  Systems out of equilibrium, in stationary as well as in nonstationary
regimes, display a linear response to energy impulses simply expressed as the
sum of two specific temporal correlation functions. There is a natural
interpretation of these quantities. The first term corresponds to the
correlation between observable and excess entropy flux yielding a relation with
energy dissipation like in equilibrium. The second term comes with a new
meaning: it is the correlation between the observable and the excess in
dynamical activity or reactivity, playing an important role in dynamical
fluctuation theory out-of-equilibrium. It appears as a generalized escape rate
in the occupation statistics. The resulting response formula holds for all
observables and allows direct numerical or experimental evaluation, for example
in the discussion of effective temperatures, as it only involves the
statistical averaging of explicit quantities, e.g. without needing an
expression for the nonequilibrium distribution. The physical interpretation and
the mathematical derivation are independent of many details of the dynamics,
but in this first part they are restricted to Markov jump processes and
overdamped diffusions.
