  We characterize all Gaussian dynamical semigroups in continuous variables
quantum systems of n-bosonic modes which have a thermal Gibbs state as a
stationary solution. This is performed through an explicit relation between the
diffusion and dissipation matrices, which characterize the semigroup dynamic,
and the covariance matrix of the thermal equilibrium state. We also show that
Alicki's quantum detailed-balance condition, based on a Gelfand-Naimark-Segal
inner product, allows the determination of the temperature dependence of the
diffusion and dissipation matrices, and the identification of different
Gaussian dynamical semigroups which shares the same thermal equilibrium state.
