Heat Conduction in a Two-Dimensional Harmonic Crystal with Disorder

We study the problem of heat conduction in a mass-disordered two-dimensional harmonic crystal. Using two different stochastic heat baths, we perform simulations to determine the system size ($L$) dependence of the heat current ($J$). For white noise heat baths we find that $J\sim 1/L^{\alpha }$ with $\alpha \approx 0.59$, while correlated noise heat baths give $\alpha \approx 0.51$. A special case with correlated disorder is studied analytically and gives $\alpha =3/2$, which agrees also with results from exact numerics.

Figure 1

Temperature at all the sites of a $8\times 8$ fully disordered lattice, from simulations and from the exact solution. The inset shows the disorder-averaged temperature profiles for different system sizes and seems to approach a linear form.

Figure 2

Plot of disorder-averaged-current versus system size for the two different heat baths and for the case of correlated disorder with white noise. For the full disorder cases, the solid lines are fits to the last three points and have slopes 0.59 and 0.51. For the case of correlated disorder, the slope from exact numerics (and also simulations) is compared to 1.5, which is what one expects analytically.