1210.3662 (Tertuliano Franco et al.)

Phase transition of a Heat equation with Robin's boundary conditions and
exclusion process    [PDF]

Tertuliano Franco, Patricia Gonçalves, Adriana Neumann

We consider the exclusion process evolving in the one-dimensional discrete torus, with a bond whose conductance slows down the passage of particles across it. We chose the conductance at that bond as $\alpha n^{-\beta}$, where $\alpha>0$, $\beta\in [0,\infty]$, and $n$ is the scale parameter. In \cite{fgn}, by rescaling time diffusively, it was proved that the hydrodynamical limit depends strongly on the regime of $\beta$. Here, firstly we derive a new proof of the hydrodynamical limit for $\beta=1$, by showing that the hydrodynamic equation, is a Heat Equation with Robin's boundary conditions that depend on $\alpha$. As a consequence, the weak solution of the hydrodynamic equation given in \cite{fgn}, involving a generalized derivative $\frac{d}{du} \frac{d}{dW}$, coincides with the weak solution of a Heat Equation with Robin's boundary conditions. Secondly, arguing by energy estimates, we prove a phase transition for the weak solution of a Heat Equation with Robin's boundary conditions. Namely, if $\alpha\to\infty$, that weak solution converges to the weak solution of the Heat Equation without boundary conditions, while if $\alpha\to 0$, the convergence is to the weak solution of the Heat Equation with Neumann's boundary conditions.

View original: http://arxiv.org/abs/1210.3662