2D Blackbody Cavity Standing Waves & Degrees of Freedom

Selected Mode (2, 2) Eigenmode (n_x, n_y)

Quadratic DoFs (Selected Mode) 2 DoFs 1 Electric (T) + 1 Magnetic (V)

Degeneracy Factor g_n 1 (Non-Deg.) Distinct pairs with same |k|

Cumulative DoFs in Cavity 32 DoFs Up to selected frequency cut-off

1. 2D Cavity Field Distribution E_z(x, y, t) k = 2.83 (π/L)

Positive (+E_z) Negative (−E_z) Nodal Lines (E_z = 0)

(n_x, n_y)	\|k\| × (L/π)	Degeneracy	Quadratic DoFs	Total Energy

1. Quadratic Degrees of Freedom per Spatial Mode ($f = 2$): Each independent spatial standing wave mode $(n_x, n_y)$ acts as an autonomous harmonic oscillator with electric energy density ($\propto E_z^2$) and magnetic energy density ($\propto B_x^2 + B_y^2$). By equipartition, each quadratic coordinate receives $\frac{1}{2}k_B T$, giving 2 degrees of freedom ($E = k_B T$) per mode.
2. Spatial Mode Counting in 2D ($N(k)$): Valid modes lie in the positive quadrant of $k$-space ($n_x \ge 1, n_y \ge 1$). The total number of independent spatial modes with wavenumber below $k$ is:
N_spatial(k) ≈ ¼ π (kL / π)² = ¼ π (n_x² + n_y²)
3. Total Thermal Degrees of Freedom ($F_{\text{total}}$): Multiplying by 2 quadratic degrees of freedom per spatial mode:
F_{\text{total}} = 2 × N_spatial(k) &implies; ⟨E_{\text{cavity}}⟩ = F_{\text{total}} × (½ k_B T) = N_spatial × k_B T
4. Degeneracy Multiplicity: Non-symmetric modes $(n_x \neq n_y)$ like $(1, 2)$ and $(2, 1)$ have the exact same frequency $\omega = c\frac{\pi}{L}\sqrt{5}$, but represent two completely orthogonal spatial modes (4 quadratic degrees of freedom total, storing $2k_B T$).