  Onsager's relations allow one to express the second law of thermodynamics in
terms of the underlying associated currents. These relations, however, are
usually valid only close to equilibrium. Using a quantum phase space
formulation of the second law, we show that open bosonic Gaussian systems also
obey a set of Onsager relations, valid arbitrarily far from equilibrium. These
relations, however, are found to be given by a more complex non-linear
function, which reduces to the usual quadratic form close to equilibrium. This
non-linearity implies that far from equilibrium, there exists a fundamental
asymmetry between entropy flow from system to bath and vice-versa. The
ramifications of this for applications in driven-dissipative quantum optical
setups are also discussed.
