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 Laplacian growth equation, which embraces numerous fundamental free boundary dynamics in nonequilibrium physics. For nonclassical scenarios we introduce virtual point sources, in which presence the action becomes the Kullback-Leibler entropy. Strikingly, this entropy is shown to be the sum of electrostatic energies of layers grown per elementary time unit. Hence the growth probability of the presented nonequilibrium process obeys the Gibbs-Boltzmann statistics, which, as a rule, is not applied out from equilibrium. Each layer's probability is expressed as a product of simple factors in an auxiliary complex plane after a properly chosen conformal map. The action at this plane is a sum of Robin functions, which solve the Liouville equation. At the end we establish connections of our theory with the $\tau$ function of the integrable Toda hierarchy and with the Liouville theory for noncritical quantum strings.

Figure 1

Conformal map $z=f\left(w\right)$ from the exterior of the unit circle to $D_{-}$, so that $\infty =f\left(\infty \right)$, the conformal radius, $r=f^{\prime }\left(\infty \right)>0$, and $A=f(1/\overline{a})$.

Figure 2

(a) Stochastic growth of a single layer, $D_{i+1}/D_i$: Here a thin line is ${\mathrm{\Gamma }}_i=\partial D_i$ formed during the first $i$ time units; a dashed line represents classical (deterministic) LG for a single source at $\infty$ during the $(i+1)\mathrm{st}$ unit, and a solid line, ${\mathrm{\Gamma }}_{i+1}$, is an external boundary of a stochastic layer, $D_{i+1}/D_i$, grown per elementary time unit, $\delta t$. This stochastic layer is equivalent to a classical layer, grown in the presence of $M$ virtual sources located at $A_1,A_2,...,A_M$. (b) Three consecutive fragments of ${\mathrm{\Gamma }}_i$, partitioned onto $N\gg 1$ equal pieces of the size $\sqrt{\hslash }$ after stochastic growth during the $(i+1)\mathrm{st}$ time unit. The heights of grown columns equal to $h_n=\sqrt{\hslash }{k}_n$.