Symmetric Exclusion Process under Stochastic Resetting
Urna Basu Anupam Kundu Arnab Pal






























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    Raman Research Institute, Bengaluru 560080, IndiaInternational Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, IndiaSchool of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, IsraelCenter for the Physics and Chemistry of Living Systems. Tel Aviv University, 6997801, Tel Aviv, IsraelThe Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel
    
    
    We study the behaviour of a Symmetric Exclusion Process (SEP) in presence of stochastic resetting where the configuration of the system is reset to a step-like profile with a fixed rate r. We show that the presence of resetting affects both the stationary and dynamical properties of SEP strongly. We compute the exact time-dependent density profile and show that the stationary state is characterized by a non-trivial inhomogeneous profile in contrast to the flat one for r=0.  We also show that for r>0 the average diffusive current grows linearly with time t, in stark contrast to the √(t) growth for r=0. In addition to the underlying diffusive current, we identify the resetting current in the system which emerges due to the sudden relocation of the particles to the step-like configuration and is strongly correlated to the diffusive current. We show that the average resetting current is negative, but its magnitude also grows linearly with time t. We also compute the probability distributions of the diffusive current, resetting current and the total current (sum of the diffusive and the resetting currents) using the renewal approach. We demonstrate that while the typical fluctuations of both the diffusive and reset currents around the mean are typically Gaussian, the distribution of the total current shows a strong non-Gaussian behaviour. 
    
    Symmetric Exclusion Process  under Stochastic Resetting
        Arnab Pal
        Received: date / Revised version: date
    =======================================================
    
    
    
    § INTRODUCTION
    
    
    Stochastic resetting, which refers to intermittent interruption and restart of a dynamical process, has been a subject of immense interest in recent years. It has found applications in a wide range of areas starting from search problems <cit.>, population dynamics <cit.>, enzymatic catalysis <cit.> to computer science<cit.>, stock markets <cit.> and biological processes <cit.>. Stochastic resetting of a single Brownian particle is the paradigmatic example where the position of the particle is reset to a fixed point in space with a certain rate <cit.>. This simple act has drastic consequences on the statistical properties of the particle — it results in a nontrivial stationary state, anomalous relaxation behaviour, as well as finite mean first passage time. 
    
    
    
    
    
    Several variations and extensions of this simple model have been explored in recent years<cit.>.  Specific examples include: resetting in presence of an external potential <cit.>, in a confinement <cit.> or to an extended region <cit.>, and resetting to already excursed positions  <cit.>.
    Stochastic resetting has also been studied in more general nonequilibrium contexts – in reaction processes <cit.>, Lévy flights <cit.>, coagulation-diffusion process <cit.>, telegraphic process <cit.>, for run-and-tumble particles <cit.> and to model nonequilibrium baths <cit.>. Studies were not only limited to a constant rate resetting, other protocols have also been investigated in great details. These include deterministic resetting <cit.>, space <cit.> or time dependent <cit.> resetting rate, resetting followed by a refractory period<cit.>, non-Markovian resetting <cit.>  and resetting sensitive to internal dynamics <cit.>.
    
    
    
    
    
    
    An important question that naturally arises is how the presence of resetting dynamics affects the systems with many interacting degrees of freedom. This issue has not been explored much so far except for a few handful of  models. These studies include dynamics of KPZ-like fluctuating interfaces<cit.>, one-dimensional quantum spin chain<cit.>, and a pair of interacting Brownian particles <cit.>. In all these cases, resetting leads to nonequilibrium stationary states, characterized by non-Gaussian fluctuations.  
    However, the effect of resetting on the behaviour of current, which plays an important role in characterizing the nonequilibrium stationary state, has not been studied yet. This question is of paramount importance, because presence of stochastic resetting introduces an additional time-scale and is expected to modify the behaviour of current significantly. The  exclusion processes <cit.>, which are simple well known models of interacting particles, provide a natural playground for exploring these questions. 
    
    
    
    
    
    In this article we study the effect of stochastic resetting on Symmetric Exclusion Process (SEP) <cit.> and explore how the presence of resetting changes the dynamical and stationary properties of SEP. The stochastic resetting is implemented by interrupting the time-evolution at some rate r, and restarting the process from a specific configuration. It turns out that the incorporation of the resetting mechanism introduces an extra current  in addition to the usual diffusive particle current J_d.
    We show that, for r>0, the average diffusive current increases linearly with time t, in contrast to the √(t) behaviour in the absence of the resetting <cit.>. Additionally, the average resetting current also shows a linear temporal growth in magnitude, although it remains negative. We also compute the distribution of the diffusive current J_d, resetting current , as well as the total current J_r=J_d+.  We observe that, while the diffusive and resetting current show Gaussian behaviour, the fluctuations of the total current are characterized by a strongly non-Gaussian distribution. 
    
    
    
    The article is organized as follows: In the next section we define our system and summarize our main results. Sec <ref> is devoted to the computation of the time evolution of the density profile under resetting. In Sec. <ref> we investigate the  behaviour of the particle current – Sec. <ref> and <ref> focus on the diffusive and resetting currents respectively, whereas the behaviour of the total current J_r is explored in Sec. <ref>. We conclude with some open questions in Sec. <ref>.  
    
    
    
    
    
    
    § MODEL AND RESULTS
    
    
    The symmetric exclusion processes (SEP) is a paradigmatic model for interacting particle systems <cit.> which have been used to describe a wide range of physical phenomena including particle transport in narrow channels, motion of molecular motors,  ion transport through porous medium etc. This process describes unbiased motion of particles on a lattice which interact via mutual local exclusion. In this section we define the dynamics of SEP with stochastic resetting and present a brief summary of our main results.
    
    
    
    Let us consider a periodic lattice of size L where each lattice site can contain at most one particle. The state of a site, say x, is characterized by a variable s_x which takes values 1 and 0 depending on 
    whether the site x is occupied or not, respectively.   The configuration of the system is characterized by C={s_x; x=0,1,2,… L-1 }. We consider the case of half-filling, , 
    the total number of particles ∑_x s_x = L/2.  The system evolves according to the following two dynamical moves:
    
    
      *  Hopping: A particle randomly hops to one of its nearest neighbouring sites with unit rate, provided the target site is empty. 
    
      *  Resetting: In addition,  the system is `reset' to some specific configuration C_0 with rate r. In the following we consider C_0 to be a step like state where all the particles are in the left-half of the lattice: 
    
     C_0:={
                       s_x=1     for 0 ≤ x ≤L/2-1,
    
                       s_x=0      otherwise..
     
