  Electronic heat transport along the flux lines in a long ballistic mesoscopic
superconductor cylinder with a radius of the order of several coherence lengths
is investigated theoretically using both semiclassical approach and the full
quantum-mechanical analysis of the Bogoliubov--de Gennes equations. The
semiclassical approach is constructed analogously to the Landauer transport
theory in mesoscopic conductors employing the idea that heat is carried by the
quasiparticle modes propagating along the vortex core. We show that the vortex
heat conductance in a mesoscopic sample is strongly enhanced as compared to its
value for a bulk superconductor; it grows as the cylinder radius decreases.
This unusual behavior results from a strongly increased number of
single-particle transport modes due to giant mesoscopic oscillations of energy
levels, which originate from the interplay between the Andreev reflection at
the vortex core boundary and the normal reflection at the sample edge. We
derive the exact quantum-mechanical expression for the heat conductance and
solve the Bogoliubov--de Gennes equations numerically. The results of numerical
computations confirm the qualitative Landauer-type picture and allow us to take
into account the partial reflections of excitations. We analyze the effect of
surface imperfections on the spectrum of core excitations. We show that the
giant oscillations of core levels and thus the essential features of the heat
transport characteristic to ideal mesoscopic samples hold for a broad class of
surface imperfections as well.
