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Heat kernel regularization of the effective action for stochastic reaction-diffusion equations.

D Hochberg1, C Molina-París, M Visser

  • 1Laboratorio de Astrofísica Espacial y Física Fundamental, Apartado 50727, 28080 Madrid, Spain. hochberg@laeff.esa.es

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 20, 2001
PubMed
Summary

This study investigates the short-distance renormalizability of stochastic reaction-diffusion equations using the heat kernel method. The findings indicate that these equations are finite in lower dimensions and renormalizable in higher dimensions, with scale-dependent parameters only in d=2.

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Area of Science:

  • * Theoretical physics and statistical mechanics
  • * Nonlinear dynamics and stochastic processes

Background:

  • * Scale dependence in stochastic differential equations arises from fluctuations and nonlinear interactions.
  • * Stochastic dynamics are often analyzed using functional integral formulations.
  • * Renormalizability is a key concept for understanding the behavior of physical systems at different scales.

Purpose of the Study:

  • * To investigate the short-distance renormalizability of a stochastic polynomial reaction-diffusion equation with additive white noise.
  • * To apply the heat kernel method for calculating the one-loop effective action and divergences.
  • * To determine the conditions under which the equation is finite or renormalizable across different spatial dimensions (d).

Main Methods:

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  • * Application of the heat kernel method.
  • * Calculation of the one-loop effective action.
  • * Analysis of ultraviolet scale-dependent divergences.

    • Main Results:

    • * The stochastic reaction-diffusion equation is one-loop finite in d=0 and d=1 dimensions for white noise.
    • * The equation is one-loop renormalizable in d=2 and d=3 space dimensions.
    • * One-loop renormalization group equations were derived, showing scale dependence only in d=2.

    Conclusions:

    • * The heat kernel method effectively probes the short-distance behavior of stochastic systems.
    • * The dimensionality of space critically influences the renormalizability of stochastic reaction-diffusion equations.
    • * Scale dependence of parameters is dimension-specific, highlighting the importance of dimensionality in stochastic dynamics.