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Heat diffusion in a two-dimensional thermal fuse model

We present numerical studies of electrical breakdown in disordered materials using a two-dimensional thermal fuse model with heat diffusion. A conducting fuse is heated locally by a Joule heating term. Heat diffuses to neighbouring fuses by a diffusion term. When the temperature reaches a given threshold, the fuse breaks and turns into an insulator. The time dynamics is governed by the time scales related to the two terms, in the presence of quenched disorder in the conductances of the fuses. For the two limiting domains, when one time scale is much smaller than the other, we find that the global breakdown time $t_r$ follows $t_r\sim I^2$ and $t_r\sim L^2$, where $I$ is the applied current, and $L$ is the system size. However, such power law does not apply in the intermediate domain where the competition between the two terms produces a subtle behaviour.