    Both the hopping and resetting dynamics conserve the total number of particles, so that the half-filling condition is respected at all times, and the global particle density remains fixed at 1/2.  
     Between two resetting events the time evolution of the system is governed by the  hopping dynamics only. The time-scale associated with the resetting mechanism is given by r^-1, which also gives a measure of the typical duration between two consecutive resetting events.  Figure <ref> shows typical examples of the time evolution for two different values of the resetting rate r.
    
    
    
    
    
    In the absence of resetting, the master equation governing the time-evolution of the probability P_0( C,t) for the system to be in the configuration C at time t is given by 
    𝕀/𝕀 t P_0( C,t) =  L_0  P_0( C,t).
    
    Here L_0 is the Markov matrix in absence of the resetting, , L_0  P_0( C,t) = ∑_ C'[W_C' → C P_0( C',t) - W_C → C' P_0( C,t)] where W_C' → C denotes the rate for the jump C → C' due to hopping dynamics  only. Note that, W_C' → C=1 only if the two configurations C and C' are connected by a single hop of a particle to a neighbouring site.  
    
    
    Let P( C,t) denote the probability of finding the system in the configuration C  at time t in presence of resetting. In this case, the master equation reads,
    𝕀/𝕀 t P( C,t)    =    L_0  P( C,t)       + r ∑_C'  C_0 P( C',t) δ_C,C_0 - r  P( C,t) (1-δ_C,C_0)    =    ( L_0-r)  P( C,t) + r δ_C,C_0, 
    where δ_C,C_0 is the Kronecker delta symbol, which takes the value unity when C is same as C_0, and is zero otherwise.
    It is straightforward to write a formal solution of Eq. (<ref>),
     P( C,t)    =    e^( L_0-r)t P( C,0) + r ∫_0^t 𝕀 s  e^( L_0-r)sδ_C,C_0   =    e^-rt P_0( C,t) + r ∫_0^t 𝕀 s  e^-rs P_0( C,s). 
    Here P_0( C,t)=e^ L_0 t P( C,0) is the probability of finding the system in configuration C at time t in the  absence of resetting given that the system was initially at C_0,, P( C,0)= P_0( C,0)=δ_ C,  C_0. Equation (<ref>) is nothing but the renewal equation for the configuration probability, which has been obtained earlier and used to study resetting phenomena in various other contexts <cit.>. Note that Eq. (<ref>) holds true irrespective of the specific choice of C_0 given in Eq. (<ref>).  
    
    
    
    
    
    
    
    
    
    
    In the absence of resetting the ordinary SEP on a ring relaxes to an equilibrium state with flat density profile and zero current. The approach to the equilibrium state, starting from the step-like initial configuration C_0, is characterized by a diffusive current flowing through the system. It has been shown that, for an infinitely large system, the time-integrated current measuring the net particle flux through the central bond up to time t, grows as √(t) for large t<cit.>. Presence of resetting is expected to affect these characteristics of SEP which we investigate in detail in this paper. A brief summary of our results is presented below. 
    
    
      *  First,  we  compute an exact expression for the evolution of the average density profile ρ(x,t) =  s_x(t) for any arbitrary value of the resetting rate r, which is given in Eq. (<ref>). 
    We observe that the evolution is non-trivially modified due to the presence of resetting which leads to an inhomogeneous stationary density profile [see Fig. <ref>(b)] in contrast to 
    the flat one for r=0. 
    
    
      *  This inhomogeneous density profile provides some characterization of the non-equilibrium state of the system. It is, however, also important to look at how the particle currents in the system are affected by the introduction of resetting. In addition to the usual diffusive current J_𝕀(t) created due to the local hopping of the particles, there is also a contribution (t) to the total current due to the global movements of the particles during the resetting events.
    
    We show that the behaviour of the diffusive current changes drastically in presence of resetting. In particular, we compute the average diffusive current J_d(t) exactly, which, in the long time limit, shows a linear growth with time t,
        J_d(t) ≃ t√(r/r+4).
      This behaviour is in stark contrast to the √(t) growth which is seen in absence of resetting <cit.>. Similar change in the dynamical behaviour  is also observed for the variance of the diffusive current,  which also grows as ∼ t in presence of resetting,  as opposed to √(t).  We explore the behaviour of the resetting current  too and show that, its average and variance also grow linearly with time. 
    We also investigate the probability distribution of (t) and demonstrate that, in the long-time regime, the typical fluctuations of J_d(t) around its mean is characterized by a Gaussian distribution. Similar Gaussian fluctuations are also expected for the resetting current .
      *  Finally, we study the behaviour of the total current J_r =  + and calculate the average J_r(t) and the second moment J_r^2(t) as functions of time t.  In the long-time limit the moments reach stationary values. In particular, we show that the average stationary current is given by
     J_r  = 1/√(r(r+4)). 
    
    We also compute the stationary probability distribution of the total current P_r^st(J_r), for small values of r, using a renewal approach. Interestingly, it turns out that, this distribution is non-Gaussian, and has very asymmetric behaviour at the two tails.
    
    
    
    
    
    
    
    § DENSITY PROFILE
    
    
    Presence of repeated resetting to the inhomogeneous configuration C_0 destroys the translational invariance in the system and a non-trivial density profile can be expected, even in the stationary state. 
    
    
    The average density ρ(x,t) =  s_x(t) is given by the probability that the site x is occupied any time t. The time-evolution equation for the density profile can be derived by multiplying Eq. (<ref>) by s_x and summing over all configurations C,𝕀/𝕀 tρ(x,t)    =   ρ(x+1,t)+ ρ(x-1,t)- 2ρ(x,t)        -rρ(x,t) + r  ϕ(x) 
    Here ϕ(x) is the  density profile corresponding to the resetting configuration C_0 which, as mentioned before, is also taken as the initial profile.  
    The exact time-dependent density profile ρ(x,t) can be obtained by solving Eq. (<ref>). To this end we introduce the discrete Fourier transform
    ρ̃(n,t) = ∑_x=0^L-1 e^i  2π n x/Lρ(x,t),  with  n=0,1,2 … L-1.   
    Substituting Eq. (<ref>) in Eq. (<ref>), we get,
    𝕀/𝕀 tρ̃(n,t) = - (λ_n+r) ρ̃(n,t) + r ϕ̃(n) 
    
    with λ_n= 2(1- cos2 π n/L) and ϕ̃(n) is the Fourier transform of the resetting (and initial) profile ϕ(x). Equation (<ref>) can immediately be solved,
    ρ̃(n,t) =  rϕ̃(n)/r+λ_n  + λ_n ϕ̃(n)/r+λ_n e^-(r+λ_n)t
    The density profile is then obtained by inverting the Fourier transform,
    ρ(x,t)    =   r/L∑_n=0^L-1ϕ̃(n)/r+λ_n  e^-i 2 π n x/L      + 1/L∑_n=0^L-1λ_n ϕ̃(n)/r+λ_n e^-(r+λ_n)t e^-i 2 π n x/L
    
    In the stationary state, the second term decays exponentially and the stationary density profile is given by,
    ρ(x)    =   r/L∑_n=0^L-1ϕ̃(n)/r+λ_n  e^-i 2 π n x/L   =   1/2 + r/L∑_n=1^L-1ϕ̃(n)/r+λ_n  e^-i 2 π n x/L
    where we have used the fact that ϕ̃(0) = ∑_x ϕ(x) = L/2. Clearly, in absence of resetting,  for r=0, we get the flat profile which corresponds to the equilibrium scenario. For non-zero r, however, the stationary profile is non-trivial and corresponds to a non-equilibrium stationary state, carrying non-zero current.  We will explore that in the next section. 
    
