Are there known examples of spatially inhimogeneous PDE appearing as hydrodynamic equations of interacting particle systems? In particular, I wonder whether a spatially inhomogeneous reaction diffusion equation such as $$u_t - \Delta u = f(x, u)$$ with $f$ non-constant in $x$ appears in the literature. Intuitively, it seems like this could be done by making the generator of the particle system depend on space (in a non-translationally invariant way), but everything I have seen in the literature treats the translationally invariant case (and presumably this is the reason the coefficients of the resulting equations depend only on the solution $u$, not on the space variable $x$).
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
A very well studied nonlinear equation of this type is the Gross–Pitaevskii equation,
$$i\hbar\frac{\partial\Psi(\mathbf{r},t)}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) + g(\mathbf{r})\vert\Psi(\mathbf{r},t)\vert^2\right)\Psi(\mathbf{r},t)$$
It's a quantum problem, so there is an imaginary unit $i$ in front of the time derivative. For a numerical solution one usually makes a Wick rotation $t\mapsto i\tau$ to remove the $i$ and work with a diffusion equation (here is an example of such a procedure). Spatial inhomogeneities play a crucial role in applications to a cold-atom trap.
answered Sep 6, 2017 at 13:05
-
$\begingroup$ I don't see how this addresses my question. The example you provided uses a finite-difference method, but I'm interested in interacting particle systems and their scaling limits, not numerics (see the review article by Claudio Landim). $\endgroup$ Commented Sep 7, 2017 at 16:54
-
$\begingroup$ well, the Gross–Pitaevskii equation does describe a strongly interacting system, a collection of atoms that condense into a collective state (Bose-Einstein condensate) at low temperatures; it can hardly get more "interacting" than that.... $\endgroup$ Commented Sep 7, 2017 at 17:32