gszep
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6 years ago After discussions with Chiara Cammarota this question can be addressed analytically using Theory of Phase Ordering Kinetrics [1]. Correlation functions have been derived for the Ising Model [2] (see Equation 6) which tell us that the size of regions grow as a function of time L ~ sqrt( t )
The motion of the boundary is given by its local curvature [1] ( @luca-cardelli see Equation 18) In particular, the detailed shape of the potentials are not important: the main role of the double-well potential V(φ) is to establish and maintain well-defined domain walls
Do these results hold for Approximate-Majority or do we seek for a reaction-diffusion extension? A good starting point is finding the form for the free energy F as given in Section 4.3 Curvature-driven Domain Growth [3] for the Approximate-Majority network.
[1] Bray, A. J. (1995). Theory of Phase Ordering Kinetics. [2] Biroli, G. (2005). A Crash Course on Aging. [3] Biroli, G. (2015). Slow Relaxations and Non-Equilibrium Dynamics in Classical and Quantum Systems.
The boundary between X (blue) and Y(yellow) seems to equilibrate as show below for equal diffusivity. For unequal diffusivity the boundary expands into the direction of the less diffusive chemical.