    
    
    
    It is worth mentioning that ρ(x,t) also satisfies a renewal equation (following directly from Eq. (<ref>)) in terms of the density profile ρ_0(x,t) in absence of resetting,
    ρ(x,t) = e^-rtρ_0(x,t) + r ∫_0^t 𝕀τ e^- r τρ_0(x,τ). 
    For the sake of completeness we have added a brief review of the density profile and its evolution for ordinary SEP in the Appendix <ref>. Using the explicit form of ρ_0(x,t) given in Eq. (<ref>) it is straightforward to check that Eq. (<ref>) leads to Eq. (<ref>). 
    
    We have not used any specific form of ϕ(x) so far; in fact, the results above are valid for resetting to any generic profile. For the specific choice of the step-like configuration given in Eq. (<ref>) we have ϕ(x)=1- Θ(x+ 1 -L/2) and
    ϕ̃(n) = 1-(-1)^n/1-e^i 2 π n/L = {
    1+i  π n/L       for odd    n 
    0            for even    n                                                        
    .
    
    
    
    
    
    
    
    
    In that case, the density profile takes the form,
    ρ(x,t)    =   ρ(x) + 1/L∑_n=1,3^L-1 e^- i  2 π n x/Lλ_n (1+i  π n/L) /r+λ_n  e^-(r+λ_n)t      
    where 
    ρ(x) = 1/2 + r/L∑_n=1,3^L-1 e^- i  2 π n x/L(1+i  cotπ n/L)/r+λ_n
    is the stationary profile.
    
    Figure <ref>(a) shows the time-evolution of the density profile ρ(x,t) for a specific resetting rate r and Fig. <ref>(b) shows stationary profiles ρ(x) for different values of r. In both cases, the analytical results (solid lines) are compared with the data obtained from numerical simulations (symbols). An excellent match confirms our analytical prediction. 
    
    
    
    § PARTICLE CURRENT
    
    
    The behaviour of current plays an important role in characterizing the interacting particle systems like exclusion processes. 
    For ordinary SEP, there is no particle current flowing through the system in the stationary (equilibrium) state. However, starting from a step-like initial configuration, the relaxation to equilibrium is characterized by the presence  of a non-vanishing particle current. In particular, the behaviour of the time-integrated current, , the net particle flux through the central bond up to time t, has been studied extensively in the past and it was shown that at long time limit, the average flux grows ∼√(t)<cit.>.
    
    
    
    In presence of resetting, there are two different kinds of particle motions, consequently the total current can be expressed as,
    
    J_r(t) = J_d(t) + (t) 
    Here  is net diffusive flux, , the net number of particles which crossed the central bond due to the nearest neighbour hopping.  denotes the contribution due to the sudden
    reset to the step-like configuration C_0. Note that, after a resetting, the system is brought back to C_0, , there are no particles to the right of the central bond, implying that the total current J_r is also reset to zero after each resetting event. Figure <ref> shows the time evolution of J_d and J_r for a typical trajectory of the system. The sudden jumps in J_r indicate the resetting events. 
    
    
    
    In the absence of resetting, the only source of current is the diffusive hopping motion. In the following we explore the behaviours of all these three different currents, in presence of resetting. 
    
    
    
    
     §.§ Diffusive current
    
    
    The diffusive current (t) measures the total number of particles which crossed the central bond (L/2 -1, L/2) during the time interval [0,t], and can be expressed as a time integral,
    
    J_d(t)    =   ∫_0^t 𝕀 s  j(s) 
    Here, j(t) denotes the instantaneous diffusive current, , the number of particles crossing the central bond during the time-interval t and t+𝕀 t. The average instantaneous current is given by,
    
    
    
     j(t)    =     s_L/2-1(1-s_L/2) -  (1- s_L/2-1)s_L/2   =    ρ(L/2 -1,t) - ρ(L/2,t )
    
    Using the explicit expression for the density from eq. (<ref>) one gets,
     j(t)  = 2/L∑_n=1,3^L-1[r/r+ λ_n+ λ_n/r+λ_ne^-(r+λ_n)t].
    
    In the limit of thermodynamically large system size, L →∞, the sum in the above expression can be converted to an integral over continuous variable q= 2 π n/L  and we get, 
     j(t)  = ∫_0^2 π𝕀 q/2 π[r/r+λ_q+ λ_q/r+λ_qe^-(r+λ_q)t]
    
    where λ_q = 2(1- cos q). In the long-time regime, the second term decays exponentially and j(t) reaches a stationary value,
    lim_t→∞ j(t)  = ∫_0^2 π𝕀 q/2 πr/r+ 2(1- cos q) = √(r/r+4). 
    The average net flux J_d(t) up to time t can be found by integrating the instantaneous current,
     J_d(t)   =   √(r/r+4) t     +   ∫_0^2 π𝕀 q/2 πλ_q/(r+λ_q)^2(1- e^-(r+λ_q)t). 
    Clearly, in the long-time regime, the second term goes to a constant and the first term dominates the behaviour of the average current which grows linearly with time,
     J_d(t)≃√(r/r+4) t.
    
    This equation is one of our main results, which shows that the behaviour of the diffusive current changes drastically by the presence of resetting; instead of the standard √(t) growth in a diffusive system, resetting yields a much faster, linear, temporal growth of the  diffusive current.
    
    The average current (t) at any time t,, before reaching the ∼ t behaviour, can be obtained from Eq. (<ref>) by evaluating the q-integral numerically. In fact, one can also derive an alternative expression which lends itself more easily to numerical evaluation. Let us recall that the density profile ρ(x,t) satisfies a renewal equation (<ref>) for any x. Then, clearly, j(t) must also satisfy the same renewal equation,
    
     j(t)   =    e^-r t j_0(t)  + r ∫_0^t 𝕀τ  e^-r τ j_0(τ) 
    where j_0(t) denotes instantaneous current through the central bond in the absence of resetting. The average instantaneous current is given by j_0(t)  = e^-2t I_0(2t) where I_0 is the Modified Bessel function of the first kind (see Appendix <ref> for the details). The average diffusive net current is obtained by integrating the above equation w.r.t. time [see Eq. (<ref>)],
    
    
    
    
    (t)    =   ∫_0^t 𝕀τ e^-r τ (1+rt - rτ)  j_0(τ) .   
    It is straightforward to show that Eq. (<ref>) is equivalent to Eq. (<ref>). Average current (t) computed from Eq. (<ref>), for different values of r, are plotted together with the same obtained from simulation in Figure <ref>(a). 
    
