angular <variance> approximately $1/x$. Consequently
$$
\lim_{h\downarrow0}\lim_{N\to\infty}m_N(h)=1,\qquad
\lim_{N\to\infty}\lim_{h\to0}m_N(h)=0.
$$
The uniform angular average remains a symmetric mixture if one sets $h=0$ first. The other order selects a definite direction. This is the continuous-orientation version of the <symmetry-breaking order of limits>; it illustrates selection without replacing the separate problem of stability against spatial fluctuations.
\Image[/continuous-symmetry-source-selection.png]
{title=One collective angle illustrates noncommuting source and size limits}
{description=Original normalized <Boltzmann distribution> integrals for the rigid-orientation model. Every finite-$N$ response curve passes through zero at zero source. At fixed positive source, increasing $N$ narrows the angular <probability density function>. These curves are model calculations, not measured data or an independent derivation of a thermodynamic ordered phase.}
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