Fast Monte Carlo Simulations and Singularities in the Probability Distributions of Nonequilibrium Systems

A numerical technique is introduced that reduces exponentially the time required for Monte Carlo simulations of nonequilibrium systems. Results for the quasistationary probability distribution in two model systems are compared with the asymptotically exact theory in the limit of extremely small noise intensity. Singularities of the nonequilibrium distributions are revealed by the simulations.

Figure 1

The first [${\Phi }_1(x)$, lower curve] and second [${\Phi }_2(x)$, upper curve] pieces of the inferred potential $\Phi (x)$ for the system (1) with $D=0.005$. The discontinuity in the gradient of ${\Phi }_2(x)$ near $x_i$ is an artifact due to a boundary effect in the calculation of the discrete probability distribution. To avoid this problem ${\Phi }_1(x)$ and ${\Phi }_2(x)$ are merged at the point $x_f$ and the initial part of ${\Phi }_2(x)$ is discarded. We normalize ${\Phi }_1(x)$ choosing ${\Phi }_1(x_m)=0$, and each successive piece of $\Phi (x)$ is normalized in order to match with the previous one at the point where they join. Inset: the inferred potential $\Phi (x)$ for the system (1) with $D=0.005$. The new technique (circles) is compared with standard Monte Carlo simulations (bold line) and with the Duffing potential $U(x)$ (thin line).

Figure 2

The whole inferred $\Phi (x,t)$ for the system (2) for $A=0.1$, $\Omega =1$, $D=0.005$. Two lines are the lines of constant probability found after the first step of simulations. The corresponding levels of probability were chosen as ${\Phi }_i=3D$ and ${\Phi }_f=5D$.

Figure 3

A time section of the inferred $\Phi (x,t=4.1)$ for the system (2) with $A=0.1$, $\Omega =1$, and different noise intensities: $D=0.005$ (diamonds); $D=0.01$ (circles); and $D=0.02$ (triangles). The theoretical predictions are shown by full lines for finite noise intensities, and by dashed line for $D=0$. Inset: oscillations of $\Phi (x,t)$ at the boundary of attraction for different noise intensities.

Figure 4

A section $(x=y)$ of the inferred $\Phi (A)$ for the system (3) with ${\omega }_0=1$, noise intensity $D=0.01$ and $\eta =0.25$ (circles); and $\eta =0.5$ (diamonds). Theoretical predictions are shown in each case for $D=0$ (dashed curves) and $D=0.01$ (full curves).