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

Abstract

The Symmetric Exclusion Process (SEP), where particles hop on a 1D 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 quenchedhas a quantitative impact on the long-time properties of tracer diffusion. However, so far, all the studies in the quenched case focused on the low-density limit of the SEP. Here, we derive the cumulant generating function of the tracer position in the 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.

Large deviations of a tracer position in the dense and the dilute limits of a single-file diffusion

“…Both annealed and quenched initial settings are considered. These results come as a straight-forward perturbation solution of a hydrodynamic approach presented earlier [2] and they confirm recent results [20][21][22] for specific model systems.…”

“…To the leading order we get the CGF in the dilute and the dense limits for a general single-file diffusion. For the special case of symmetric exclusion process, our results confirm the expression of CGF obtained by microscopic solutions [20][21][22].…”

“…There are no analogous results available for general single-file systems. For a fixed (quenched) initial state of the symmetric exclusion process there are exact results for half-filling [3] and for the high density limit [21]. Low density limit of the symmetric exclusion process corre- * jagannath.rana@tifr.res.in FIG.…”

“…Although the hydrodynamic formulation is applicable to arbitrary initial profile, we consider an example of step density profile ρ(x) = ρ a Θ(−x) + ρ b Θ(x) which has been frequently studied for tracer diffusion [20][21][22]33]. For ρ a = ρ b , bulk of the system evolves towards an asymptotic equilibrium effectively biasing the tracer in one direction.…”

Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process

“…In the meanwhile, Grabsch et al [44] made a major breakthrough in the understanding of large deviations in single file systems such as the SEP. Without using neither integrability nor the MFT, they intuited and unveiled recursively a closed equation for the final optimal profile, allowing them to determine that profile and the corresponding large deviations (see [45][46][47] for precursory works in the same group).…”