Stochastic Laplacian growth

Abstract

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 non-equilibrium physics. For non-classical 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 non-equilibrium 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 non-critical quantum strings. The goal of this work is to unify two fundamental highly non-equilibrium processes, Laplacian growth (LG) [1][2][3][4][5], which is deterministic interface dynamics, and diffusion-limited aggregation (DLA) [6] -a discrete universal stochastic fractal growth. These remarkable processes have a lot in common and were suspected to be deeply related [7][8][9][10][11].Laplacian growth raised enormous interest in physics because of (i) its impressively wide applicability ranging from solidification and oil recovery to biological growth [1], (ii) remarkable universal asymptotic shapes, it exhibits [1,[12][13][14][15][16], and (iii) discoveries of deep intriguing connections of LG to quantum gravity [2] and the quantum Hall effect [17]. In mathematics the Laplacian growth appears so exciting because it possesses beautiful and powerful properties, unusual for most of nonlinear PDEs, such as infinitely many conservation laws [18] and closed form exact solutions [16,[19][20][21][22]. A new splash of intense activity in LG (see [3] for a review) was provoked by the work [2], where strong connections of LG with major integrable hierarchies and the theory of random matrices were established.Mathematical formulation of LG is (deceptively) simple: a droplet of air, D + (t), where t is time, is surrounded by a viscous fluid, D − (t) = C/D + (t), called D(t) for simplicity. Both liquids are sandwiched between two parallel close plates. Fluid velocity in D(t) obeys the Darcy law, v = −∇p (in scaled units), where p(z,z) is pressure and z = x + iy is a complex coordinate on the plane. Because of incompressibility, ∇·v = 0, then ∇ 2 p = 0 in D, except points with sources, which provide growth. Also, p = 0 at the interface, Γ(t) = ∂D(t), between two fluids, if...

Statistical mechanics of stochastic growth phenomena

“…The fluctuation theory of the Laplacian growth also has a rich mathematical structure if one uses its dual description (known as the inverse potential problem) in terms of the conformal maps. The growth probabil-ity (30) can be obtained also by considering the "entropy of the layer" that is composed of particles which attach to the boundary of the domain per time unit [22,23]. This observation allows to relate the Laplacian growth problem to the theory of random partitions [24], and might have far reaching consequences.…”

“…The symmetry relations between the harmonic moments suggest to chose T k 's as independent variables. Variations of T k 's also cause the deviation of the chemical potential for newly incoming particles from its "deterministic" value (22). Since u n (ζ)|dζ| = Ṡ(ζ)dζ/2i, we transform the line integral in (22) in the contour one.…”

“…Variations of T k 's also cause the deviation of the chemical potential for newly incoming particles from its "deterministic" value (22). Since u n (ζ)|dζ| = Ṡ(ζ)dζ/2i, we transform the line integral in (22) in the contour one. The Laplacian growth equation (17) relates the change of the Schwarz function with the variation of the electric potential, π Ṡ+ (z) = − q m /(z − A m ).…”

Integrability-preserving regularizations of Laplacian Growth

“…On the other hand, DLA is in turn marked by its formulation as a discrete aggregation process, and by the difficulty of defining probability measures on spaces of curves. When attempting to describe DLA by starting from purely discrete probability measures (such as in the recent works [1][2][3]), the main difficulty stems from the fact that do not have yet a generalized Central Limit Theorem for aggregation processes of these types.…”

“…Let now K be a compact set in C, and condensers of special type C R = {|z| ≥ R, K} for R → ∞. The function 1 cap C R − 1 2π log R increases with increasing R and the limit cap…”