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Mark Mineev-Weinstein

Publications and source records attributed to Mark Mineev-Weinstein.

15 recordsLinked to original sources
arXiv2024

Variational Pattern Selection

We address pattern selection problems in nonlinear interface dynamics by maximizing the entropy of the most probable (classical) scenario associated with the processes. This variational principle we applied to well-known selection problems in a Hele-Shaw cell: stationary Saffman-Taylor finger in a channel and self-similar finger in a wedge. The obtained results excellently agree with experiments. We also address the universal fjord opening angle. This principle complements a non-variational selection developed earlier. Surface tension is not needed for selection in both approaches, contrary to the common belief.

nlin.PS↗
arXiv2024

Multiple bubble dynamics and velocity selection in Laplacian growth without surface tension

A new selection phenomenon in nonlinear interface dynamics is predicted. A generic class of exact regular unsteady multi-bubble solutions in a Hele-Shaw cell is presented. These solutions show that the case where the asymptotic bubble velocity, $U$, is twice greater than the velocity, $V$, of the uniform background flow, i.e., $U = 2V$, is the only attractor of the dynamics. Contrary to common belief, the predicted velocity selection requires neither surface tension nor other external regularization.

physics.flu-dyn↗
arXiv2019

Bubble dynamics in a Hele-Shaw channel and velocity selection without surface tension

Inviscid bubble dynamics in a viscous fluid, moving with velocity $V$ far from the bubble, is considered. The Cauchy problem of recovering the bubble evolution from its initial shape is completely solved without surface tension. The well-posed (after a Tikhonov regularization) dynamical system provides an extensive list of unsteady closed form solutions due to integrability. A new (rational) class of solutions is obtained and added to the logarithmic class found earlier (Phys. Rev. E 89, 061003(R), 2014). The only attractor selects the observable bubble shape and velocity $U = 2V$ from the continuum of possible solutions. The attractor is asymptotically stable. Numerical results illustrate the most salient aspects of the bubble dynamics.

physics.flu-dyn↗
arXiv2017

Statistical mechanics of stochastic growth phenomena

We develop statistical mechanics for stochastic growth processes as applied 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 (quasi-static) 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 planar domains. Transitions between these domains are described by the stochastic Laplacian growth equation, while the transitional probabilities coincide with the free-particle propagator on the infinite dimensional complex manifold with the Kähler metric.

cond-mat.stat-mech↗
arXiv2016

Theory of stochastic Laplacian growth

We generalize the diffusion-limited aggregation by issuing many randomly-walking particles, which stick to a cluster at the discrete time unit providing its growth. Using simple combinatorial arguments we determine probabilities of different growth scenarios and prove that the most probable evolution is governed by the deterministic Laplacian growth equation. A potential-theoretical analysis of the growth probabilities reveals connections with the tau-function of the integrable dispersionless limit of the two-dimensional Toda hierarchy, normal matrix ensembles, and the two-dimensional Dyson gas confined in a non-uniform magnetic field. We introduce the time-dependent Hamiltonian, which generates transitions between different classes of equivalence of closed curves, and prove the Hamiltonian structure of the interface dynamics. Finally, we propose a relation between probabilities of growth scenarios and the semi-classical limit of certain correlation functions of "light" exponential operators in the Liouville conformal field theory on a pseudosphere.

cond-mat.stat-mech↗
arXiv2016

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 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.

cond-mat.stat-mech↗
arXiv2014

Laplacian Growth I: Finger Competition and Formation of a Single Saffman-Taylor Finger without Surface Tension: An Exact Result

We study the exact non-singular zero-surface tension solutions of the Saffman-Taylor problem for all times. We show that all moving logarithmic singularities a_k(t) in the complex plane ω= e^{iϕ}, where ϕis the stream function, are repelled from the origin, attracted to the unit circle and eventually coalesce. This pole evolution describes essentially all the dynamical features of viscous fingering in the Hele-Shaw cell observed by Saffman and Taylor [Proc. R. Soc. A 245, 312 (1958)], namely tip-splitting, multi-finger competition, inverse cascade, and subsequent formation of a single Saffman-Taylor finger.

patt-sol↗
arXiv2013

Selection of the Taylor-Saffman Bubble does not Require Surface Tension

A new general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex-plane. It is then demonstrated that the only stable fixed point (attractor) of the non-singular bubble dynamics corresponds precisely to the selected pattern. This thus shows that, contrary to the established theory, bubble selection in a Hele-Shaw cell does not require surface tension. The solutions reported here significantly extend previous results for a simply-connected geometry (finger) to a doubly-connected one (bubble). We conjecture that the same selection rule without surface tension holds for Hele-Shaw flows of arbitrary connectivity. We also believe that this mechanism can be found in other, similarly described, selection problems.

nlin.PS↗
arXiv2009

Planar elliptic growth

The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for area-preserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integration for closed form solutions are identified as the singularities of the Schwarz function, which are located both inside and outside the moving contour. Well-posedness of the recovery of the elliptic operator governing the process from the continuum of interfaces parametrized by time is addressed and two examples of exact solutions of elliptic growth are presented.

nlin.SI↗
arXiv2008

Random Matrices in 2D, Laplacian Growth and Operator Theory

Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.

nlin.SI↗
arXiv2002

A note on the extension of the polar decomposition for the multidimensional Burgers equation

It is shown that the generalizations to more than one space dimension of the pole decomposition for the Burgers equation with finite viscosity and no force are of the form u = -2 viscosity grad log P, where the P's are explicitly known algebraic (or trigonometric) polynomials in the space variables with polynomial (or exponential) dependence on time. Such solutions have polar singularities on complex algebraic varieties.

nlin.SI↗
arXiv2000

Integrable Structure of Interface Dynamics

We establish the equivalence of a 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.

nlin.SI↗
arXiv1999

Whitham-Toda Hierarchy in the Laplacian Growth Problem

The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version of the string equation that selects the solution to the hierarchy.

solv-int↗
arXiv1998

Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: an Exact Result

Using our exact time-depending solutions, we solve the Saffman-Taylor finger selection problem in the absence of surface tension by showing that an arbitrary interface in a Hele-Shaw cell evolves to a single uniformly advancing finger occupying one half of the channel width. This result contradicts the generally accepted belief that surface tension is indispensable for the selection of the one-half-width finger.

patt-sol↗