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Thermal and potential conductivities of ideal Maxwellian, Fermi and Bose gases are derived by considering the small corrections due to the wave character of gas particles. Potential conductivity is regarded as conductivity due to any potential gradient like electrical, gravitational or chemical ones. A long rectangular channel is considered as a transport domain. The size of the domain in the transport direction is much longer than the mean free path of particles l while the sizes in transverse directions are shorter than l. On the other hand, all sizes of the domain are assumed to be larger than the thermal de Broglie wavelength of particles. Therefore, quantum size effects (QSE) are weak enough to be considered as small corrections on conventional terms. Corrections on thermal and potential conductivities are examined. It is seen that the size and shape of the transport domain become additional control parameters on both conductivities. Since the size dependencies of thermal and electrical conductivities are different, the Lorenz number becomes size and shape dependent and deviations from the Wiedemann–Franz law may be expected in nanoscale due to QSE. Variations of the corrections with chemical potential are analysed.

We point out an apparently overlooked consequence of the boundary conditions obeyed by the electric displacement vector at air-metal interfaces: the continuity of the normal component combined with the quantum mechanical penetration of the electron gas in the air implies the existence of a surface on which the dielectric function vanishes. This, in turn, leads to an enhancement of the normal component of the total electric field. We study this effect for a planar metal surface, with the inhomogenous electron density accounted for by a Jellium model. We also illustrate the effect for equilateral triangular nanoislands via numerical solutions of the appropriate Maxwell equations, and show that the field enhancement is several orders of magnitude larger than what the conventional theory predicts

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