Switching Interacting Particle Systems: Scaling Limits, Uphill Diffusion and Boundary Layer
Simone Floreani, Cristian Giardinà, Frank den Hollander, Shubhamoy Nandan, Frank Redig
Abstract
Abstract This paper considers three classes of interacting particle systems on $${{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> : independent random walks, the exclusion process, and the inclusion process. Particles are allowed to switch their jump rate (the rate identifies the type of particle) between 1 ( fast particles ) and $$\epsilon \in [0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> ( slow particles ). The switch between the two jump rates happens at rate $$\gamma \in (0,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by $$N^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , time by $$N^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , the switching rate by $$N^{-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> , and letting $$N\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Fick’s law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on $$[N]=\{1,\ldots ,N\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>N</mml:mi> <mml:mo>]</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , adding boundary reservoirs at sites 1 and N of fast and slow particles, respectively. Inside [ N ] particles move as before, but now particles are injected and absorbed at sites 1 and N with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Fick’s law made possible by the switching between types. We rescale the microscopic steady-state density profile and steady-state current and obtain the steady-state solution of a boundary-value problem for the double diffusivity model.