Diffusion with Local Resetting and Exclusion

Stochastic resetting models diverse phenomena across numerous scientific disciplines. Current understanding stems from the renewal framework, which relates systems subject to global resetting to their non-resetting counterparts. Yet, in interacting many-body systems, even the simplest scenarios involving resetting give rise to the notion of local resetting, whose analysis falls outside the scope of the renewal approach. A prime example is that of diffusing particles with excluded volume interactions that independently attempt to reset their position to the origin of a 1D lattice. With renewal rendered ineffective, we instead employ a mean-field approach whose validity is corroborated via extensive numerical simulations. The emerging picture sheds first light on the non-trivial interplay between interactions and resetting in many-body systems.

In spite of its centrality to the study of resetting phenomena, the renewal framework's applicability is limited. Specifically, in many-body systems, it only applies when resetting acts to bring the entire system back to its initial configuration, a process which we hereby term "global resetting" [51][52][53]. Yet, generically, resetting will have a more local nature that can, in turn, be affected by interactions. As a concrete example, consider diffusive particles, subject to repulsive short-ranged interactions, that move along a 1D track immersed in a fluid. In this case, resetting can be used to model the unbinding of a particle from the track and its subsequent rebind-1 r X Figure 1. An illustration of diffusion with exclusion and local resetting on a ring lattice. Each particle attempts to hop to its left and right neighboring sites with rate 1, and to reset its position to the origin with rate r. In both cases, an attempt is successful only if the target site is vacant.
ing at a uniquely favorable location. However, due to interactions, rebinding cannot occur if this location is already occupied by a different particle. This, in fact, is precisely the picture that arises in bio-polymerization, where RNA polymerases and ribosomes play the role of resetting particles. Since a resetting event leaves the system's configuration mostly intact, aside from the new position of the resetting particle, the renewal framework does not apply and alternative approaches must be sought. The absence of a unifying theoretical framework renders the study of "local" resetting in interacting many-body systems extremely challenging.
In this Letter, we take a first step towards establishing an understanding of local resetting within an analytically tractable model. In particular, we study a 1D ring lattice of L sites, occupied by diffusive particles which interact via volume exclusion and independently attempt to reset their position to the origin site, if it is vacant (see Fig. 1). These dynamics give rise to a non-trivial steady-state density profile, whose analysis lies well beyond the scope of renewal theory. Instead, taking a mean-field (MF) approach, we are able to derive a closed-form solution for the stationary density profile and analyze its scaling properties in the limit of large L. The MF description is corroborated against extensive numerical simulations to remarkable accuracy. The results established herein pave the way to the extension of current experimental studies of single-particle resetting [54,55], to interacting many-body systems.
The Model -Consider a 1D periodic lattice of L sites labeled = 0, ..., L − 1, occupied by N particles of average density ρ ≡ N/L. The particles are subject to hardcore, exclusion interactions by which each site may hold one particle at most [56][57][58][59]. The system then evolves in continuous time via the dynamical rules illustrated in Fig. 1: Each particle attempts to hop to its left and right neighboring sites with rate 1 and to reset its position to site = 0 with rate r. In both cases, an attempt is successful only if the target site is vacant.
These dynamics can be formulated in terms of a Markov chain. Let τ (t) denote the occupation of site at time t, taking the value 0 if the site is vacant and 1 otherwise. The model's dynamics imply the following evolution of τ (t) where Γ (t) is given by and w.p. abbreviates "with probability". Clearly, a distinction must be made between site = 0, which experiences an influx of resetting particles, and the remaining sites = 0. Equation (1) for Γ (t) indeed serves as short-hand notation for σ = −1 and R = rτ ( Note that the expression for R 0 is not arbitrary, originating from the model's particle conservation, i.e.
Main Results -Using an analytical mean-field approach and extensive numerical simulations, we show that the stationary density profile of resetting and interacting random walkers behaves very differently from that found in the absence of interactions. As in the noninteracting case, resetting acts to concentrate particles at the origin site = 0. However, here, exclusion prevents the origin from being occupied by more than a single particle. Moreover, resetting cannot occur when the origin is occupied, giving rise to a non-trivial interplay between exclusion, diffusion, and resetting.
