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Non-Equilibrium Noise in V-Shape Linear Well Profiles

Noah M. MacKay

Abstract

Non-equilibrium noise is characterized as noise realizations where external agitations disrupt the harmonic equilibrium of Brownian motion. Excitations in a particle's random walk into a so-called Lévy flight changes the distribution of the noise from Gaussian to the fat-tailed Lévy distribution. Generalization between Gaussian and Lévy distributions is the $α$-stability distribution, where $1<α\leq2$. In this study, the $α$-stability distributed noise is subjugated into the Langevin and fractional Fokker--Planck equations that correspond to a V-shaped linear potential well $V(x)=F|x|$. From these equations, an Euler scheme for computational simulation via iterations is presented, and a probability density function that is normalizable under any $α\in(1,2]$ is obtained. This study is focused more on the theoretical framework of non-equilibrium noise in V-shaped linear well profiles, which is intended to be applied to systems known to exhibit self-organized criticality.

Non-Equilibrium Noise in V-Shape Linear Well Profiles

Abstract

Non-equilibrium noise is characterized as noise realizations where external agitations disrupt the harmonic equilibrium of Brownian motion. Excitations in a particle's random walk into a so-called Lévy flight changes the distribution of the noise from Gaussian to the fat-tailed Lévy distribution. Generalization between Gaussian and Lévy distributions is the -stability distribution, where . In this study, the -stability distributed noise is subjugated into the Langevin and fractional Fokker--Planck equations that correspond to a V-shaped linear potential well . From these equations, an Euler scheme for computational simulation via iterations is presented, and a probability density function that is normalizable under any is obtained. This study is focused more on the theoretical framework of non-equilibrium noise in V-shaped linear well profiles, which is intended to be applied to systems known to exhibit self-organized criticality.

Paper Structure

This paper contains 10 sections, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. The Langevin Equation
  4. Numerics and Kick Generating
  5. The Fractional Fokker--Planck Equation
  6. $\alpha=1$
  7. $\alpha=2$
  8. Normalizing the Mittag--Leffler Function for $1<\alpha<2$
  9. Discussion
  10. Conclusion

Figures (4)

  • Figure 1: A particle (blue marble) is randomly kicked up the V-shaped linear well. The Lévy flight along the slope projects a horizontal "kick length" $\lambda=x_1-x_0$. The particle then slides back down with a constant force $F$.
  • Figure 2: Gaussian noise ($\alpha=2$) in the V-shaped linear well via the Euler scheme. Due to microscopic reversibility, forward and reverse flows of time are indistinguishable.
  • Figure 3: Lévy noise (using $\alpha=1.5$) in the V-shaped linear well via the Euler scheme. (a) A noise signal is simulated with a distinct flow of time (blue arrow). (b) By reversing the blue time arrow, the playback under a new flow of time (red arrow) reveals a characteristic SOC event.
  • Figure 4: Non-equilibrium pdfs for specific values of $\alpha$ in the V-linear profile ($F=0.75$ and $\sigma=1.25$): $\alpha=1.99$ in blue, $\alpha=1.50$ in orange, $\alpha=1.30$ in green, and $\alpha=1.05$ in red.