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Phase transition of a Heat equation with Robin's boundary conditions and exclusion process | | Tertuliano Franco
; Patricia Gonçalves
; Adriana Neumann
; | | Date: |
13 Oct 2012 | | Abstract: | 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 $alpha n^{-eta}$, where
$alpha>0$, $etain [0,infty]$, 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 $eta$.
Here, firstly we derive a new proof of the hydrodynamical limit for
$eta=1$, 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
$alpha oinfty$, that weak solution converges to the weak solution of the
Heat Equation without boundary conditions, while if $alpha o 0$, the
convergence is to the weak solution of the Heat Equation with Neumann’s
boundary conditions. | | Source: | arXiv, 1210.3662 | | Services: | Forum | Review | PDF | Favorites | |
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