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Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, "a class period lasts 50 min," or "the gas tank in my car holds 65 L," or "the distance between the two posts is 100 m." A physical quantity that can be specified completely in this manner is called a scalar quantity. The word "scalar" is a synonym for "number." Time, mass, distance, length, volume, temperature, and energy are some examples of scalar quantities.
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Experimental Manipulation of Body Size to Estimate Morphological Scaling Relationships in Drosophila
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Growing interfaces uncover universal fluctuations behind scale invariance.

Kazumasa A Takeuchi1, Masaki Sano, Tomohiro Sasamoto

  • 1Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan. kazumasa@daisy.phys.s.u-tokyo.ac.jp

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Researchers solved the Kardar-Parisi-Zhang (KPZ) equation for growing interfaces. They discovered universal distributions of interface positions in liquid-crystal turbulence, revealing new insights beyond scaling laws.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Statistical Mechanics

Background:

  • Brownian motion of points is well-understood due to scale invariance and universal Gaussian fluctuations.
  • Stochastic motion of lines (interface growth) is less understood, lacking experimental validation and analytic solutions for key models like the Kardar-Parisi-Zhang (KPZ) equation.

Purpose of the Study:

  • To address the lack of quantitative experiments and analytic solutions for stochastic line motion.
  • To investigate the universality of growing interfaces beyond established scaling laws.

Main Methods:

  • Developed an experiment for quantitative comparison with theoretical models of growing interfaces.
  • Provided an exact analytic solution for the Kardar-Parisi-Zhang (KPZ) equation.
  • Studied liquid-crystal turbulence as a model system for interface growth.

Main Results:

  • Demonstrated unprecedented universality in growing interfaces, extending beyond simple scaling laws.
  • Identified universal distributions for interface positions in liquid-crystal turbulence.
  • Showed that these distributions follow largest-eigenvalue distributions of random matrices, differing for curved and flat interfaces but remaining universal within each case.

Conclusions:

  • The study provides the first quantitative experimental and theoretical framework for understanding the stochastic motion of lines.
  • The exact solution of the KPZ equation explains the observed universal behaviors in interface growth.
  • Findings reveal a deeper level of universality in complex systems governed by the KPZ equation.