Exact Fluctuations of Nonequilibrium Steady States from Approximate Auxiliary Dynamics

We describe a framework to reduce the computational effort to evaluate large deviation functions of time integrated observables within nonequilibrium steady states. We do this by incorporating an auxiliary dynamics into trajectory based Monte Carlo calculations, through a transformation of the system’s propagator using an approximate guiding function. This procedure importance samples the trajectories that most contribute to the large deviation function, mitigating the exponential complexity of such calculations. We illustrate the method by studying driven diffusion and interacting lattice models in one and two spatial dimensions. Our work offers an avenue to calculate large deviation functions for high dimensional systems driven far from equilibrium.

Figure 1

Large deviation function for the entropy production of a driven Brownian particle on a periodic potential with $v_o=2$, $f=12.5$. The main figure shows the functions computed with exact diagonalization (red) and DMC (black) calculation. The inset shows the fraction of correlated walkers without GDF (blue), or with GDF from an instanton approximation to the auxiliary process (red) or the exact auxiliary process (black).

Figure 2

Large deviation function for the mass current of an open simple exclusion process. The main figure shows the functions computed with exact diagonalization (red) and DMC (black) calculations. The inset shows the fraction of correlated walkers without guiding functions (blue), or with GDF from approximations to the auxiliary process using a uniform GDF (green) or clusters of 1 (red), 2 (orange), 4 (cyan), and 8 (black) sites.

Figure 3

Large deviation function for the mass current of a closed 2D asymmetric exclusion process. (a) Large deviation function computed from DMC calculations with importance sampling. (b) Susceptibility for current fluctuations as a function of $\lambda$. (c) Ratio of the fraction of independent walkers with importance sampling $f_I^{\mathrm{IS}}$ and without importance sampling $f_{\mathrm{I}}$.