### 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 language | English |
---|---|

Journal | Physical Review E |

Volume | 94 |

Issue number | 6 |

DOIs | |

Publication status | Published - 19 Dec 2016 |

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### Scopus subject areas

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

### Cite this

*Physical Review E*,

*94*(6). https://doi.org/10.1103/PhysRevE.94.060103

}

*Physical Review E*, vol. 94, no. 6. https://doi.org/10.1103/PhysRevE.94.060103

**Stochastic Laplacian growth.** / Alekseev, Oleg; Mineev-Weinstein, Mark.

Research output

TY - JOUR

T1 - Stochastic Laplacian growth

AU - Alekseev, Oleg

AU - Mineev-Weinstein, Mark

PY - 2016/12/19

Y1 - 2016/12/19

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85006356543&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.94.060103

DO - 10.1103/PhysRevE.94.060103

M3 - Article

AN - SCOPUS:85006356543

VL - 94

JO - Physical Review E

JF - Physical Review E

SN - 1539-3755

IS - 6

ER -