Casimir self-entropy of an electromagnetic thin sheet

Casimir entropies due to quantum fluctuations in the interaction between electrical bodies can often be negative, caused either by dissipation or by geometry. Although generally such entropies vanish at zero temperature, consistent with the third law of thermodynamics (the Nernst heat theorem), there is a region in the space of temperature and separation between the bodies where negative entropy occurs, while positive interaction entropies arise for large distances or temperatures. Systematic studies on this phenomenon in the Casimir-Polder interaction between a polarizable nanoparticle or atom and a conducting plate in the dipole approximation have been given recently. Since the total entropy should be positive according to the second law of thermodynamics, we expect that the self-entropy of the bodies would be sufficiently positive as to overwhelm the negative interaction entropy. This expectation, however, has not been explicitly verified. Here we compute the self-entropy of an electromagnetic $\delta$-function plate, which corresponds to a perfectly conducting sheet in the strong coupling limit. The transverse electric contribution to the self-entropy is negative, while the transverse magnetic contribution is larger and positive, so the total self-entropy is positive. However, this self-entropy vanishes in the strong-coupling limit. In that case, it is the self-entropy of the nanoparticle, which we recalculate in the perfect conducting limit, that is just sufficient to result in a non-negative total entropy.

Figure 1

The entropy of interaction between a perfectly conducting sphere and a perfectly conducting plane (Sph-Pl) and between two perfectly conducting spheres (Sph-Sph) normalized with respect to the corresponding high-temperature values is displayed as a function of the product of distance $Z$ and temperature $T$. The entropy has been evaluated within the dipole and single-scattering approximation [12]; the ratio of entropies is independent of the radii of the spheres. The inset shows the behavior of the entropy for small $TZ$. We call the negative entropy region perturbative for the sphere-plane configuration and nonperturbative for the sphere-sphere configuration.

Figure 2

The dimensionless scalar entropy for a single $\delta$-function plate, which is the same as the TE electromagnetic entropy in the plasma model, shown as a solid black line, as a function of the temperature divided by the coupling strength. The dotted blue line shows the strong-coupling, low-temperature limit, while the dashed red line depicts the behavior at high temperature or weak coupling. The entropy satisfies the third law of thermodynamics, in that the entropy vanishes as the temperature goes to zero, but it is, surprisingly, always negative.

Figure 3

Semilog plot of the transverse magnetic entropy $s^{\mathrm{TM}}$ as a function of $y=1/x=4\pi T/{\lambda }_0$ for the plasma model. It is compared with the leading asymptotic expansions for large and small $x$ given in Eqs. (4.26), in the dotted and dashed lines, respectively. The TM entropy is always positive, and overwhelms the negative TE entropy seen in Fig. 2 except for large $x$.

Figure 4

The total self-entropy $s=s^{\mathrm{TE}}+s^{\mathrm{TM}}$ of a electromagnetic $\delta$-function plate in the plasma model, plotted as a function of $x={\lambda }_0/(4\pi T)$. Shown for comparison is the TM contribution (red, dashed line) and the TE contribution (black, dotted line).