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Bernoulli's Equation for Flow Along a Streamline01:30

Bernoulli's Equation for Flow Along a Streamline

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Left passage probability of Schramm-Loewner Evolution.

M N Najafi1

  • 1Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran. morteza.nattagh@gmail.com

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 16, 2013
PubMed
Summary

We investigated the left passage probability for a variant of Schramm-Loewner Evolution (SLE(κ,ρ[over arrow])). The study found this probability depends significantly on boundary conditions, with a general solution for large boundary values.

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Area of Science:

  • Mathematical Physics
  • Probability Theory
  • Complex Analysis

Background:

  • Schramm-Loewner Evolution (SLE) describes random curves in 2D. SLE(κ,ρ[over arrow]) is a variant lacking conformal invariance but possessing self-similarity due to preferred boundary points.
  • Understanding the behavior of these curves is crucial for various statistical physics models, including loop-erased random walks and Abelian sandpile models.

Purpose of the Study:

  • To investigate the left passage probability (LPP) of SLE(κ,ρ[over arrow]) using a field theoretical framework.
  • To derive and solve the differential equation governing the LPP for specific cases and present a general perturbative solution.

Main Methods:

  • Utilized a field theoretical framework to analyze the left passage probability (LPP) of SLE(κ,ρ[over arrow]).
  • Derived a differential equation governing the LPP.
  • Numerically solved the equation for the special case κ=2, h(ρ)=0 and presented a perturbative general solution for large boundary values.

Main Results:

  • The derived differential equation governs the left passage probability (LPP) of SLE(κ,ρ[over arrow]).
  • For the case κ=2 and h(ρ)=0, the LPP was numerically solved, revealing a significant dependence on the difference between initial and boundary points (x(0)-ξ(0)).
  • A perturbative general solution for large boundary values was obtained and applied to SLE(κ,κ-6) as a prototype.

Conclusions:

  • The field theoretical approach provides a robust method for studying the left passage probability of SLE variants.
  • The dependence of LPP on boundary conditions is a key feature of SLE(κ,ρ[over arrow]), with implications for related models.
  • The derived solutions offer insights into the statistical properties of these non-conformal random curves.