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Stochastic porous-medium equation in one dimension.

Maximilien Bernard1,2, Andrei A Fedorenko3, Pierre Le Doussal1

  • 1l'Ecole Normale Supérieure, Laboratoire de Physique de , CNRS, ENS and PSL Université, Sorbonne Université, Université Paris Cité, 24 rue Lhomond, 75005 Paris, France.

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This summary is machine-generated.

We studied the porous medium equation (PME) with white noise, predicting growth exponents using the functional renormalization group. Simulations revealed anomalous scaling and multiscaling, explained by a Bessel process random walk model.

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Area of Science:

  • Stochastic Partial Differential Equations
  • Statistical Physics
  • Interface Growth Phenomena

Background:

  • The porous medium equation (PME) models various physical phenomena, including fluid flow and heat transfer.
  • Understanding interface dynamics in the presence of noise is crucial for materials science and fluid dynamics.
  • Stochastic growth models are essential for describing random processes in nature.

Purpose of the Study:

  • To investigate the one-dimensional porous medium equation (PME) with additive nonconservative white noise.
  • To interpret the PME as a stochastic growth equation for an interface height field.
  • To predict and analyze growth exponents (α and β) and scaling behaviors.

Main Methods:

  • Application of the functional renormalization group (FRG) to predict growth exponents.
  • Extensive numerical simulations to validate theoretical predictions and explore scaling properties.
  • Analysis of the stationary measure using a random walk model related to a Bessel process.

Main Results:

  • The functional renormalization group successfully predicted the growth exponents α and β.
  • Numerical simulations confirmed the predicted exponents but also revealed anomalous scaling with a local exponent α_{loc}.
  • Evidence of multiscaling was observed, stemming from broad distributions of local height differences.
  • The stationary measure of the stochastic PME was accurately described by a Bessel process-related random walk model.

Conclusions:

  • The study provides a comprehensive analysis of the stochastic porous medium equation in one dimension.
  • The functional renormalization group is a powerful tool for predicting exponents in stochastic growth models.
  • Anomalous scaling and multiscaling phenomena are significant features of this system, requiring advanced modeling.
  • The Bessel process random walk model offers valuable insights into the multiscaling properties of the stochastic PME.