Simulation of stationary Gaussian noise with regard to the Langevin equation with memory effect

We present an efficient method for simulating a stationary Gaussian noise with an arbitrary covariance function, and then we study numerically the impact of time-correlated noise on the time evolution of a ($1+1$)-dimensional generalized Langevin equation by comparing also to analytical results. Finally, we apply our method to the generalized Langevin equation with an external harmonic and double-well potential.

Figure 1

Comparison of the numerical simulation of $\langle \xi \left(0\right)\xi \left(t\right)\rangle$ with the analytical result for two different covariance functions ($C_1,D=2,\tau =2$ and $C_2,D=2,\alpha =2$) based on 8000 realizations.

Figure 2

Numerical evolution of the mean kinetic energy (26) for the exponential covariance function $C_1\left(t\right)$ with $\tau =2$, the Gaussian covariance function $C_2\left(t\right)$ with $\alpha =1$, and the theoretical limit of $\tau \to 0,\alpha \to 0$ corresponding to $\delta$-correlated white noise. The initial conditions for both simulations are given in (27).

Figure 3

The spectral density of the velocity for different correlation times. For small values of $\tau$, the peak becomes broader and is shifted to higher frequencies.

Figure 4

Comparison of the kinetic energy between the analytical expression [Eq. (42)] and the numerical average over 8000 realizations.

Figure 5

Numerical results for the kinetic energy and different values of $\tau$. As expected, with increasing correlation times the system takes longer to approach the equilibrium state.

Figure 6

Spectral density of the position $x$ in a harmonic oscillator. The simulation parameters are $T=1,D=1,m=0.2$, and ${\omega }_0=\sqrt{5}$.

Figure 7

Behavior of the peak frequency ${\omega }_{\mathrm{peak}}$ as a function of $\tau$. In a narrow frequency range, the peak vanishes completely. With increasing values of $\tau$ it approaches the frequency of the harmonic oscillator, whereas in the limit $\tau \to 0$ its frequency converges to the white-noise limit. The same simulation parameters have been used as in Fig. 6.

Figure 8

Simulation for the probability to pass a square barrier compared to the analytic result (50) [27]. We use the same representation of the barrier height as in this reference, i.e., $\kappa =K/B_{\mathrm{eff}}$ with $B_{\mathrm{eff}}=B{\omega }_0/{\lambda }_1^2$.

Figure 9

A total of 400 independent realizations of the Langevin equation (52) with a symmetric double-well potential and initial conditions as defined in (53). Shown is the total energy and position of each particle at the time $t=5$.

Figure 10

Kinetic energy of the Langevin equation (52) with a symmetric double-well potential for the covariance function $C_1$.

Figure 11

Relative number of particles located on the right side of the symmetric potential for different correlation times $\tau$ and two different time ranges.

Figure 12

Characteristic time ${\tau }_{\mathrm{eq}}$ for the system to reach equilibrium in the symmetric double-well potential as a function of the correlation time $\tau$.

Figure 13

A total of 400 independent realizations of the Langevin equation (52) with an asymmetric double-well potential and initial conditions as defined in (54). Shown is the total energy and position of each particle at the time $t=5$.

Figure 14

Relative number of particles located on the right side of the potential well for different temperatures $T$ and correlation time $\tau =0.1$.

Figure 15

Same as Fig. 14 but for the correlation time $\tau =2$.

Figure 16

Relative number of particles located on the right side of the asymmetric potential well as a function of the temperature $T$ at fixed times $t=4$ (upper figure) and $t=20$ (lower figure). With increasing temperature, more particles overcome the potential well in a shorter time. For very high temperatures, the particle's energy becomes large enough to overcome the potential barrier as easily from the right as from the left such that the curve converges to 0.5. The correlation time has a larger impact on the relative number of particles for smaller times $t$.