  The driving force of the dynamical system can be decomposed into the gradient
of a potential landscape and curl flux (current). The fluctuation-dissipation
theorem (FDT) is often applied to near equilibrium systems with detailed
balance. The response due to a small perturbation can be expressed by a
spontaneous fluctuation. For non-equilibrium systems, we derived a generalized
FDT that the response function is composed of two parts: (1) a spontaneous
correlation representing the relaxation which is present in the near
equilibrium systems with detailed balance; (2) a correlation related to the
persistence of the curl flux in steady state, which is also in part linked to a
internal curvature of a gauge field. The generalized FDT is also related to the
fluctuation theorem. In the equal time limit, the generalized FDT naturally
leads to non-equilibrium thermodynamics where the entropy production rate can
be decomposed into spontaneous relaxation driven by gradient force and house
keeping contribution driven by the non-zero flux that sustains the
non-equilibrium environment and breaks the detailed balance.
