  We construct lower bounds to the spectral gap of a family of Lindblad
generators known as Davies maps. These maps describe the thermalization of
quantum systems weakly coupled to a heat bath. The steady state of these
systems is given by the Gibbs distribution with respect to the system
Hamiltonian. The bounds can be evaluated explicitly, when the eigenbasis and
the spectrum of the Hamiltonian is known. A crucial assumption is that the
spectrum of the Hamiltonian is non-degenerate. Furthermore, we provide a
counterexample to the conjecture, that the convergence rate is always
determined by the gap of the associated Pauli master equation. We conclude,
that the full dynamics of the Lindblad generator has to be considered. Finally,
we present several physical example systems for which the bound to the spectral
gap is evaluated.
