October 18, 2017
Mark Mineev-Weinstein (UFRN, Brasil)
Stochastic Laplacian growth
A point source on a plane constantly emits particles which rapidly diffuse and then stick to a growing cluster. The growth probability of a cluster is presented as a sum over all possible scenarios leading to the same final shape. The classical point for the action, defined as a minus logarithm of the growth probability, describes the most probable scenario and reproduces the integrable Laplacian growth equation, which embraces numerous fundamental nonlinear free boundary dynamics in non-equilibrium physics. Strikingly, the entropy for non-classical scenarios is shown to be linearly proportional to the electrostatic energies of Coulomb interaction of charged liquid, uniformly occupying the grown domain, with itself (minus a non-significant integral). Hence the growth probability of the presented non-equilibrium process obeys the Gibbs-Boltzmann statistics, which is known to be inapplicable far from equilibrium. The domain growth probability is expressed as a product of simple factors in an auxiliary complex plane after a properly chosen conformal map.
Based on these result, I will develop the statistical mechanics for a stochastic Laplacian growth. If time permits, I will also outline the program of dynamical pattern selection in Laplacian growth and share the plan of obtaining the fractal spectrum of grown clusters in the long time asymptotics