Article

Journal of Statistical Physics

, 144:1308

First online:

Schloegl’s Second Model for Autocatalysis on a Cubic Lattice: Mean-Field-Type Discrete Reaction-Diffusion Equation Analysis

  • Chi-Jen WangAffiliated withAmes Laboratory—USDOE, Iowa State UniversityDepartments of Mathematics, Iowa State University
  • , Xiaofang GuoAffiliated withAmes Laboratory—USDOE, Iowa State UniversityDepartments of Mathematics, Iowa State UniversityDepartment of Physics & Astronomy, Iowa State University
  • , Da-Jiang LiuAffiliated withAmes Laboratory—USDOE, Iowa State University
  • , J. W. EvansAffiliated withAmes Laboratory—USDOE, Iowa State UniversityDepartments of Mathematics, Iowa State UniversityDepartment of Physics & Astronomy, Iowa State University Email author 

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Abstract

Schloegl’s second model for autocatalysis on a hypercubic lattice of dimension d≥2 involves: (i) spontaneous annihilation of particles at lattice sites with rate p; and (ii) autocatalytic creation of particles at vacant sites at a rate proportional to the number of diagonal pairs of particles on neighboring sites. Kinetic Monte Carlo simulations for a d=3 cubic lattice reveal a discontinuous transition from a populated state to a vacuum state as p increases above p=p e . However, stationary points, p=p eq (≤p e ), for planar interfaces separating these states depend on interface orientation. Our focus is on analysis of interface dynamics via discrete reaction-diffusion equations (dRDE’s) obtained from mean-field type approximations to the exact master equations for spatially inhomogeneous states. These dRDE can display propagation failure absent due to fluctuations in the stochastic model. However, accounting for this anomaly, dRDE analysis elucidates exact behavior with quantitative accuracy for higher-level approximations.

Keywords

Schloegl’s second model Generic two-phase coexistence Discrete reaction-diffusion equations Interface propagation