Method for direct analytic solution of the nonlinear Langevin equation using multiple timescale analysis: Mean-square displacement

We consider a class of nonlinear Langevin equations with additive, Gaussian white noise. Because of nonlinearity, the calculation of moments poses a serious problem for any direct solution of the Langevin equation. Based on multiple timescale analysis we introduce a scheme for directly solving the equations. We first derive the equations for the fast and slow dynamics, in the spirit of the Blekhman perturbation method in vibrational mechanics, the fast motion being described by the Brownian motion of a harmonic oscillator whose effect is subsumed in the slow motion resulting in a parametrically driven nonlinear oscillator. The multiple timescale perturbation theory is then used to obtain a secular divergence-free analytic solution for the slow nonlinear dynamics for calculation of the moments. Our analytical results for mean-square displacement are corroborated with direct numerical simulation of Langevin equations.

Figure 1

The mean-square displacement $\langle x^2\rangle$ is plotted against time $t$ for the Duffing oscillator for the following parameter set: ${\omega }_0=1.0$, $\gamma =0.01,\alpha =1.0,\epsilon =0.01$, and $D=0.01$ (analytical, dotted line; numerical, solid line). For numerical simulation averaging is done over ${10}^5$ trajectories. The inset refers to the variation of $\langle v^2\rangle$ vs time $t$, calculated numerically to ensure the attainment of thermal equilibrium (units arbitrary).

Figure 2

Same as in Fig. 1 but for $D=0.001$.

Figure 3

Same as in Fig. 1 but for $D=0.005$.

Figure 4

The variation of $X^2$, $\langle {\psi }^2\rangle$ along with $\langle x{\left(t\right)}^2\rangle$ (numerical) and $\langle x{\left(t\right)}^2\rangle$ (theoretical) against time is plotted after the initial transients die down for the parameter set ${\omega }_0=1.0$, $\gamma =0.01,\alpha =1.0,\epsilon =0.01$, $a=b=5.0$, and $D=0.01$, corresponding to the Duffing oscillator. The inset in the figure refers to the variation of $X^2$ vs $t$ over a short period of time to show the complex oscillations (units arbitrary).

Figure 5

The variation of $\langle x{\left(t\right)}^2\rangle {|}_{t\to \infty }$ as a function of $D$ is plotted for ${\omega }_0=1.0$, $\gamma =0.01,\alpha =1.0,\epsilon =0.01$ and for $t=1800$ corresponding to the Duffing oscillator for numerical (cross) and analytical (line) values.

Figure 6

The mean-square displacement $\langle x^2\rangle$ is plotted against time $t$ for the bistable oscillator for the following parameter set: ${\omega }_0=1.0,\gamma =0.5,\alpha =1.0,\epsilon =0.01$, and $D=0.01$ (analytical, dotted line; numerical, solid line). For numerical simulation averaging is done over ${10}^5$ trajectories. The inset in the figure refers to the variation of $\langle v^2\rangle$ vs time $t$, calculated numerically to ensure the attainment of thermal equilibrium (units arbitrary).

Figure 7

Same as in Fig. 6 but for $D=0.005$.

Figure 8

The mean-square displacement $\langle x^2\rangle$ plotted against time $t$ for the periodic potential for the parameter set $a=1$, $\gamma =0.01$, $D=0.001$ with the initial conditions $x\left(0\right)=0$, $\dot{x}\left(0\right)=0.25$ (analytical, dotted line; numerical, solid line). For the numerical simulation averaging is done over ${10}^5$ trajectories. The inset displays the variation of $X^2$ vs $t$ calculated analytically using the solution given in [48] (units arbitrary).