We consider the exclusion process in the one-dimensional discrete torus with \(N\) points, where all the bonds have
conductance one, except a finite number of slow bonds, with conductance \(N^{−\beta}\), with \(\beta\in[0,\infty)\).
We prove that the time evolution of the empirical density of particles, in the diffusive scaling,
has a distinct behavior according to the range of the parameter \(\beta\). If
\(\beta\in [0, 1)\), the hydrodynamic limit is given by the usual heat equation. If \(\beta = 1\), it is given by a parabolic equation involving an
operator \(\frac{d}{dx}\frac{d}{dW}\), where \(W\) is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic
point related to the slow bond. If \(\beta\in (1, \infty)\), it is given by the heat equation with Neumann’s boundary conditions, meaning no passage through the slow bonds in the continuum.

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Franco_IHP.pdf