For a fixed mean densityρ, the model's exclusion interactions and diffusive dynamics yield a density profile that is a scaling function of /L, asserting that the density substantially deviates fromρ throughout the entire system. This stands in stark contrast to the noninteracting picture, where the width of the profile is ∼ D/r, with D denoting the diffusion coefficient [2]. Here we find that, for large systems, the density profile is entirely independent of the resetting rate r. In fact, it only depends on the mean densityρ, as seen in Eqs. (12)- (14). A different picture emerges when the number of particles N =ρL is instead kept fixed, in which case the density profile is shown to be a function of alone, as obtained in Eq. (9). These MF predictions are verified, to remarkable precision, in data collapses obtained from extensive numerical simulations. Figure  2 demonstrates the density profile's scaling form for a fixed mean density while Fig. 3 shows its behavior for fixed particle number N . Additional data is provided in the supplemental material (SM) for different parameters.
MF analysis -To evoke the MF approximation, we first average over the Markov chain in Eqs. (1) and (2), replacing the mean occupation τ by the corresponding density field ρ ∈ [0, 1]. The MF approximation is then manifested in the factorization of products of the form τ τ k ≈ τ τ k → ρ ρ k . The rates R =0 and R =0 in Eq. (3) respectively become R =0 = r L−1 m=1 ρ m and R =0 = rρ . While it is generally quite difficult to rigorously justify the MF approximation, this approach has proven to be remarkably successful for analyzing numerously many lattice models with exclusion interactions [60][61][62][63][64][65][66]. As we show below, MF also provides a remarkably accurate description of the present model.
In the limit dt → 0, the MF approximation yields the following equation for ρ (t), Our interest lies in the stationary behavior of the density profile. To this end, we set ∂ t ρ = 0 and separately analyze Eq. (4) at sites = 0, termed the "bulk" equation, and at site = 0, which we call the "boundary" equation. Solving the stationary bulk equation gives where c 1,2 are constants and we have defined and Using the dynamic's symmetry around site = 0, i.e. ρ = ρ L− , gives c 2 = c 1 A −L + . The particle conservation condition is then used to determine c 1 , such that ρ becomes Although we have obtained a formal solution for the MF density profile ρ , our work is not yet done since ρ still depends on the density at site = 0 through a in Eq. (7). To determine ρ 0 , we revisit Eq. (4) for the density profile and consider its behavior at = 0. In the stationary limit, this equation can be written as where N − ρ 0 = L−1 m=1 ρ m and we have again used the → L − symmetry to replace ρ 1 + ρ L−1 → 2ρ 1 . To make progress, we separately consider two distinctly different physical scenarios: the case of a constant particle densityρ and the case of a constant particle number N .
Fixed density ρ -In this case the number of particles in the system N grows linearly with system-size L. For large L, and a correspondingly large number of particles, there is a small time gap between the instance site = 0 is vacated by a particle hopping to a vacant neighboring site, and the time it is reoccupied due to a resetting event. Reoccupation due to hopping from neighboring sites is negligible, since it is attempted with rate 1 while resetting events are attempted with rate Lρr ≡ N r 1. In the limit of L → ∞, where the system contains infinitely many particles, the total rate of resetting attempts becomes infinite and reoccupation is immediate. We thus expect that the density near the origin be unity, up to small finite-L corrections. We thus consider the ansatz generally allowing for a different scaling with L at sites = 0 and = 0. Substituting this ansatz into Eq. (10) yields the relation µ = 1 + ν. We can then determine the value of ν by substituting Eq. (11) into Eq. (9) for ρ , evaluated at site = 1. This self-consistency requirement sets ν = 1 and, correspondingly, µ = 2 (see the SM for details). With this, we obtain the density profile at sites = 0 as The parameter α is determined by demanding that the ansatz for ρ 0 in Eq. (11) be consistent with the density profile ρ in Eq. (12) at site = 0. This gives providing a transcendental equation for α that we must numerically solve, given the value of ρ. With this, the density profile assumes the simpler form Although the relation between α andρ cannot be inverted analytically, we can still obtain important insight regarding the dependence of α onρ in the asymptotic limits of α → 0 and α → ∞. A straightforward analysis yields as demonstrated in Fig. 4. We conclude that for a fixed mean density ρ, ρ becomes a scaling function of /L, at large L. This implies that the density profile spans the entire system, as is clearly observed in Fig. 2. This scaling behavior also appears for r ∝ L, as shown in the SM. Fixed particle number N -We next consider the case where the particle number N is fixed, finite, and independent of L. The time gap between the instance site = 0 is vacated and the instance it is reoccupied now remains finite, even as L is increased. Correspondingly, both ρ 0 and ρ 1 are asymptotically independent of L.
