This paper considers three classes of interacting particle systems on Z: 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 ϵ∈ [0 , 1] (slow particles). The switch between the two jump rates happens at rate γ∈ (0 , ∞). 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, time by N2, the switching rate by N 2, and letting N→ ∞. The limit equations for the macroscopic densities associated to the fast and slow particles is the wellstudied double diffusivity model. This system of reactiondiffusion 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 outofequilibrium properties, we analyse the system on [N] = { 1 , … , N} , 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 steadystate density profile and the steadystate 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 singletype particle system, is a violation of Fick’s law made possible by the switching between types. We rescale the microscopic steadystate density profile and steadystate current and obtain the steadystate solution of a boundaryvalue problem for the double diffusivity model.
Switching Interacting Particle Systems: Scaling Limits, Uphill Diffusion and Boundary Layer / Floreani, Simone; Giardinà, Cristian; Frank den Hollander, ; Nandan, Shubamoy; Redig, Frank.  In: JOURNAL OF STATISTICAL PHYSICS.  ISSN 00224715.  186:3(2022), pp. 145. [10.1007/s10955022028787]
Switching Interacting Particle Systems: Scaling Limits, Uphill Diffusion and Boundary Layer
Cristian Giardinà;
2022
Abstract
This paper considers three classes of interacting particle systems on Z: 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 ϵ∈ [0 , 1] (slow particles). The switch between the two jump rates happens at rate γ∈ (0 , ∞). 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, time by N2, the switching rate by N 2, and letting N→ ∞. The limit equations for the macroscopic densities associated to the fast and slow particles is the wellstudied double diffusivity model. This system of reactiondiffusion 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 outofequilibrium properties, we analyse the system on [N] = { 1 , … , N} , 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 steadystate density profile and the steadystate 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 singletype particle system, is a violation of Fick’s law made possible by the switching between types. We rescale the microscopic steadystate density profile and steadystate current and obtain the steadystate solution of a boundaryvalue problem for the double diffusivity model.File  Dimensione  Formato  

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