Analytical and numerical verification of the Nernst theorem for metals

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersion model gives consistent results for the Casimir free energy at low temperatures. Specifically, for the free energy near $T=0$ we find the leading term proportional to $T^2$ and the next-to-leading term proportional to $T^{5∕2}$. These terms give rise to zero Casimir entropy as $T\to 0$ and are thus in accordance with Nernst’s theorem.

Figure 1

Squared reflection coefficients $A$ and $B$ of the metal interfaces as defined in Eq. (7) for the TM and TE modes, as a function of $\zeta ∕c$ and the transverse momentum $k_{\perp }$. (a) $A$ for the TM mode, (b) $B$ for the TE mode, and (c) $B$ for $k_{\perp }$ and $\zeta$ close to zero.

Figure 2

Numerical evaluation of the free energy (6) between two gold half-spaces as a function of temperature. The inset gives details for low $T$.

Figure 3

Plot of the ratio $R$ defined in Eq. (33).