Casimir-Lifshitz force out of thermal equilibrium

We study the Casimir-Lifshitz interaction out of thermal equilibrium, when the interacting objects are at different temperatures. The analysis is focused on the surface-surface, surface-rarefied body, and surface-atom configurations. A systematic investigation of the contributions to the force coming from the propagating and evanescent components of the electromagnetic radiation is performed. The large distance behaviors of such interactions is discussed, and both analytical and numerical results are compared with the equilibrium ones. A detailed analysis of the crossing between the surface-surface and the surface-rarefied body, and finally the surface-atom force is shown, and a complete derivation and discussion of the recently predicted nonadditivity effects and asymptotic behaviors is presented.

Figure 1

Schematic figure of the surface-surface system out of thermal equilibrium. Here the two bodies occupy infinite half-spaces.

Figure 2

Thermal component (only $l$-dependent part) of the pressure out of equilibrium for fused silica-silicon system in the configuration ($T_1=300\mathrm{K}$, $T_2=0\mathrm{K}$) (solid) and in the configuration ($T_1=0\mathrm{K}$, $T_2=300\mathrm{K}$) (dashed). We plot also the thermal part of the force at thermal equilibrium at $T=300\mathrm{K}$ (dotted).

Figure 3

Relative contribution of the thermal component of the pressure (only $l$-dependent part) out of equilibrium for fused silica-silicon system in the configuration ($T_1=300\mathrm{K}$, $T_2=0\mathrm{K}$) (solid), ($T_1=0\mathrm{K}$, $T_2=300\mathrm{K}$) (dashed), and at thermal equilibrium at $T=300\mathrm{K}$ (dotted).

Figure 4

Same of Fig. 2, for the sapphire-fused silica system.

Figure 5

Same as Fig. 3, for the sapphire-fused silica system.

Figure 6

Schematic figure of the two-slab system out of thermal equilibrium.

Figure 7

Dimensionless function $f\left(v\right)$ [see Eqs. (118, 149)] describing the transition between additive and nonadditive regimes. The dashed line presents the asymptotic limit at small $v$.

Figure 8

Relevant length scales and asymptotic behaviors of the surface-rarefied body pressure out of thermal equilibrium. There is a first region given by Eq. (123) where the pressure is additive and coincides with Eq. (122), and a second region, satisfying Eq. (121), where the pressure is given by Eq. (120) and is no longer additive.

Figure 9

Function $f_{\mathrm{PW}}\left(v\right)$ (solid) [Eq. (131)] and its asymptotic limits (dashed) at small [Eq. (135)] and large [Eq. (133)] values of $v$.

Figure 10

Function $f_{\mathrm{EW}}^{\left(1\right)}\left(v\right)$ (solid) [Eq. (140)]. The asymptotic limit at large values of $v$ [Eq. (143)] is shown by the dashed line. The inset demonstrates $v^3$ behavior at small $v$.

Figure 11

Function $f_{\mathrm{EW}}^{\left(2\right)}\left(v\right)$ (solid) [Eq. (141)] and its asymptotic limits (dashed) at small and large [Eq. (145)] values of $v$.