Stochastic Laplacian growth

Oleg Alekseev, Mark Mineev-Weinstein

    Research output

    3 Citations (Scopus)

    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 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 τ function of the integrable Toda hierarchy and with the Liouville theory for noncritical quantum strings.

    Original languageEnglish
    JournalPhysical Review E
    Volume94
    Issue number6
    DOIs
    Publication statusPublished - 19 Dec 2016

    Fingerprint

    Point Source
    Scenarios
    point sources
    Non-equilibrium
    Entropy
    entropy
    Conformal Map
    Liouville Equation
    Liouville equations
    free boundaries
    Probable
    logarithms
    Free Boundary
    Ludwig Boltzmann
    Logarithm
    Electrostatics
    Argand diagram
    hierarchies
    strings
    Strings

    Scopus subject areas

    • Statistical and Nonlinear Physics
    • Statistics and Probability
    • Condensed Matter Physics

    Cite this

    Alekseev, Oleg ; Mineev-Weinstein, Mark. / Stochastic Laplacian growth. In: Physical Review E. 2016 ; Vol. 94, No. 6.
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    Stochastic Laplacian growth. / Alekseev, Oleg; Mineev-Weinstein, Mark.

    In: Physical Review E, Vol. 94, No. 6, 19.12.2016.

    Research output

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    AB - 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 τ function of the integrable Toda hierarchy and with the Liouville theory for noncritical quantum strings.

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