Temperature gradient and Fourier's law in gradient-mass harmonic systems

The heat flow and thermal profile in a one-dimensional (1D) harmonic lattice with coordinate-dependent masses have been calculated in the thermodynamic limit. It is shown in the particular example of a 1D harmonic lattice with linearly increasing masses that in standard Langevin conditions of contact, a temperature gradient can form, and Fourier's law can be obeyed.

Figure 1

Schematic of a harmonic system with coordinate-dependent masses.

Figure 2

Dispersion relation of the vibration energy in a harmonic chain with linearly increasing masses (5) plotted vs the wave vector for different relations between the boundary masses ${\omega }_{0,1}=\sqrt{2K/M_{0,1}}$. For energies $\omega <\sqrt{2}{\omega }_0$, the system operates with delocalized phonons, and for $\sqrt{2}{\omega }_0<\omega <\sqrt{2}{\omega }_1$, with localized gradons.