Decay mechanisms of ensemble averages of nonlinear oscillators

The ensemble average of noninteracting particles in a nonlinear oscillator system is investigated. Depending on the initial phase-space distribution, the nonlinearity-induced dephasing mechanism can lead to temporal decays of the average particle position that can be quite different from the standard exponential decay. In fact, the approach to the equilibrium can be Gaussian or even nonmonotonic in time. In the long-time limit, it is possible to construct a single differential equation for the time evolution of the average position. Unlike the infinite set of coupled nonlinear differential equations derived from the standard approach based on the Liouville equation, this equation can be even linear. We also show that the predicted dephasing mechanisms have their direct counterpart in the corresponding dynamics of quantum mechanical wave packets.

Figure 1

The time evolution of the (Lorentzian) ensemble's average position $X\left(t\right)$, velocity $dX/dt$, and acceleration $d^{2}X/d{t}^2$, as well as a phase-space portrait $\left\{X\left(t\right),dX/dt\right\}$. We averaged over 20 000 orbits with $X_0=-3$ and $\mathrm{\Delta }{X}_0=0.1$. Each single-particle orbit fulfills $d^{2}x/d{t}^2=-x^3$ with $x\left(t=0\right)=x_0$ and $dx/dt\left(t=0\right)=0$.

Figure 2

Comparison of the time evolution of the approximate ensemble averages $X_{\mathrm{L}}\left(t\right), X_{\mathrm{G}}\left(t\right)$ and $X_{\mathrm{U}}\left(t\right)$ with the exact averages over 10 000 orbits of the cubic nonlinear oscillator, where $X_0=-40$ and $\mathrm{\Delta }{X}_0=2$.

Figure 3

Comparison of the time evolution of the quantum mechanical average values $X\left(t\right)\equiv \langle x\rangle \left(t\right)$ with the analytical expressions Eqs. (4.3), which are based on classical mechanical ensemble averages. The initial wave functions were chosen to be $\mathrm{\Psi }\left(x,t=0\right)=\rho {\left(x\right)}^{1/2}$ with the corresponding densities of Eq. (4.2). The Schrödinger equation was solved on a $9000\times 300$ space-time grid with equidistant spacings $\mathrm{\Delta }x=0.0033$ and $\mathrm{\Delta }t=0.066$. The initial densities were Lorentzian ${\rho }_{\mathrm{L}}\left(x\right)$ from Eq. (4.2a) with $X_0=-7$ and $\mathrm{\Delta }{X}_0=0.2$, Gaussian ${\rho }_{\mathrm{G}}\left(x\right)$ (4.2b) with $X_0=-7$ and $\mathrm{\Delta }{X}_0=0.2$, and uniform ${\rho }_{\mathrm{U}}\left(x\right)$ (4.2c) with $X_0=-10$ and $\mathrm{\Delta }{X}_0=0.5$.

Figure 4

The time evolution of the ensemble average $X_{\mathrm{G}}\left(t\right)$ obtained from the truncated set of coupled eight and 28 first-order differential equations (B3) for $X_0=-3$ and $\mathrm{\Delta }{X}_0=0.1$. It is compared to the exact graph obtained by averaging over 10 000 orbits.