No abstract

Two- and three-dimensional Helmholtz equations in wedge-shaped and conical domains are addressed by the random walk method. The solutions of the Dirichlet problems in such domains are represented as mathematical expectations of specified functionals on trajectories of multidimensional random motions whose radial components run in a complex space while the angular components remain real valued. This technique is applied to the Sommerfeld problem of diffraction by a semi-infinite screen which is explicitly solved here in the probabilistic form. The numerical results confirm the efficiency of the random walk approach to the analysis of wave propagation.

Heat exchange between closely positioned bodies has become an important issue for many areas of modern technology including, but not limited to, integrated circuits, atomic force microscopy and high-density magnetic recording, which deal with bodies separated by gaps as narrow as a few nanometers. It is now recognized that heat transport across a gap of sub-micron width noticeably exceeds the limit set by the conventional theory of radiative heat transfer. This papers shows than if the gap's width is below a certain value, estimated as about 10 nanometers for silicon at room temperature, then, in addition to electromagnetic radiation, significant heat is a also carried by acoustic waves. Moreover, as the width of the gap decreases below about 5 nanometers, acoustic waves rapidly become the dominant heat carrier.

This paper generalizes the well-known acoustic mismatch theory of Kapitsa interface thermal resistance by taking into consideration a broad class of thermal vibrations that were excluded from that theory by the imposition of the Sommerfeld radiation condition, which is required for the theory of sound but is not relevant for the analysis of heat transport. This extension preserves the main ideas of the acoustic mismatch theory but provides much more reasonable estimates for the interface resistance. The predictions of the new theory are compared with various published experimental results for the thermal resistance between liquid helium at low temperatures and several different metals (Ag, Au, Cu, Pb and Pt). The computations are straightforward and require only well-known material parameters. The predictions agree with the experiments to within their stated range of accuracy.

Planck's law of thermal radiation is limited to equilibrium systems that have a definite temperature and do not carry any heat flux. Here we extend it to steady‐state systems with a constant heat flux. The obtained formulas explicitly describe the spectrum of thermal radiation in every direction and provide a sound basis for the self‐consistent analysis of radiative heat transport across interfaces, gaps, layered and other important structures.