Diffusion with local resetting and exclusion

Simple Exclusion Processes with Local Resetting

“…In both cases, the proper coordinate is y = l/N . Summarizing, we have shown that the stationary state of SSEP-LR and TASEP-LR in the thermodynamic limit depends crucially on how the resetting rate r scales with the system size L. In SSEP-LR with finite density we find a small resetting, purely diffusive regime if rL 2 → 0, an in- termediate resetting regime if rL 2 tends to a positive constant, and the large resetting regime investigated in [10] if rL 2 → ∞. In the vanishing density case similar considerations apply, with the driving parameter rN 2 .…”

“…The above results suggest that the stationary density profile depends on r and L only through λ (except for the prefactor ρ), and hence only through the combination rL 2 . This provides some insight into the transition from the homogeneous profile that is obtained in the purely diffusive case (r = 0) to the nonuniform profile, with a maximum at the origin, obtained in [10] for r of order 1 and r ∝ L. Indeed, 3 different regimes can be found, depending on the behaviour of rL 2 (or equivalently rN 2 ) in the thermodynamic limit L → ∞. (i) Small resetting: if r tends to 0 faster than L −2 (or equivalently N −2 ), then λ → 0, ρ 0 = ρ and ρ(x) = ρ, the purely diffusive case.…”

“…Another natural step forward would be to introduce local resetting in TASEP (or, more generally, ASEP) with OBCs, and in model with additional interactions [14][15][16][17]. Regarding methods, based on the agreement between MF and KMC found in [10] and in our work, it would be interesting to rigorously assess whether, and to what extent, MF results can become exact in the thermodynamic limit. It would also be worth investigating the relaxation towards the stationary state, in which dynamical transitions without a static counterpart have been found in (T)ASEP with OBCs [18][19][20], also (at least with approximate methods) in the presence of Langmuir kinetics [21,22] or additional interactions between particles [23,24].…”

“…In [10] the authors studied the SSEP-LR stationary state with r of order 1 and r ∝ L, in the (i) fixed density and (ii) fixed number of particle cases, using both mean-field (MF) approximation and Kinetic Monte Carlo (KMC) simulations. Their crucial finding is that in the thermodynamic limit the stationary density profile is independent of the resetting rate r, and depends only on the position (in case (i) scaled by the system size) and (i) the average density or (ii) the number of particles.…”

“…(i) Small resetting: if r tends to 0 faster than L −2 (or equivalently N −2 ), then λ → 0, ρ 0 = ρ and ρ(x) = ρ, the purely diffusive case. (ii) Large resetting: if rL 2 → ∞, which includes the cases studied in [10], ρ 0 tends to 1 (more precisely λ stays finite) and the profile has a maximum at the origin. In this regime local resetting dominates over diffusion.…”