Direct Evaluation of Large-Deviation Functions

We introduce a numerical procedure to evaluate directly the probabilities of large deviations of physical quantities, such as current or density, that are local in time. The large-deviation functions are given in terms of the typical properties of a modified dynamics, and since they no longer involve rare events, can be evaluated efficiently and over a wider ranges of values. We illustrate the method with the current fluctuations of the Totally Asymmetric Exclusion Process and with the work distribution of a driven Lorentz gas.

Figure 1

Space-time diagram for a ring of $N=100$ sites. Top: $\lambda =-50$ and density 0.5; the shock is dense and does not advance. Bottom: $\lambda =-30$ and density 0.3; the shock drifts to the right.

Figure 2

Plot of $\mu (\lambda )$ vs $\lambda$ for the TASEP at density one-half. Numerical results and analytic results of Ref. 3, with points and solid line, respectively.

Figure 3

The billiard. The radii are $R_1=0.39$, $R_2=0.79$. We also show an example of trajectory for the external field $\overrightarrow{E}=(1,0)$.

Figure 4

Plot of $\mu (\lambda )$ vs $\lambda$ for the driven Lorentz gas. Data for $\overrightarrow{E}=(E,0)$, $E=1,2$, and noise intensity $\Delta ={10}^{-3},{10}^{-4}$. The Gallavotti-Cohen theorem implies the symmetry around $\lambda =-1/2$. The best fit (continuous lines) is quadratic for $E=1$ (Gaussian behavior), and a 4th order polynomial for $E=2$.