Statistical mechanics of stochastic growth phenomena

We develop statistical mechanics for stochastic growth processes and apply it to Laplacian growth by using its remarkable connection with a random matrix theory. The Laplacian growth equation is obtained from the variation principle and describes adiabatic (quasistatic) thermodynamic processes in the two-dimensional Dyson gas. By using Einstein's theory of thermodynamic fluctuations we consider transitional probabilities between thermodynamic states, which are in a one-to-one correspondence with simply connected domains occupied by gas. Transitions between these domains are described by the stochastic Laplacian growth equation, while the transitional probabilities coincide with a free-particle propagator on an infinite-dimensional complex manifold with a Kähler metric.

Figure 1

(a) The electronic droplet in the external potential; the pointlike charges (impurities) are marked by $\otimes$. (b) The support of eigenvalues $D_e$ (solid line), and the background charge occupying the domain $D$ (dashed line).