  We present a numerical study of thermodynamical properties of dimerized
frustrated Heisenberg chains down to extremely low temperatures with
applications to CuGeO$_3$. A variant of the finite temperature density matrix
renormalization group (DMRG) allows the study of the dimerized phase previously
unaccessible to ab initio calculations. We investigate static dimerized systems
as well as the instability of the quantum chain towards lattice dimerization.
The crossover from a quadratic response in the free energy to the distortion
field at finite temperature to nonanalytic behavior at zero temperature is
studied quantitatively. Various physical quantities are derived and compared
with experimental data for CuGeO$_3$ such as magnetic dimerization, critical
temperature, susceptibility and entropy.
