Cumulant generating functions of a tracer in quenched dense symmetric exclusion processes

The symmetric exclusion process (SEP), where particles hop on a one-dimensional lattice with the restriction that there can only be one particle per site, is a paradigmatic model of interacting particle systems. Recently, it has been shown that the nature of the initial conditions—annealed or quenched—has a quantitative impact on the long-time properties of tracer diffusion. However, so far, the cumulant generating function in the quenched case was only determined in the low-density limit and for the specific case of a half-filled system. Here, we derive it in the opposite dense limit with quenched initial conditions. Importantly, our approach also allows us to consider the nonequilibrium situations of (i) a biased tracer in the SEP and (ii) a symmetric tracer in a step of density. In the former situation, we show that the initial conditions have a striking impact, and change the very dependence of the cumulants on the bias.

Figure 1

Biased tracer in a SEP with different types of initial conditions. Different realizations at $t=0$ are displayed along the red axis. The bath particles (gray) perform symmetric random walks with exclusion. The random walk of the tracer (blue) is biased. (a) Annealed initial conditions: the initial state of the system is drawn at random for each realization. (b) Quenched initial conditions: the initial state is drawn once for all realizations. (c) Deterministic initial conditions, which is an important example of quenched initial conditions: the empty sites are equally spaced.

Figure 2

Cumulants ${\kappa }_1^Q$, ${\kappa }_2^Q$, ${\kappa }_3^Q$ of a biased tracer in the quenched SEP at density $\rho =0.95$ for biases $s=p_1-p_{-1}=0.2$, 0.5, 0.8, and 1 (blue to red). The numerical simulations (circles) are performed with deterministic initial conditions. We denote ${\rho }_0=1-\rho$ and we compute the scalings for ${\kappa }_2^Q$ and ${\kappa }_3^Q$ from Eq. (10): $f_2\left(s\right)=\left[1+s^2(1-\sqrt{2})\right]/\sqrt{2\pi }$ and $f_3\left(s\right)=\left[6\sqrt{2}(3+s^2)arctan\left(2^{-1/2}\right)/\pi -1-3\sqrt{2}-3(\sqrt{2}-1)s^2\right]/\sqrt{4\pi }$. In the three subfigures, the dashed gray line corresponds to our prediction, $\sqrt{2t}$.

Figure 3

Cumulants ${\kappa }_1^Q$, ${\kappa }_2^Q$, ${\kappa }_3^Q$ of a tracer in the quenched SEP with densities ${\rho }_{-}$ behind the tracer and ${\rho }_{+}$ in front of it. From blue to red, $({\rho }_{-},{\rho }_{+})=(0.96,0.9),(0.98,0.96),(0.99,0.95),(0.99,0.98)$. The numerical simulations (circles) are performed with deterministic initial conditions. The gray lines are the predictions deduced from Eq. (17). We denote ${\rho }_0=1-\rho$.