    
    
    An explicit form for J_d(t) can be derived for small r ≪ 1.
    Using a variable transformation w=r τ, and using the exact form for j_0(τ) we get,
    (t)    =   1/r∫_0^rt𝕀 w   (1 + rt - w)  e^-w e^-2 w/r I_0(2 w/r).   
    For small r, the argument of I_0 is large and one can use the asymptotic form for the Modified Bessel function given in Eq. (<ref>),
     (t)    ≃   1/r∫_0^rt𝕀 w   (1 + rt - w)  e^-w1/2 √(π w/r)   =   1/2 √(r)[(r t+ 1/2) erf(√(rt))+ √(rt/π)e^-rt]. 
    In the short time-regime this function grows as √(t) which is reminiscent of the free SEP and crosses over to the linear behaviour for t ≫ r^-1. Figure <ref>(b) shows plot of √(r)(t) as a function of r t for different small values of r which shows a perfect collapse and matches with the scaling function given by the above equation. 
    
    To characterize the fluctuation of the diffusive current we next calculate the second moment of J_d. The above renewal equation method cannot be applied directly to compute higher order moments. To this end, we now adopt a different approach.  Let us assume that there are n resetting events during the time interval [0,t]; moreover, let t_i denote the interval between (i-1)^th and i^th events, so that ∑_i=1^n+1 t_i = t. Note that, t_n+1 denotes the time-interval between the last reset and the final time t. Let us also recall that between two consecutive resetting events the system evolves following ordinary SEP dynamics.
    The diffusive current during the interval [0,t] can, then, be expressed as,
    
    J_d = ∑_i=1^n+1 J_0(t_i), 
    where, J_0(t_i) are independent of each other. For notational convenience, we denote J_i ≡ J_0(t_i). The probability density that the diffusive current will have a value  in time t is then given by
    
    P(J_d,t)    =   ∑_n=0^∞∫_0^t ∏_i=1^n+1𝕀 t_i    P_n({t_i };t)        ×∫∏_i=1^n+1𝕀 J_i P_0(J_i,t_i) 
       δ(J_d-∑_i J_i),   where,
     P_n({t_i };t)    =    r^n e^-r∑_i=1^n+1 t_iδ(t-∑_i t_i) 
    denotes the probability of having n resetting events with duration t_i within the interval [0,t]. The distribution of the individual J_i-s are denoted by P_0(J_i,t_i) which is exactly the distribution of the diffusive current in SEP, in the absence of resetting. 
    
    
    
    
    
    
    
    To handle the constraints presented by the δ-functions, it is convenient to calculate the Laplace transform w.r.t. time t of the moment generating function e^λ J_d,
    Q(s,λ)    =    L_t → s [ e^λ J_d]= ∫_0^∞𝕀 t  e^-s t e^λ J_d   =   ∫_0^∞𝕀 t  e^-s t∫𝕀 J_d  e^λ J_d P(J_d,t).
    
    Using Eq. (<ref>), and performing the integrals over 
    and t, we get,
    
    Q(s,λ)   =   ∑_n=0^∞ r^n ∫_0^∞∏_i^n+1𝕀 t_i exp[-(r+s) ∑_i=1^n+1 t_i]
     [0.2em]
          ×∫∏_i^n+1𝕀 J_i exp[λ∑_i=1^n+1 J_i] P_0(J_i,t_i)    =   ∑_n=0^∞ r^n h(s,λ)^n+1
    where, we have denoted,
    
    h(s,λ) = ∫_0^∞𝕀τ e^-(r+s)τ∫𝕀 J_0   e^λ J_0 P_0(J_0,τ)
    Performing the sum in Eq. (<ref>), we get,
    
    Q(s,λ) = h(s,λ)/1- r h(s,λ)
    which gives a simple relation between the moment generating functions of the current in the presence and absence of resetting. 
    To calculate h(s,λ) we need the current distribution P_0(J_0,τ) for the ordinary SEP, which is not known in general for arbitrary values of τ. However, for small values of r and s, the τ-integral in Eq. (<ref>) is dominated by large values of τ, and in that case one can use the result of Ref. <cit.> where the authors have derived an expression for the moment generating function of J_0(τ) in the large time limit.
    Adapting their result to our specific case (see Appendix <ref> for the details), we have,
    ∫𝕀 J_0   e^λ J_0 P_0(J_0,τ) = e^λ J_0≃ e^√(τ)F(λ), 
    with, 
    
    F(λ) = -1/√(π)Li_3/2(1-e^λ).
    Here Li_α(z) denotes the Poly-Logarithm function (see Ref. <cit.>, Eq. 25.12.10). Substituting Eq. (<ref>) in Eq. (<ref>) and performing the integral over τ, we get, for small r and s,
    h(s,λ)    =   1/r+s[1+ √(π)F(λ)/2√(r+s)e^F(λ)^2/4(r+s)(1+erf[F(λ)/2√(r+s)]) ]      
    One can easily extract the Laplace transforms of the moments using Eq. (<ref>) along with (<ref>). First, we have,
     L_t → s[(t) ]    =   𝕀/𝕀λ Q(s,λ)|_λ=0   =   √(r+s)/2s^2
    The average current can be obtained by inverting the Laplace transform,
    μ_𝕀(t) ≡(t)    =    L^-1_s → t[√(r+s)/2s^2]
    
    The inversion can be performed exactly using Mathematica, and yields, 
    μ_𝕀(t)   =   1/2 √(r)[(r t+ 1/2) erf(√(rt))+ √(rt/π)e^-rt] 
    Note that the above equation is the same as Eq. (<ref>), which was obtained using a different method.  
    
