Large deviations of current for the symmetric simple exclusion process on a semi-infinite line, and on an infinite line with a slow bond

TL;DR

The paper derives the full counting statistics for current in the semi-infinite symmetric simple exclusion process using macroscopic fluctuation theory, revealing an exact semi-infinite SCGF μ_si(λ,ρ_a,ρ_b) obtained via a mapping to the infinite-line problem and symmetry of the MFT equations. The result unifies finite and infinite geometry insights and is verified by cloning simulations, while enabling exact results for related problems: infinite-line with a defect bond, fast single-site injection, and tracer survival with absorbing boundaries. The approach highlights the power of MFT in solving inhomogeneous transport problems and yields practical benchmarks for non-equilibrium fluctuations. The work also links to kinetically constrained models and spin dynamics, predicting stretched-exponential relaxation and offering a framework for extensions to quenched initial states and higher dimensions.

Abstract

Two influential exact results in classical one-dimensional diffusive transport are about current statistics for the symmetric simple exclusion process: one in the stationary state on a finite line coupled with two unequal reservoirs at the boundaries, and the other in the non-stationary state on an infinite line. We present the corresponding result for the intermediate geometry of a semi-infinite line coupled with a single reservoir. This result is obtained using the fluctuating hydrodynamics approach of macroscopic fluctuation theory and confirmed by rare event simulations using a cloning algorithm. We apply our exact result for solving several related challenging problems, namely, the full counting statistics in presence of a defect bond, exclusion process with localized injection, survival of a tagged particle in presence of an absorbing boundary, and the stretched exponential decay in a kinetically constrained model.

Figures (6)

• Figure 1: SSEP on a semi-infinite lattice coupled with a reservoir of density $\rho_a$ with a coupling strength $\gamma>0$.
• Figure 2: Scgf for semi-infinite SSEP: Solid red line represents the theoretical result of scgf \ref{['musi_complete']} for $\rho_{a(b)}=0.5$, with blue dots representing the corresponding simulation result obtained by the cloning algorithm. The results closely match in the range $-1\leq\lambda\leq1$, with the deviations shown in the inset. For comparison, Gaussian approximation and the scgf for infinite line $\mu_{\text{inf}}$2009_Derrida_Current are shown in dotted magenta and black dashed line, respectively.
• Figure 3: Scgf for semi-infinite SSEP: The scgf of current for the semi-infinite SSEP for densities $(\rho_a,\rho_b)\equiv(0.9,0.1)$. The solid red line represents the theoretical result \ref{['musi_complete']}, while the data points are obtained by the cloning algorithm with $T=500$ and $N_c=50,000$. The dashed line represents $R(\omega(\lambda,\rho_a,\rho_b))$ in \ref{['IntegralRep']} as a function of $\lambda$ which differs from the solid line for the parameter regime $\omega(\lambda,\rho_a,\rho_b)<-1/2$, which corresponds to $\lambda\lesssim-1.0$ in the figure.
• Figure 4: Scgf for infinite SSEP with a slow bond: The solid blue line denotes the scgf given in \ref{['eq:scfg slow inf']} for $\rho_{a(b)}=0.5$ and $\Gamma=1$. The blue markers denote the corresponding numerical results obtained by a continuous-time cloning algorithm with $T=500$ and $N_c=10^4$, for $\Gamma=1$. The solid red line denotes the $\Gamma \to \infty$ limit result 2009_Derrida_Current.
• Figure 5: Scgf for semi-infinite SSEP with slow boundary coupling: The solid lines denote the scgf \ref{['eq:scfg slow si']} for $\rho_{a(b)}=0.5$ and different values of $\Gamma$ (green for $\Gamma=1$, blue for $\Gamma=2$, and red for $\Gamma\to \infty$). The markers denote the corresponding numerical results obtained by the cloning algorithm with $T=500$ and $N_c=10^4$.