  A theory of temperature dynamics in many-body systems driven by
time-dependent external sources is introduced. The formalism based on the
combination of the perturbation theory and the fluctuational-electrodynamics
approach in many-body systems. By using response theory, explicit formula for
the temperature and phase shifts is derived and expressed in terms of the
amplitude and phase of external power sources. Although the proposed method is
highly efficient because it can skip the transient response, it is valid when
external powers are weak. As an illustration of this theoretical framework, we
have shown the dynamics of temperatures in one, two, and three degree of
freedom systems driven by sine wave input powers. Finally, we highlighted some
emergent phenomena arising from purely dynamical many-body effects, including
amplification, attenuation, delaying or accelerating temperature responses.
This work could find important applications in the domain of dynamical thermal
management at the nanoscale.
