Anomalous Dynamics of Inertial Systems Driven by Colored Lévy Noise

Abstract

Dynamics of underdamped particles subjected to colored Lévy noise is investigated analytically. The probability density distribution of the Lévy particle in the harmonic potential is exactly obtained by solving the fractional Fokker–Planck equation, which is also a Lévy type one with width depends on both correlation time $\tau _c$ and damping coefficient $\gamma$ and converges to the distribution for the white-noise, overdamped case in the limit $\tau _c\rightarrow 0$, $\gamma \rightarrow \infty$. Moreover, we obtain an analytical expression of escape rate for the underdamped particle escaping from a metastable potential by using the reactive flux method. It is shown that the stationary escape rate exhibits nonmonotonic dependence on the Lévy index, while an increase of the noise correlation time leads to the monotonic decrease of escape rete.

Appendix: Solution of a, b and c

Substituting Eq. (7) into Eq. (6) and introducing a new variable $t'=t-s$, we get the relations

$$\begin{aligned} \begin{array}{ll} a(0)=0,~~b(0)=0,~~c(0)=\frac{1}{\tau _c},\\ \frac{\partial a(t')}{\partial t'}=b(t'),\\ \frac{\partial b(t')}{\partial t'}+\gamma b(t')+\omega ^2a(t')-\frac{c(t')}{m}=0,\\ \frac{\partial c(t')}{\partial t'}+\frac{c(t')}{\tau _c}=0. \end{array} \end{aligned}$$

(A1)

The Laplace transforms of $a(t')$, $b(t')$ and $c(t')$ are given by

$$\begin{aligned} \begin{array}{ll} \hat{a}(p)=\frac{1}{m\tau _c\left( p^2+\gamma p+\omega ^2\right) (p+\frac{1}{\tau _c})},~\hat{b}(p)=\frac{p}{m\tau _c\left( p^2+\gamma p+\omega ^2\right) (p+\frac{1}{\tau _c})},~ c(p)=\frac{1}{\tau _c(p+\frac{1}{\tau _c})}. \end{array} \end{aligned}$$

(A2)

Based the above relations, we have

$$\begin{aligned} a(t')= & {} -\frac{(\lambda _2-\lambda _3)e^{\lambda _1t'}+(\lambda _3-\lambda _1)e^{\lambda _2t'} +(\lambda _1-\lambda _2)e^{\lambda _3t'}}{m\tau _c(\lambda _1-\lambda _2)(\lambda _2-\lambda _3)(\lambda _3-\lambda _1)},\nonumber \\ b(t')= & {} -\frac{(\lambda _2-\lambda _3)\lambda _1e^{\lambda _1t'}+(\lambda _3-\lambda _1)\lambda _2e^{\lambda _2t'} +(\lambda _1-\lambda _2)\lambda _3e^{\lambda _3t'}}{m\tau _c(\lambda _1-\lambda _2)(\lambda _2-\lambda _3)(\lambda _3-\lambda _1)},\nonumber \\ c(t')= & {} \frac{1}{\tau _c}e^{\lambda _3t'},~ \text {for}~ \lambda _1\ne \lambda _2\ne \lambda _3\ne \lambda _1; \end{aligned}$$

(A3)

$$\begin{aligned} a(t')= & {} \frac{(\lambda _1-\lambda _3)e^{\lambda _1t'}t'+e^{\lambda _3t'} -e^{\lambda _1t'}}{m\tau _c(\lambda _1-\lambda _3)^2}, \nonumber \\ b(t')= & {} \frac{(\lambda _1-\lambda _3)\lambda _1e^{\lambda _1t'}t'+\lambda _3(e^{\lambda _3t'} -e^{\lambda _1t'})}{m\tau _c(\lambda _1-\lambda _3)^2},\nonumber \\ c(t')= & {} \frac{1}{\tau _c}e^{\lambda _3t'},~ \text {for}~ \lambda _1=\lambda _2\ne \lambda _3; \end{aligned}$$

