  It is known that the origin of the deviations from standard thermodynamics
proceed from the strong coupling to the bath. Here, it is shown that these
deviations are related to the power spectrum of the bath. Specifically, it is
shown that the system thermal-equilibrium-state cannot be characterized by the
canonical Boltzmann's distribution in quantum mechanics. This is because the
uncertainty principle imposed a lower bound of the dispersion of the total
energy of the system that is dominated by the spectral density of the bath.
However, in the classical case, for a wide class of systems that interact via
central forces with pairwise-self-interacting environment, the system thermal
equilibrium state is exactly characterized by the canonical Boltzmann
distribution. As a consequence of this analysis and taking into account all
energy scales in the system and reservoir interaction, an effective coupling to
the environment is introduced. Sample computations in different regimes
predicted by this effective coupling are shown. Specifically, in the strong
coupling effective regime, the system exhibits deviations from standard
thermodynamics and in the weak coupling effective regime, quantum features such
as stationary entanglement are possible at high temperatures. Moreover, it is
known that the spectrum of thermal baths depends on the non-Markovian
character. Hence, non-Markovian interactions have a important role in the
thermal equilibrium state of physical systems. For example, in quantum
optomechanics is looked up to cool the mechanical system through an auxiliary
system which generally is a cavity. This cooling process takes into account the
non-Markovian interaction and as it is shown here, it is more effective than if
we use only the Markovian approximation in the equation of motion for the
different modes.