With this insight, we return to Eq. (9) in order to relate ρ at = 0 to ρ 0 for finite N , in the limit of large L. Substituting ρ = N/L and a = r (1 − ρ 0 ) into Eq. (9), setting = 0 and equating to ρ 0 asymptotically yields the polynomial equation where we have used the fact that A −L + and A L − both vanish as L → ∞, for any L-independent a (see Eq. (6)). This equation for ρ 0 has three roots, of which only one satisfies ρ 0 ∈ [0, 1]. While its precise analytical form is rather involved and is thus deferred to the SM, in the limit 1 N L, where the system contains many particles but the average density is low, it reduces to ρ 0 = 1 − 4r −1 N −2 + O N −4 . By Eq. (7), this translates into a ∼ = 4/N 2 which, when substituted into Eq. (9), reveals that the density profile has a width of ∼ N/2 around site = 0. Since N is fixed, the density profile remains a function of alone, implying that it only extends over a finite ∼ O (N ) region near the origin, as demonstrated in Fig. 3.
Conclusions -In this Letter, we have gone beyond the popular renewal framework to study the effect of local resetting on a many-body interacting system. Employing a MF approach, we derived the stationary density profile of N particles diffusing on a 1D ring lattice of L sites while being subject to exclusion interactions and local resetting with rate r. For large systems occupied by many particles, we find that the profile's width is independent of r, being solely a function of the position and average densityρ (or the number of particles N , depending on which was kept fixed while taking the large system limit). This intriguing behavior directly follows from the delicate interplay between local resetting and exclusion interactions, which prohibit resetting if the origin is already occupied. Indeed, for non-interacting particles the density profile is known to adopt a width ∼ r −1/2 [2], irrespective ofρ or N , which stands in stark contrast to our findings here.
Two recent experimental studies explored resetting in the context of diffusive single-particle systems, clearly marking interacting many-body systems as the next frontier [54,55]. Exclusion is one complication that is sure to arise in such systems, and the results established herein are thus especially well-posed to serve as a benchmark for comparison. Moreover, the approach we presented can be extended further, as required, to capture more realistic scenarios featuring excluded volume interactions and local resetting. These will be considered elsewhere. Acknowledgments

I. ADDITIONAL COMPARISON BETWEEN SIMULATIONS AND MEAN-FIELD PREDICTIONS
In this section we further establish the effectiveness of the mean-field (MF) approach in the study of the model presented in the main text. We do so by extending the comparison provided in the main text, between the MF result for the density profile ρ of the main text Eq. (14) and direct numerical simulations of the model, for additional parameters.
The main text Fig. 2 compares a data collapse (for different values of the system size L) of the density profile versus x = /L to our MF results, for a resetting rate r = 1 and a mean densityρ = 0.2. Figure S1 complements this by providing the same comparison for the mean densityρ = 0.6 and the resetting rate r = 1. In Fig. S2 we provide an additional comparison for a resetting rate which grows linearly with L and a mean density ρ = 0.1. The remarkable agreement between MF theory and numerical simulations of the density profile versus x = /L reassures that MF provides a very good description for a broad range of model parameters and implies that the profile indeed remains a scaling function of /L, even when r ∝ L.

II. SYSTEM-SIZE SCALING OF ρ
Here we set the values of ν and µ in the ansatz ρ 0 = 1 − α L µ and ρ 1 = 1 − β L ν that appear in the main text Eq. (11). To this end, we first substitute them into the stationary boundary equation As L → ∞, this can only be satisfied if µ > 1, in which case Eq. (17) becomes Correspondingly, we conclude Having related µ to ν, we next set out to derive the value of ν. To do this, we substitute the ansatz at site = 0, i.e. ρ 0 = 1 − α L 1+ν , into the main text Eq. (9) for the density profile ρ at site = 1 and demand that it identifies with our ansatz, Recall that A ± is provided in the main text Eq. (6) as A ± = 1 + a 2 1 ± 1 + 4 a and that a = r (1 − ρ 0 ). Using the ansatz for ρ 0 , we can rewrite A L ± in the limit of large L as Depending on the value of ν, A L ± exhibits one of three distinct behaviors as L → ∞: For 0 < ν < 1, it exponentially decays/blows-up, while for ν > 1 we get . Yet, for ν = 1, we obtain the L-independent expression A L ± ∼ = e ± √ rα . We finally use this understanding to derive the value of ν. It is straightforward to show that the leading, large-L behavior of ρ 1 in Eq. (20) is This can only self-consistently agree with the ansatz as L → ∞ if ν = 1, which immediately also allows us to deduce µ = 2. The large-L scaling of the deviation of the density ρ 0 from unity is numerically verified in Fig. S3.