Unsteady two-temperature heat transport in mass-in-mass chains

We investigate the unsteady heat (energy) transport in an infinite mass-in-mass chain with a given initial temperature profile. The chain consists of two sublattices: the $\beta$-Fermi-Pasta-Ulam-Tsingou (FPUT) chain and oscillators (of a different mass) connected to each FPUT particle. Initial conditions are such that initial kinetic temperatures of the FPUT particles and the oscillators are equal. Using the harmonic theory, we analytically describe evolution of these two temperatures in the ballistic regime. In particular, we derive a closed-form fundamental solution and solution for a sinusoidal initial temperature profile in the case when the oscillators are significantly lighter than the FPUT particles. The harmonic theory predicts that during the heat transfer the temperatures of sublattices are significantly different, while initially and finally (at large times) they are equal. This may look like an artifact of the harmonic approximation, but we show that it is not the case. Two distinct temperatures are also observed in the anharmonic case, even when the heat transport regime is no longer quasiballistic. We show that the value of the nonlinearity coefficient required to equalize the temperatures strongly depends on the particle mass ratio. If the oscillators are much lighter than the FPUT particles, then a fairly strong nonlinearity is required for the equalization.

Figure 1

The mass-in-mass chain, consisting of the $\beta$-FPUT chain and additional nonlinear oscillators.

Figure 2

Acoustic (blue lines) and optical (black lines) branches of the dispersion relation (a) and corresponding group velocities (b) for different mass ratios: $\gamma =2$ (solid line), $\gamma =1$ (dashed line), $\gamma =\frac{1}{2}$ (dash-dotted line), and $\gamma =\frac{1}{5}$ (dotted line). The bandgap as a function of the mass ratio $\gamma$ (c).

Figure 3

Contributions of acoustic (a) and optical (b) branches of dispersion relation to the fundamental solution [Eq. (18)] for $\gamma =2$(solid line), $1$(dashed line), $1/2$(dash dotted line), and $1/10$(dotted line). Vertical asymptotes at $x=tmax{v}_j^g$ (red dashed lines) are also shown.

Figure 4

Decay of the amplitudes $A_{11}$ (a) and $A_{22}$ (b) of the kinetic temperatures for $\gamma =2$. Analytical (red solid line) and numerical (red circles) solutions are shown. Contributions of acoustic (black solid line) and optical (blue line) branches of dispersion relation are shown.

Figure 5

Decay of the amplitudes $A_{11}$ (a) and $A_{22}$ (b) of the kinetic temperatures for $\gamma =1/10$. Analytical (red solid line) and numerical (red circles) solutions are shown. Contributions of acoustic (black solid line) and optical (blue line) branches of dispersion relation are shown.

Figure 6

The complete solution [Eq. (26), red solid line] and approximate solution [Eq. (30), black dashed line] for the amplitudes $A_{11}$ (a) and $A_{22}$ (b) at $\gamma =1/10$.

Figure 7

The amplitudes $A_{11}$ (a) and $A_{22}$ (b) of sinusoidal temperature profiles in the weakly anharmonic case for $\gamma =2$. The analytical solution [Eq. (26), solid line] and numerical simulation results for $\tilde{\beta }=0.05$ (blue circles), $\tilde{\beta }=0.1$ (red asterisks) are shown.

Figure 8

The amplitudes $A_{11}$ (blue line) and $A_{22}$ (black line) of sinusoidal temperature profiles and their difference (red line) in case of strong anharmonicity ($\tilde{\beta }=2$) for $\gamma =2$.

Figure 9

The amplitudes $A_{11}$ (a) and $A_{22}$ (b) of sinusoidal temperature profiles in the weakly anharmonic case for $\gamma =1/10$. The analytical solution [Eq. (26), solid line] and numerical simulation results for $\tilde{\beta }=0.05$ (blue circles), $\tilde{\beta }=0.1$ (red asterisks) are shown.

Figure 10

The amplitudes $A_{11}$ (blue line) and $A_{22}$ (black line) of sinusoidal temperature profiles and their difference (red line) in the case of strong anharmonicity [$\tilde{\beta }=2$ (a), $\tilde{\beta }=100$ (b)] for $\gamma =1/10$.

Figure 11

Dependence of the maximum difference of temperatures on the nonlinearity coefficient for $\gamma =2$ (blue circles) and $\gamma =0.1$ (black squares). Dashed horizontal lines correspond to maximum difference in the harmonic case $\tilde{\beta }=0$.