  We investigate the thermodynamic properties of a two-dimensional d-wave
superconductor in the vortex state using a semiclassical approach, and argue
that such an approach is valid for the analysis of the experimental data on
high-temperature superconductors. We develop a formalism where the spatial
average of a physical quantity is written as an integral over the probability
density of the Doppler shift, and evaluate this probability density for several
model cases. The approach is then used to analyse the behavior of the specific
heat and the NMR spin-lattice relaxation rate in a magnetic field. We compare
our results with the experimental measurements, and explain the origin of the
discrepancy between the results from different groups. We also address the
observability of the recently predicted fourfold oscillations of the specific
heat for the magnetic field parallel to the copper oxide planes. We consider
both the orbital and the Zeeman effects, and conclude that at experimentally
relevant temperatures Zeeman splitting does not appreciably reduce the
anisotropy, although it does change the field dependence of the anisotropic
specific heat. We predict a scaling law for the non-exponentially decaying NMR
magnetization, and discuss different approaches to the effective relaxation
rate.
