Energy density of the cubic and spherical cavities with low adiabatic invariant

The Planck radiation spectrum of ideal cubic and spherical cavities with low adiabatic invariants, ... , is discrete and strongly dependent on the cavity geometry and temperature. This behavior is the consequence of the random distribution of the state weights in the cubic cavity and of the random overlapping of the successive multiplet components, in the case of spherical cavity. The total energy density of cavities with low adiabatic invariant, y (obtained by summing up the exact contributions of the eigenvalues and their weights) does not obey any longer Stefan-Boltzmann law. The new law includes a corrective factor depending on rand imposes an exponential-type decrease of the total energy density to zero, when ... . This special quantum regime, defined by limits of principal quantum number or of adiabatic invariant, appears to be similar for cubic and spherical cavities. The total energy density of cavities with low y shows important macroscopic quantum effects over quite large domains of volumes and temperatures.