Temperature dependence of Van der Waals forces in classical electrodynamics with classical electromagnetic zero-point radiation

Abstract

Retarded dispersion forces are calculated at finite temperature in the theory of classical electrodynamics with classical electromagnetic zero-point radiation. Expressions are given at all separations for the forces between a polarizable particle and a conducting wall and between two polarizable particles where the particles are represented by electric dipole oscillators immersed in thermal radiation. Also given are the simple expressions holding in the unretarded short-distance limit, and in the asymptotic large-distance limit. At any finite nonzero temperature $T$, the asymptotic large-distance Van der Waals force between two polarizable particles separated by a distance $R\ensuremath{\gg}\frac{\ensuremath{\hbar}c}{\mathrm{KT}}$ is given by the potential $U(R)=\ensuremath{-}\frac{3{\ensuremath{\alpha}}^{2}\mathrm{KT}}{{R}^{6}}$ rather than the Casimir-Polder form for zero temperature, $U(R)=\ensuremath{-}(\frac{23}{4\ensuremath{\pi}})\frac{{\ensuremath{\alpha}}^{2}\ensuremath{\hbar}c}{{R}^{7}}$, where $\ensuremath{\alpha}$ is the static polarizability of each particle. Also the ${R}^{\ensuremath{-}6}$ form holds at high temperatures for any fixed separation. The finite-temperature analysis presented follows directly from earlier classical work at zero temperature since within the classical theory, classical zero-point radiation and classical thermal radiation are treated on the same footing. The classical calculations are easier than those of quantum-electrodynamic perturbation theory but have been shown generally to reproduce the quantum results for dipole-oscillator systems. The forces in the high-temperature limit are shown to agree with the results of classical statistical mechanics and with the use of the Rayleigh-Jeans law for the thermal radiation spectrum. The new results for polarizable particles fit nicely with earlier work by other authors on the finite-temperature corrections to the Casimir effect, the force between uncharged conducting parallel plates. It is emphasized that the force between conducting plates may be regarded as due to the classical boundary conditions at the conductors rather than to any discrete quantum aspects; the Rayleigh-Jeans spectrum also leads to a force between conducting plates and this force is in agreement with the high-temperature limit of previous calculations including zero-point radiation.