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Stochastic gyration driven by dichotomous noises

Timothée Herbeau, Leonid Pastur, Pascal Viot, Gleb Oshanin

Abstract

We consider stochastic dynamics of a particle on a plane in presence of two noises and a confining parabolic potential - an analog of the experimentally-relevant Brownian Gyrator (BG) model. In contrast to the standard BG model, we suppose here that the time-evolution of the position components is driven not by Gaussian white-noises, but by two statistically-independent dichotomous noises. We calculate analytically the position variances and cross-correlations, as well as the mean angular momentum, which permits us to establish the conditions in which a spontaneous rotational motion of the particle around the origin takes place. We also present a numerical analysis of the mean angular velocity. Lastly, we calculate analytically some marginal position probability density functions revealing a remarkably rich behavior that emerges in such a system of two coupled linear stochastic differential equations. We show that depending on the values of parameters characterizing noises these distributions approach the steady-state forms defined on a finite support, having very unusual shapes, possessing multiple maxima and minima, plateaus and exhibiting a discontinuous behavior.

Stochastic gyration driven by dichotomous noises

Abstract

We consider stochastic dynamics of a particle on a plane in presence of two noises and a confining parabolic potential - an analog of the experimentally-relevant Brownian Gyrator (BG) model. In contrast to the standard BG model, we suppose here that the time-evolution of the position components is driven not by Gaussian white-noises, but by two statistically-independent dichotomous noises. We calculate analytically the position variances and cross-correlations, as well as the mean angular momentum, which permits us to establish the conditions in which a spontaneous rotational motion of the particle around the origin takes place. We also present a numerical analysis of the mean angular velocity. Lastly, we calculate analytically some marginal position probability density functions revealing a remarkably rich behavior that emerges in such a system of two coupled linear stochastic differential equations. We show that depending on the values of parameters characterizing noises these distributions approach the steady-state forms defined on a finite support, having very unusual shapes, possessing multiple maxima and minima, plateaus and exhibiting a discontinuous behavior.
Paper Structure (24 sections, 100 equations, 18 figures)

This paper contains 24 sections, 100 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Model and basic notations
  3. Moments of positions of the SG, specific angular momentum and specific angular velocity
  4. Moments of position components
  5. Specific angular momentum
  6. Specific angular velocity
  7. Probability density functions of $\mathcal{S} = x + y$ and $\mathcal{D} = x - y$ in the steady-state: General results
  8. Moments of $P(\mathcal{S})$
  9. White-noise limit: Standard BG model
  10. Behavior of $P(\mathcal{S})$ of eq. \ref{['pU']} at $\mathcal{S}=0$.
  11. The probability density function $P(\mathcal{S})$ at $\mathcal{S} = \pm \Omega$ and $\mathcal{S} = \pm \Sigma$
  12. The probability density function $P(\mathcal{S})$: Particular cases
  13. Case A: $\alpha_x$ and $\alpha_y$ are both less than $1$
  14. Case B: $\alpha_x < 1$ and $\alpha_y = 1$, or vice versa
  15. Case C: $\alpha_x < 1$ and $\alpha_y > 1$, or vice versa
  16. ...and 9 more sections

Figures (18)

  • Figure 1: Three randomly-generated SG trajectories (blue curves) shown in the $(x,y,t)$ space for $v_x=1$ and $v_y=3$, with switching rates $\lambda_x = \lambda_y = 1$ (left panel), $1/3$ (middle panel), and $0.2$ (right panel). The red curves represent the projections of these trajectories onto the $xy$-plane.
  • Figure 2: The variances (second moments) of the $x$- and $y$-components in the limit $t \to \infty$ (left and middle panels), as well as the cross-moment $x(t) y(t)$ (right panel) as functions of the coupling parameter $u$ for $\lambda_x=0.5$, $\lambda_y=2.5$, $v_x=1.0$, and $v_y=2$. The solid curves correspond to the exact analytical expressions, while the stars indicate the numerical results.
  • Figure 3: The mean specific angular momentum $\mathbb{E}_{x,y}\left[ L \right]$, (eq.\ref{['eq:L']}, green curve) as function of $u$ for $\lambda_x =\lambda_{y}=0.25$ (left panel), $\lambda_x =\lambda_{y}=0.5$ (middle panel) and $\lambda_x =\lambda_{y}=2.5$ (right panel) with fixed non-equal amplitudes of noises, $v_x=1$ and $v_y=3$. The magenta curve depicts the behavior of the mean specific angular momentum in the Gaussian white-noise (GWN) limit, (see eq. \ref{['LWGN']} and Viot2024). Stars depict the results of numerical simulations.
  • Figure 4: Numerically-evaluated mean specific angular velocity $\mathbb{E}_{x,y} [W]$ as function of $u$. Left panel: $\lambda = \lambda_x = \lambda_y =5$, $1$ and $0.1$ (see the inset) with fixed amplitudes of noises $v_x=1$ and $v_y=3$. The red solid curve depicts the exact expression for the Gaussian white-noise (see the expression in, e.g., Viot2024). Other curves are merely guides to the eye obtained by a freehand interpolation of symbols depicting the numerical results. Middle panel: equal amplitudes of noises $v_x = v_y = 3$, $\lambda_x = 1$ and variable $\lambda_y$ (see the inset). Right panel: fixed ratios $D_x =0$, $D_y =1$, $\lambda_x = 1$ and variable $\lambda_y$. The solid red curves depicts the exact expression for the Gaussian white-noise.
  • Figure 5: Logarithms of the probability density functions $P(\mathcal{S})$ in eq. \ref{['pU']} (black solid curve) and the white-noise-limit asymptotic form in eq. \ref{['Gauss']} (blue dashed curve) as functions of $\mathcal{S}$ for $u=3/4$, $v_x = v_y = 10$ and $\lambda_x = \lambda_y = 9.7$.
  • ...and 13 more figures