Joint Seminar on Wednesdays

Joint seminar on Mathematical Physics
of National Research University HSE and Skoltech Center for Advanced Studies
on Wednesdays at 17.30 at aud. 110 of the Mathematics Department of NRU HSE (6 Usacheva)

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