Analysis of discrete reaction-diffusion equations for autocatalysis and continuum diffusion equations for transport
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We analyze both the spatiotemporal behavior of non-linear "reaction" models utilizing reaction-diffusion equations, and spatial transport problems on surfaces and in nanopores utilizing the relevant diffusion or Fokker-Planck equations.
The non-linear "reaction" models involve spatial discrete systems where "particles" reside at the sites of a periodic lattice: particles, X, spontaneously annihilate (X->O) at a specified rate p, and are autocatalytically created given the presence of nearby pairs of particles (O+2X->3X) at rates depending on the local configuration. [This reaction model is equivalent to a spatial epidemic model where sick individuals spontaneously recover (S->H), and healthy individuals are infected by pairs of sick neighbors (H+2S->3S).] The model exhibits a non-equilibrium phase-transition from a populated state to a vacuum state (with no particles) with increasing p. Near this transition, one can consider the propagation of interfaces separating the two states. Planar interfaces exhibit an orientation-dependence (leading to so-called generic two-phase coexistence), and curved interfaces enclosing droplets exhibit even richer behavior. These phenomena are analyzed utilizing the appropriate set of discrete reaction-diffusion equations (corresponding to lattice differential equations).
Diffusive transport of particles between islands or clusters of particles on a surface leads to coarsening of island arrays which can be analyzed by solution of an appropriate boundary value problem for the surface diffusion equation. We extend previous treatments to strongly anisotropic systems. Diffusion and passing of pairs of overdamped Langevin molecules in narrow nanopores can be described by the appropriate Fokker-Planck equations (corresponding to a high-dimensional diffusion equation). We provide the first analysis of this problem focusing on a characterization of the propensity of passing as a function of pore diameter.