    
    
    
    
    The Laplace transform of the second moment is obtained from the second derivative of Q(s,λ), L_t → s^2(t)    =   𝕀^2/𝕀λ^2 Q(s,λ)|_λ=0
       =   1/π s^2+ b √(r+s)/2s^2 + r/2 s^3
    where b=(1-1/√(2)). Fortunately, the inverse Laplace transform can be performed exactly  in this case also, and it yields, for small r,^2(t)    =   1/4 π[t(π r t+4)+2 b √(π t) e^-r t         + b π/√(r)(1+2 rt)erf(√(rt))] 
    Note that, the above expression is expected to be valid for large t, as we have assumed s to be small. The variance of the diffusive current σ^2_d(t) =^2(t)  -(t) ^2 can be obtained using Eqs. (<ref>) and (<ref>). In particular, in the long time limit, the variance increases linearly with time t, and is given by,
    σ_d^2(t) ≃ t [4-π/4 π + √(r)/2(1- 1/√(2))]
    
    Figure <ref>(c) shows a plot of σ^2_d(t) vs t for different values of r, obtained from numerical simulations; all the curves show linear growth in the long time regime. The curves corresponding  to small values of r ≪ 1 are compared with the analytical result (solid lines) which shows a perfect match for t > 10.
     Distribution of : It is interesting to investigate the probability distribution of the diffusive current J_d(t).  From Eq. (<ref>) we observe that J_d(t) is the sum of the hopping currents J_0(t_i)  between successive resetting events. Since the time-evolution of the system is Markovian and after each resetting event the system is brought back to the initial configuration, the variables J_0(t_i) are independent and distributed identically [Of course, the time durations { t_i} are different.]. As mentioned earlier, the distribution of J_0(t_i) is known <cit.> and  has finite moments. 
    Over a large time interval t, the number n of resetting events, which is also a random quantity, is typically large  and on an average grows linearly with time t; in fact, n = rt.  For t ≫ r^-1,(t) is a sum of large number of independent random variables. Hence, by central limit theorem, one can expect that for large t, the typical distribution of J_d would be a Gaussian: 
    
    P(,t) ≃1/√(2 πσ_d^2(t))exp(-[J_d-μ_𝕀(t)]^2/2 σ_d^2(t)) 
    where the mean μ_𝕀(t) and the variance σ_d^2(t) are given in Eqs. (<ref>) and (<ref>) respectively. This prediction is verified in Fig. <ref>(a) where the Gaussian form of P(,t) is compared to the data obtained from numerical simulations for a set of (large) values of t and fixed r. Clearly, the analytical curves are indistinguishable from the simulation data, which confirms our prediction.   Fig. <ref>(b) shows the same data plotted against the scaled variable (-μ_d(t))/σ_d(t) and compared with the standard normal distribution (solid black line). 
    
    
    
    
    
    
     §.§ Resetting current
    
    
    Presence of the resetting dynamics gives rise to a resetting current  [see Eq. (<ref>)] which measures the flow of particles due to the sudden change in the configuration of the system. In this Section we investigate the properties of this resetting current. 
    Let us remember that the number of particles crossing the central bond (from right to left) at the resetting event is exactly same as the hopping current (from left to right) during the period after the previous resetting event. Net resetting current during a time interval [0,t], then, can be expressed as,
     = - ∑_i=1^nJ_0(t_i)
    where, as before, n denotes the number of resetting events in time t and t_i denotes the interval between (i-1)^th and i^th resetting events. 
    Note that, the upper limit of the sum is n in the above  Eq. (<ref>) as there is no contribution to the resetting current after the last resetting event.
    
    
    
    To calculate the moments of  we follow the same method as in Sec. <ref> and calculate the Laplace transform of the moment generating function of ,
    K(s,λ) = ∫_0^∞𝕀 t  e^-s t∫𝕀 e^λ𝒫(,t).  
    Here 𝒫(,t) denotes the probability that the resetting current has a value  at time t, and is given by,
    𝒫(,t)    =   ∑_n=0^∞∫_0^t ∏_i=1^n+1𝕀 t_i    P_n({t_i };t)     ×   ∫∏_i=1^n𝕀 J_i P_0(J_i,t_i) 
       δ(+∑_i=1^n J_i).  
    with P_n({t_i };t) given in Eq. (<ref>). As before, we have used J_i ≡ J_0(t_i). Using Eq. (<ref>) in Eq. (<ref>) and performing the integrals over t and , we get, 
    
    K(s,λ)    =   1/(r+s)∑_n=0^∞ r^n h(s,-λ)^n   =   1/(r+s)[1- rh(s,-λ)] 
    where h(s,λ) is given by Eq. (<ref>). 
    As mentioned already, it can computed exactly for small values of r, s and is given by Eq. (<ref>). 
    
    
    
    Next we calculate the moments of the resetting current using Eqs. (<ref>) along with Eq. (<ref>). First, we have the Laplace transform of the average resetting current,
     L_t→ s [(t)]    =   𝕀/𝕀λ K(s,λ)|_λ=0   =    -r/2s^2√(r+s).
    
    The inverse transform can be performed exactly to obtain, 
    (t) = - 1/2√(r)[(rt-1/2) erf(√(rt))+ √(rt/π)e^-rt].  
    Note that the above expression is expected to be valid 
    for small values of r ≪ 1 and large t ≫ 1.
    Equation (<ref>) is very similar to Eq. (<ref>), which gives the average diffusive current (t), except, of course, the fact that the average reseting current is negative. In fact, at very long-times t ≫ r^-1, we see a linear growth in magnitude, 
    (t) = -(t)≃ - √(r)t/2,
    
    At short-times, however, a different behaviour is seen. From Eq. (<ref>), for t ≪ r^-1, we have,
    (t) = -2r t^3/2/3 √(π)+  O(t^5/2).
    
    Clearly, at short-times, the resetting current grows much faster than the diffusive current. Figure <ref>(a) shows a plot of (t) as a function of t for different values of r which illustrates these features. 
    
    
    It is also interesting to look at the fluctuations of . From Eq. (<ref>) we can find the Laplace transform of the second moment,
     L_t→ s [^2(t)]   =   𝕀^2 /𝕀λ^2 K(s,λ) |_λ=0   =   br/2 s^2 √(r+s)+ r(π r+2s)/2 π s^3(r+s)
    where, as before, we have used b= 1- 1/√(2).
    Once again, the Laplace transform can be inverted exactly and yields, for r ≪ 1 and t ≫ 1,^2(t)   =   1/4 π r[2e^-rt(2-π + br √(π t)) + π -r + 4 rt    +   π(rt-1)^2 + b π√(r)(2rt-1)erf(√(rt))].
    The variance of the resetting current σ_reset^2(t)=^2(t) -(t)^2 can be computed from Eqs. (<ref>) and (<ref>) and it turns out that the variance also increases linearly at the long time limit t ≫ r^-1. In fact, it is straightforward to show that, in this limit, σ_reset^2(t)=σ_𝕀^2(t) [see Eq. (<ref>)]. Figure <ref>(b) shows σ_reset^2(t) for different values of r obtained from numerical simulations together with the analytical prediction for small r.
    
