= Dimensional crossover of thermal cavity radiation
{title2=$U_{2D}=\frac{\zeta(3)}\pi\frac{A(k_BT)^3}{\hbar^2c^2}$}
In a thin <rectangular electromagnetic cavity>, the window $\hbar c\pi/L_x,\hbar c\pi/L_y\ll k_BT\ll\hbar c\pi/L_z$ freezes the nonzero thickness indices while retaining many in-plane <normal modes>. The zero-thickness-index family has one <polarization of an electromagnetic wave> and <density of states> $dN=A k\,dk/(2\pi)=2\pi A\nu\,d\nu/c^2$. Multiplying by $h\nu/(e^{h\nu/k_BT}-1)$ and integrating gives $U_{2D}=4\pi\zeta(3)A(k_BT)^3/(h^2c^2)$. Thus thermal <energy> scales as <area> times $T^3$. The <Riemann zeta function> appears through $\int_0^\infty x^2/(e^x-1)\,dx=2\zeta(3)$.
This is an intermediate low-<temperature> regime of an anisotropic finite cavity. When $k_BT$ also falls below the first in-plane excitation, thermal <energy> becomes exponentially small. If a different model excludes every zero thickness index, its first transverse mode has a finite gap and the proposed $T^3$ regime does not occur. In the three-dimensional continuum regime the corresponding <density of states> is proportional to <volume (thermodynamics)> times $\nu^2$ and thermal <energy> is proportional to <volume (thermodynamics)> times $T^4$.
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