  Non-equilibrium systems under temperature modulation are investigated in the
light of the stochastic thermodynamics. We show that, for small amplitudes of
the temperature oscillations, the heat flux behaves sinusoidally with time, a
result that allows the definition of the complex heat capacity. The real part
of the complex heat capacity is the dynamic heat capacity, and its imaginary
part is shown to be proportional to the rate of entropy production. We also
show that the poles of the complex heat capacity are equal to imaginary unit
multiplied by the eigenvalues of the unperturbed evolution operator, and are
all located in the lower half plane of the complex frequency, assuring that the
Kramers-Kronig relations are obeyed. We have also carried out an exact
calculation of the complex heat capacity of a harmonic solid and determined the
dispersion relation of the dynamic heat capacity and of the rate of entropy
production.