    
    We conclude the discussion about the resetting current with a brief comment about the probability distribution 𝒫(,t). Since , similar to ,  is also a sum of a set of independent variables J_0(t_i), we can use the central limit theorem to predict the behaviour of the corresponding distribution. In fact, for rt ≫ 1, one can expect that 𝒫(,t) is similar to P(,t) and has a Gaussian behaviour around the mean value,
    𝒫(,t) ≃1/√(2 πσ^2_reset(t))exp[-(-(t))^2/2 σ_reset^2(t)].
    
     §.§ Total current
    
    
    
    In this section we investigate the behaviour of the total current J_r, as defined in Eq. (<ref>). J_r(t) measures the net number of particles which have crossed the central bond towards right (by hopping, or due to resetting) up to time t. As already mentioned, J_r is set to zero after every resetting event; the contribution to the total current comes only from the diffusion of the particles after the last resetting event. 
    Consequently, one can write a renewal equation for P_r(J_r,t), the probability that, at time t, the total current will have a value J_r,
    P_r(J_r,t) = e^-r t P_0(J_r,t) + r ∫_0^t 𝕀 s  e^-rs P_0(J_r,s). 
    Here P_0(J_r,s) denotes the probability that,  starting from 
    C_0, in absence of resetting, J_r number of particles cross the central bond until time s. We will use the above equation to explore P_r(J_r,t), but, first it is useful to investigate the mean and the variance of the total current. 
    
    
    It is easy to see that all moments of J_r should also satisfy a renewal equation similar to Eq. (<ref>). In particular, the average total current must satisfy, 
     J_r(t)  = e^-rt J_0(t) + r∫_0^t 𝕀τ e^-rτ J_0(τ) 
    where J_0(t) is the average current in absence of resetting, and is  given by Eq. (<ref>). 
    
    Unfortunately, the above integral in Eq. (<ref>) cannot be computed analytically. However, it is possible to numerically evaluate the integral and get J_r (t) for any time t. 
    This is shown in Fig. <ref> for different values of r and compared with numerical simulations (symbols) which matches perfectly at all times.
    
    
    
    
    
    
    For small values of r, a more explicit expression for the average total current J_r(t) can be derived. In that case, it is convenient to rewrite Eq. (<ref>) as
     J_r(t) = e^-rt J_0(t)  + ∫_0^rt𝕀 u   e^-u J_0 (u/r) 
    The integral is dominated by the contribution from small u ∼O(1); consequently, u/r is large for small r, and we can use the asymptotic expression J_0(u/r) ≃√(u/r π). Substituting that in the above equation, and performing the integral, we get, for large t, J_r (t)  = 1/2√(r)erf(√(rt)). 
    Equation (<ref>) provides an explicit expression for the average total current for small r, and in the large time regime. Note that, J_r(t) given by the above equation is same as (t)  + (t), as clearly seen from Eqs. (<ref>) and (<ref>). This is expected as the total current is a sum of the diffusive current and the resetting current [see Eq. (<ref>)]. 
    
    We have also measured the total current J_r from numerical simulations. Figure <ref>(b) shows a plot of √(r) J_r (t) as a function of rt for different (small) values of r, as obtained from numerical simulation; the solid line corresponds to erf(√(rt)). The perfect collapse of all the curves verifies our analytical prediction.
    
    From Eq. (<ref>) it can be seen that for t ≪ r^-1 the average total current grows as √(t), which is a signature of the ordinary SEP. On the other hand, in the large time limit J_r reaches a stationary value  1/2 √(r). 
    
    In fact, the stationary value of the average total current  J_r can be calculated exactly from Eq. (<ref>) for any value of r. As we have already seen, at large times t,J_0(t) ∼√(t), hence, the first term in Eq. (<ref>) decays exponentially and the large-time behaviour of the average total current is dominated by the second integral in the above equation. Recalling Eq. (<ref>) and using the series expansion of the Modified Bessel functions I_0 and I_1, (see Ref. <cit.>, Eq. 10.25.2) we have,
          ∫_0^t 𝕀τ  e^-rτ J_0(τ)   =   ∑_m=0^∞1/m!(m+1)!1/(r+2)^2m+3[Γ_2m+3-Γ_2m+3[(r+2)t]       +  (r+2)(m+1)(Γ_2m+2-Γ_2m+2[(r+2)t]) ],
    
    where Γ_n and Γ_n(x) are the Gamma function and the incomplete Gamma function, respectively. Γ_n(x) decays to zero for large x for all values of n, and hence, in the  long time limit we have the contributions only from the t-independent terms,
    lim_t →∞ J_r    ≃    r∑_m=0^∞(2m+1)!/m!(m+1)!(m+1)(r+2)+2m+2/(r+2)^2m+3   =   1/√(r(r+4)). 
    Clearly, in the long-time limit the average total current reaches a stationary value μ_r = 1/√(r(r+4)) which decreases as the resetting rate r is increased. For small r,μ_r ≈1/2√(r) which is same as what we obtained by taking t →∞ limit in Eq. (<ref>). On the other hand, for large r ≫ 1,μ_r approaches 1/r.
    
    Physically, the limiting behaviours of the stationary value of the average total current can be understood from the following argument. Since the value of J_r is reset to zero after each resetting event, the final contribution to J_r comes only from the diffusion of particles after last resetting event. Moreover, the typical duration since the last resetting event is τ_r ∼ 1/r. For small r, this typical duration is long, and the average diffusive current (without resetting) during this period is  ∼√(τ_r) = 1/√(r). On the other hand, for large r,τ_r is small, the diffusive current is ∼τ_r = 1/r. 
    
    
    
    Next we calculate the second moment of the total current J_r(t)^2 . As mentioned already, the second moment also satisfies a renewal equation of the form,
     J_r(t)^2  = e^-rt J^2_0(t) + r ∫_0^t 𝕀τ  e^-r τ J^2_0(τ). 
    The above equation is valid at all times and for all values of r. Unfortunately, however, the behaviour of J^2_0(τ) is known only for long time τ (see Eq. (<ref>)) so we are not able to calculate an exact analytical expression for J_r(t)^2 for any arbitrary time t.   Nevertheless, one can use Eq. (<ref>) along with Eqs. (<ref>) and (<ref>) to calculate J_r(t)^2 for small values of r, where the integral in  
    Eq. (<ref>) is dominated by the contribution from large τ≫ r^-1. This exercise leads to a simple analytical formula for the second moment for small r (and large time t),
     J^2_r(t)≃1/π r(1 - e^-rt)+ 1/2√(r)(1- 1/√(2)) erf(√(r t))     
    
    
    Figure <ref>(c) shows the plot of J_r^2(t) as a function of t for different values of r. The curves for small r are compared with the analytical result Eq. (<ref>) which show an excellent match. Similar to the average total current, the second moment J_r^2(t) also eventually reaches a stationary value which, for small values of r, can be obtained by taking lim t →∞ in Eq. (<ref>), 
     J_r^2  = 1/π r + 1/2√(r)(1- 1/√(2)). 
    One can immediately calculate the stationary value of the variance σ_r^2 =  J_r^2  -  J_r ^2; as r → 0,σ_r^2 ≃ (4- π)/ 4 π r. 
    
