Casimir energy and entropy in the sphere-sphere geometry

We calculate the Casimir energy and entropy for two spheres described by the perfect-metal model, plasma model, and Drude model in the large-separation limit. We obtain nonmonotonic behavior of the Helmholtz free energy as a function of separation and temperature for the perfect-metal and plasma models, leading to parameter ranges with negative entropy, and also we obtain nonmonotonic behavior of the entropy as a funtion of temperature and the separation between the spheres. This nonmonotonic behavior has not been found for the Drude model. The appearance of this anomalous behavior of the entropy as well as its thermodynamic consequences are discussed.

Figure 1

Casimir energy between perfect-metal spheres as a function of $d/{\lambda }_T$ compared with the energy at zero temperature. The dotted curve is the quantum limit, the dashed curve is the classical limit, and the solid curve is the asymptotic finite-temperature Casimir energy. Note that these curves are independent of the radius of the spheres but are only valid in the large-separation limit.

Figure 2

Casimir entropy between perfect-metal spheres as a function of $d/{\lambda }_T$ compared with the classical limit. The dashed curve is the classical limit and the solid curve is the asymptotic finite-temperature Casimir entropy. Note the regions of negative entropy and negative slope of the entropy with the parameter $d/{\lambda }_T$.

Figure 3

Log-log plot of the absolute value of the entropy divided by its classical limit for perfect metal spheres at constant $r$ as a function of $d/{\lambda }_T$. Starting from the asymptotic solid blue curve ($r\to 0$), we increase $r$ in steps of $0.1$ up to $r=0.4$ (magenta curve for $r=0.1$, yellow for $r=0.2$, red for $r=0.3$, and $r=0.4$ is the dotted curve). Dashed curve is the case $r=0.41$, where the interval of negative entropy has disappeared. The curve for $r=0.45$ (solid green curve) is also shown.

Figure 4

Asymptotic force divided by its zero-temperature limit as a function of $d/{\lambda }_T$. The dotted curve is the zero-temperature result. The dashed curve is the classical limit, and the solid curve is the result at finite temperatures. The nonmonotonic behavior of the force with temperature results from the negative slope of the solid curve at constant separation.

Figure 5

Entropy between plasma spheres as a function of $d/{\lambda }_T$ divided by the classical limit for the two limit cases. The solid curve is the entropy in the perfect-metal limit ${\lambda }_R\ll R$, where a region of negative entropy and nonmonotonicities of entropy with separation and temperature are present; the dashed curve is the entropy in the transparent limit ${\lambda }_R\gg R$, where the usual monotonic behavior of entropy is shown.

Figure 6

$(d/{\lambda }_T,{\lambda }_P/R)$ map of zeros of the entropy (solid curve), the temperature derivative of the entropy (dashed curve), and the separation derivative of the entropy (dotted curve).

Figure 7

Entropy as a function of the dimensionless parameter $d/{\lambda }_T$ divided by its classical limit for the large-separation limit of the system of Drude spheres. Entropy is positive and monotonic in temperature and separation.