Exact Current Statistics of the Asymmetric Simple Exclusion Process with Open Boundaries

Nonequilibrium systems are often characterized by the transport of some quantity at a macroscopic scale, such as, for instance, a current of particles through a wire. The asymmetric simple exclusion process (ASEP) is a paradigm for nonequilibrium transport that is amenable to exact analytical solution. In the present work, we determine the full statistics of the current in the finite size open ASEP for all values of the parameters. Our exact analytical results are checked against numerical calculations using density matrix renormalization group techniques.

Figure 1

Dynamical rules for the ASEP with open boundaries. The rate of forward jumps has been normalized to 1. Backward jumps occur with rate $q<1$. All other parameters are arbitrary.

Figure 2

Third (E3, red) and fourth (E4, blue) cumulants in the maximal current phase, with $q=0.5$, $a_{+}=b_{+}=0.65$, and $a_{-}=b_{-}=0.6$; the full lines represent the corresponding large size asymptotic behaviors.

Figure 3

Second (E2, red) and third (E3, blue) cumulants in the high density phase, with $q=0.5$, $a_{+}=0.28$, $b_{+}=1.15$, $a_{-}=-0.48$, and $b_{-}=-0.27$.