    On the other hand, for short time t, we have, 
     J_r^2 (t)≃t/π+ √(t/π)(1- 1/√(2))
    At very short-times, one expects a √(t) behaviour which crosses over to a linear behavior as t is increased. This is also seen in Figure <ref>(c),  where the approach to the stationary value appears predominantly linear.
     Correlation between  and : 
    The computation of the second moment of the total current 
    J_r provides a way to estimate the correlation between the diffusive and resetting components of the current. 
    From the definition of the total current Eq. (<ref>), we get,
     J_r^2(t)  = ^2(t)  + ^2(t)  + 2 (t)(t) . 
    
    The connected correlation C(t)= (t) (t)  - (t)(t) is then given by,
    
     C(t)= 1/2[σ_r^2(t) - σ_𝕀^2(t) - σ_reset^2(t)]
    
    where σ_r^2,σ_𝕀^2 and σ_reset^2 are the variances of the total, diffusive and resetting currents, respectively. Using Eqs.  (<ref>), (<ref>), and (<ref>) along with Eqs. (<ref>), (<ref>) and (<ref>), we get, for small values of r,
    C(t)    =   1/4 π r[4-π + rt e^-2rt   +    e^-rt(π -4 -2 r√(π t)[b-√(r)t erf(√(rt))])   -    rt(4-π+π rt)-bπ√(r)(2rt-1)erf(√(rt))    +   π(r^2t^2-1/4)erf(√(rt))^2]. 
    Clearly, the diffusive and resetting currents are strongly correlated. To understand the nature of this correlation we look at the limiting behaviour of C(t). 
    At long times t ≫ r^-1, we get a linear temporal growth from  Eq. (<ref>),
    
    C(t) ≃ - σ_𝕀^2(t) ≃ -t [4-π/4π + b√(r)/2].
    
    On the other hand, for short-times t ≪ r^-1 (but t ≫ 1) we get,
    
    C(t) = - 2 b r/3 √(π)t^3/2 +  O(t^2).
    
    In fact, the correlation remains negative at all times. Figure <ref> shows a plot of -C(t) vst for different values of r obtained from numerical simulations (symbols) along with the analytical prediction (solid lines) for small values of r.
    
    
    
    The presence of a non-trivial correlation between the diffusive and resetting currents suggests that even though the fluctuations of both these components of current are Gaussian in nature, the distribution of the total current need not be so. In the following we investigate this issue and show that, indeed the fluctuations of J_r are characterized by a strongly non-Gaussian distribution.
     Probability distribution of J_r:  In this Section we explore the behaviour of the probability distribution of the total current P_r(J_r,t) using  the renewal equation (<ref>). 
    
    In the absence of resetting, the fluctuations of the total (diffusive) current are characterized by a Gaussian distribution in the long-time limit (see Appendix <ref> for more details). Using the Gaussian form of P_0(J_r,t) one can calculate the total current distribution P_r(J_r,t) for small values of r (for small r the integral is dominated by the large t contribution). It is particularly interesting to look at the stationary distribution,
    
    
    P_r^st(J_r) = r ∫_0^∞𝕀τe^- r τ/√(2 πσ_τ^2)exp[-(J_r - μ_τ)^2/2 σ_τ^2]
    where μ_τ = √(τ/π) and σ_τ^2=  √(τ/π)(1-1/√(2)) are the mean and the variance of the current in absence of resetting, respectively. Clearly, for any finite value of J_r, the Gaussian part of the integrand, , exp[-(J_r-μ_τ)^2/2σ_τ^2], vanishes both at τ→ 0 and τ→∞ limits, ensuring that the integral is convergent.  One can then use the series expansion of e^-rτ in Eq. (<ref>) to express P_r^st(J_r) as an infinite sum of integrals,
    
    P_r^st(J_r) = r ∑_n=0^∞(-r)^n/√(2π)n!∫_0^∞𝕀ττ^n/σ_τexp[-(J_r - μ_τ)^2/2 σ_τ^2].
    Because of the asymptotic properties of exp[-(J_r-μ_τ)^2/2σ_τ^2] mentioned above, each of these integrals converge. It turns out that, these integrals can be evaluated exactly for all values of n and yields an explicit expression for the stationary distribution  in the form of an infinite series,
    
    
    
    
    
    P_r^st(J_r)    =   2 √(2) r/π^1/4√(b)exp(J_r/b)      ×∑_n=0^∞(-r)^n/n!(√(π)J_r)^2n+3/2K_2n+3/2(J_r/b).
    We here have used b=(1-1/√(2)) for brevity, and  K_ν(z) is the Modified Bessel function of the second kind <cit.> (see Eq. 10.31.1 therein). Convergence of the original integral in Eq. (<ref>) ensures that the series is also convergent for any finite J_r. Hence, the stationary distribution P_r^st(J_r) can be computed to arbitrary accuracy using Eq. (<ref>). This is demonstrated in Fig. <ref>(a) where the theoretical computation is plotted together with the simulation results. 
    
    
    
    
    
    The stationary distribution has some interesting features which are visible from Fig. <ref>(a). First, it is apparent that P_r^st(J_r) is vanishingly small for negative values of J_r.  This can be understood in the following way. Let us recall that, at any time, the total current is nothing but the net number of particles hopping across the central bond since the last resetting event, ,  after being brought to the configuration C_0 where the left half of the lattice is filled-up. To produce a negative current, the number of particles crossing the central bond from left to right should be lower than that from right to left , there should be a net flux of the particles to the left. Since, the particles are allowed to hop only to empty neighbouring sites, starting from the configuration C_0, this is an extremely unlikely event and has a vanishingly small probability.
    
    
    
    Secondly, it also appears that P_r^st(J_r) is strongly non-Gaussian which is manifest in the asymmetric behaviour of the two tails, as seen in Fig. <ref>(a). 
    
    To characterize this asymmetry and the non-Gaussian nature quantitatively we look at the decay of P_r^st(J_r) at the two tails, namely, near J_r=0 and large J_r. Near J_r=0, for small values of r,  the behaviour is dominated by the n=0 term in  Eq. (<ref>). One can then use the asymptotic behaviour of  K_3/2(z) near z=0
    
    
    to get
    
    P_r^st(J_r) ≈ 2 π r  (J_r + 1 - 1/√(2)) +  O(r^2). 
    Clearly, for small values of r, the probability distribution of the total current J_r decays linearly near  J_r=0.
    
