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Non-equilibrium systems under temperature modulation are investigated in thelight of the stochastic thermodynamics. We show that, for small amplitudes ofthe temperature oscillations, the heat flux behaves sinusoidally with time, aresult that allows the definition of the complex heat capacity. The real partof the complex heat capacity is the dynamic heat capacity, and its imaginarypart is shown to be proportional to the rate of entropy production. We alsoshow that the poles of the complex heat capacity are equal to imaginary unitmultiplied by the eigenvalues of the unperturbed evolution operator, and areall located in the lower half plane of the complex frequency, assuring that theKramers-Kronig relations are obeyed. We have also carried out an exactcalculation of the complex heat capacity of a harmonic solid and determined thedispersion relation of the dynamic heat capacity and of the rate of entropyproduction.

Complex heat capacity and entropy production of temperature modulated systems