Dispersion relation and frequency-dependent thermal conductivity of the two-temperature systems

We investigate analytically the complex-valued dispersion relation for two-temperature systems with coupling. Based on this dispersion relation, we obtain and analyze the real and imaginary parts of the wave number as well as phase velocity and penetration depth. Furthermore, an effective apparent thermal conductivity is introduced, which depends on the frequency of external thermal disturbances due to coupling effects. It is shown that values of thermal conductivity at high frequencies are drastically reduced compared to low frequencies. The onset of the decrease occurs at a frequency threshold of the order of inverse of characteristic time for energy exchange between subsystems (coupling time). At this frequency, the energy exchange between the subsystems reaches its maximum value and the local nonequilibrium (non-Fourier) effects play the most important role. This work establishes a theoretical basis and opens possibilities for controlling and manipulating heat transfer in heterogeneous systems including composite and thermal metamaterials.