# Complex heat capacity and entropy production of temperature modulated systems

## TL;DR

This paper explores the behavior of heat flux and entropy production in temperature-modulated non-equilibrium systems, introducing the concept of complex heat capacity and analyzing its properties through stochastic thermodynamics.

## Contribution

It defines the complex heat capacity for temperature-modulated systems and links its imaginary part to entropy production, providing exact calculations for harmonic solids.

## Key findings

- Real part of complex heat capacity is the dynamic heat capacity.
- Imaginary part of complex heat capacity is proportional to entropy production rate.
- Poles of the complex heat capacity relate to eigenvalues of the system's evolution operator.

## Abstract

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.