Application of importance sampling to the computation of large deviations in nonequilibrium processes

We present an algorithm for finding the probabilities of rare events in nonequilibrium processes. The algorithm consists of evolving the system with a modified dynamics for which the required event occurs more frequently. By keeping track of the relative weight of phase-space trajectories generated by the modified and the original dynamics one can obtain the required probabilities. The algorithm is tested on two model systems of steady-state particle and heat transport where we find a huge improvement from direct simulation methods.

Figure 1

Plot of $P(Q)$ for $\tau =15$ for the one-site SSEP model with $\alpha =\beta =3.0,\gamma =\delta =3.0$. MC refers to direct Monte Carlo simulations. Left bias corresponds to ${\alpha }^{\text{'}}={\beta }^{\text{'}}=3.8,{\gamma }^{\text{'}}={\delta }^{\text{'}}=2.2$ and right bias to ${\alpha }^{\text{'}}={\beta }^{\text{'}}=2.2,{\gamma }^{\text{'}}={\delta }^{\text{'}}=3.8$.

Figure 2

Plot of $P(Q)$ for $\tau =15$ for the three-site SSEP model with $\alpha =\beta =4.0,\gamma =\delta =2.0$. MC refers to direct Monte Carlo simulations. For left (right) bias simulations, the particles in bulk hop to the left (right) with rate $4$ and to the right (left) with unit rate. The boundary rates are kept unchanged.

Figure 3

Plot of $P(Q)$ for $\tau =200$ for heat conduction across a single free particle with ${\gamma }_1=0.8$, ${\gamma }_2=0.2$, $T_1=1.1875$, and $T_2=0.25$. The parameters have been chosen to correspond to a region in parameter space where the fluctuation theorem is not satisfied [6]. MC refers to direct Monte Carlo simulations. The left bias corresponds to ${\gamma }_1^{\text{'}}={\gamma }_1$, ${\gamma }_2^{\text{'}}={\gamma }_2/20$, $T_1^{\text{'}}=T_1$, $T_2^{\text{'}}=20T_2$.

Figure 4

Plot of $P(Q)$ for $\tau =100$ for heat conduction across two particles connected by a harmonic spring with unit spring constant and ${\gamma }_1={\gamma }_2=\sqrt{2},T_1=10,T_2=12$. MC refers to direct Monte Carlo simulations. The left bias corresponds to ${\gamma }_1^{\text{'}}={\gamma }_1,{\gamma }_2^{\text{'}}={\gamma }_2/2,T_1^{\text{'}}=T_1,T_2^{\text{'}}=2T_2$ and right bias to ${\gamma }_1^{\text{'}}={\gamma }_1/2,{\gamma }_2^{\text{'}}={\gamma }_2,T_1^{\text{'}}=2T_1,T_2^{\text{'}}=T_2$.