(A4)

$$\begin{aligned} a(t')= & {} \frac{t'^2}{2m\tau _c}e^{\lambda _1 t'}, \nonumber \\ b(t')= & {} \frac{t'}{m\tau _c}e^{\lambda _1 t'}+\frac{\lambda _1 t'^2}{2m\tau _c}e^{\lambda _1 t'},\nonumber \\ c(t')= & {} \frac{1}{\tau _c}e^{\lambda _3t'},~ \text {for}~ \lambda _1=\lambda _2=\lambda _3; \end{aligned}$$

(A5)

where $\lambda _3=-\frac{1}{\tau _c}$, $\lambda _{1,2}=-\frac{\gamma }{2}\pm \frac{\sqrt{\gamma ^2-4\omega ^2}}{2}$ are the eigenvalues of the equation $\lambda _0^2+\gamma \lambda _0+\omega ^2=0$.

In the absence of inertial term, Eq. (A2) yields

$$\begin{aligned} \begin{array}{ll} \hat{a}(p)=\frac{1}{m\tau _c\gamma \left( p+\frac{\omega ^2}{\gamma }\right) \left( p+\frac{1}{\tau _c}\right) },~\hat{b}(p)=\frac{p}{m\tau _c\gamma \left( p+\frac{\omega ^2}{\gamma }\right) \left( p+\frac{1}{\tau _c}\right) },~ c(p)=\frac{1}{\tau _c\left( p+\frac{1}{\tau _c}\right) }. \end{array} \end{aligned}$$

(A6)

Then we recover the results for the overdamped particle in the harmonic potential subjected to the colored Lévy noise

$$\begin{aligned} a(t')= & {} \frac{(e^{-\omega ^2 t'/\gamma }-e^{\lambda _3t'})}{m\tau _c\gamma (-\omega ^2/\gamma -\lambda _3)},~ b(t')=\frac{\left( -\frac{\omega ^2}{\gamma } e^{-\omega ^2 t'/\gamma } -\lambda _3e^{\lambda _3t'}\right) }{m\tau _c\gamma (-\lambda /\gamma -\lambda _3)},\nonumber \\ ~c(t')= & {} \frac{e^{\lambda _3t'}}{\tau _c},~ \text {for}~\lambda _3\ne -\omega ^2/\gamma \nonumber \\ a(t')= & {} \frac{t'e^{\lambda _3 t'}}{m\tau _c\gamma }, ~b(t')=\frac{(\lambda _3t'+1)e^{\lambda _3t'}}{m\tau _c\gamma }, ~c(t')=\frac{e^{\lambda _3t'}}{\tau _c},~ \text {for}~\lambda _3=-\omega ^2/\gamma . \end{aligned}$$

(A7)

Besides, for the underdamped particle in the harmonic potential subjected to white Lévy noise $(\tau _c=0)$, Eq. (A2) take the form

$$\begin{aligned} \begin{array}{ll} \hat{a}(p)=\frac{1}{m\left( p^2+\gamma p+\omega ^2\right) },~\hat{b}(p)=\frac{p}{m\left( p^2+\gamma p+\omega ^2\right) },~ c(p)=1. \end{array} \end{aligned}$$

(A8)

Thus, we obtain

$$\begin{aligned} a(t')= & {} \frac{(e^{\lambda _1t'}-e^{\lambda _2t'})}{m(\lambda _1-\lambda _2)},~ b(t')=\frac{(\lambda _1e^{\lambda _1t'}-\lambda _2e^{\lambda _2t'})}{m(\lambda _1-\lambda _2)}, ~c(t')=\delta (t'),~ \text {for}~\lambda _1\ne \lambda _2 \nonumber \\ a(t')= & {} \frac{t'e^{\lambda _1t'}}{m}, ~b(t')=\frac{(\lambda _1t'+1)e^{\lambda _1t'}}{m}, ~c(t')=\delta (t'),~ \text {for}~\lambda _1=\lambda _2. \end{aligned}$$

(A9)