    To determine how P_r^st(J_r) decays for large J_r we use the asymptotic behaviour of K_ν(z); for large values of the argument z, we have (see Ref. <cit.>, Eq. 10.40.2),
    lim_z →∞ K_ν(z) ≈√(π/2z) e^-z. 
    Using that in Eq. (<ref>) and performing the sum over n, we get,
    
    P_r^st(J_r) ≈ 2 π r J_r e^-π r J_r^2 + O(r^2). 
    Note that the above expression holds true to the leading order in r, higher order corrections can be systematically calculated by including higher order terms in (<ref>). 
    
    
    
    
    We conclude the discussion about P_r^st(J_r)  with one final remark. From our numerical data, we observe a surprising collapse of the current distribution when plotted as a function of the scaled variable Y=(J_r-μ_r)/σ_r where μ_r and σ_r are, respectively, the mean and the variance of 
    J_r.
    The collapse is shown in Fig. <ref>(b) where the scaled distribution P̃(Y)  appears to be independent of r as the curves corresponding to different values of r from Fig. <ref>(a) fall on top of each other. 
    
    
    To understand this collapse, let us look at P̃(Y) predicted from Eqs. (<ref>) and (<ref>). 
    Recalling that for small values of r,μ_r ≃1/2√(r) and σ_r ≃√((4-π)/4 π r),  
    we get from Eq. (<ref>),
    P̃(Y)    ≈   1/2(√(π(4-π))+ (4-π)Y) e^-1/4 (√(π)+ √(4-π) Y)^2           +  O(r^3/2).
    Clearly, to the leading order in r,P̃(Y) calculated from Eq. (<ref>) (corresponding to large values of J_r) is independent of r, and is consistent with the scaling collapse observed in Fig. <ref>(b). 
    On the other hand, it can be easily seen, that Eq.(<ref>) does not lend itself to a similar form; P̃(Y) derived from Eq.(<ref>) depends explicitly on r,P̃(Y)    ≈   1/2√(π(4-π))(1+ √(4-π/π)Y + (2- √(2))√(r))        + O(r^3/2).
    Hence, while for large positive J_r (≳μ_r+σ_r), the distribution P̃(Y) becomes independent of r, it is not the case in the J_r → 0 limit. 
     Indeed, as seen from Eq. (<ref>), P̃(Y) explicitly depends on r.  However, notice that the 
     r-dependence in Eq. (<ref>) comes in the form of an additional term proportional to √(r), which is vanishingly small for r ≪ 1.
    This makes the expected mismatch in the collapse at the left tail in Fig. <ref>(b) practically invisible 
    
     where an apparent collapse is also observed.
    
    
    § CONCLUSION
    
    
    In this article, we explore the effect of stochastic resetting on interacting many particle systems. To this end, we study the dynamical properties of a canonical set-up, namely, the symmetric exclusion process in the presence of stochastic resetting. The resetting is implemented by interrupting the dynamical evolution of the exclusion process with some rate r and restarting it from a step-like configuration where all the particles are clustered together in the left-half of the system. 
    
    We find that the presence of resetting strongly affects the behaviour of the system. The key findings are as follows. First, in a finite size system, the density profile evolves to an inhomogeneous stationary profile in contrast to the flat profile in the  absence of resetting. We have exactly calculated the full time-dependent density profile for arbitrary resetting rate r. Secondly, we find that, in a thermodynamically large system the resetting mechanism drastically changes the √(t) growth of the diffusive current to linear in t.  We have explicitly  computed the mean and variance of the diffusive current, the latter is also shown to have a linear growth in the long-time regime. Apart from the diffusive current, we also identify the another component of the current which  arises due the resetting move and show that this resetting current is negative, with a linear temporal growth in magnitude.   
    The moments of the total current, , the sum of the diffusive and resetting current, are also calculated using the renewal approach.
    
    We also have investigated the probability distribution of the diffusive current , resetting current , as well the total current J_r. We have found that that while the typical fluctuations of  and  are Gaussian in nature, the distribution of J_r is strictly non-Gaussian. The non-Gaussian nature is manifest in the asymmetric asymptotic behaviour of the distribution at the two tails, which we also demonstrate.
    
    
    Our study opens up a new direction in the area of stochastic resetting and gives rise to a wide range of further questions. For example, it would be interesting to study the effect of stochastic resetting in other interacting particle systems, e.g., the asymmetric exclusion process, driven and equilibrium lattice gas models etc. Furthermore, it would also be interesting to study behavior of these interacting particle systems under various other resetting mechanisms like resetting at power-law times or time-dependent resetting etc.  
    
    Apart from these theoretical questions, 
    the framework of stochastic resetting in exclusion processes can also be relevant in the context of certain biophysical systems. For example, stochastic motion of backtracked RNA polymerases can be modelled as an interacting many particle random walk on the DNA template, with RNA cleavage playing the role of resetting dynamics <cit.>.  
    
    Similarly, motion of two-headed molecular motors such as kinesin and Myosin V moving on a polymeric track can be modelled as an energy driven hopping process in the presence of backward jumps (or resetting)<cit.>. We believe that the formalism introduced in the present work will be useful in understanding such systems.
    
    
    The authors acknowledge useful discussions with Christian Maes and Sanjib Sabhapandit. U.B. acknowledges support from Science and Engineering Research Board (SERB), India under Ramanujan Fellowship (Grant No. SB/S2/RJN-077/2018). A. K. acknowledges support from DST 
    grant under project No. ECR/2017/000634. A.P. gratefully acknowledges support from  the Raymond and Beverly Sackler Post-Doctoral Scholarship at Tel-Aviv University.
    
    
    
    § DENSITY PROFILE OF SEP IN ABSENCE OF RESETTING
    
    In this section we present a brief account of the dynamical evolution of the density profile and current for ordinary SEP, starting from the step-like configuration C_0. In absence of resetting, the time evolution of the system is governed by the free Markov matrix L_0 which yields, for the density profile,
    𝕀/𝕀 tρ_0(x,t)    =   ρ_0(x+1,t)+ ρ_0(x-1,t)- 2ρ_0(x,t).   
    The corresponding Fourier components ρ̃_0(n,t) evolve following,
    
    𝕀/𝕀 tρ̃_0(n,t) = - λ_n ρ̃_0(n,t) 
    
    where, as before, λ_n= 2(1- cos2 π n/L), with n=0,1,2,… L-1. The above equation is  immediately solved to obtain,
    ρ̃_0(n,t) = e^-λ_n tϕ̃(n). 
    where ϕ̃(n) corresponds to the initial profile ϕ(x). Note that λ_0=0 and hence ρ̃_0(0,t)= ϕ̃(0) = L/2 does not evolve with time. 
    
    
    The spatial density profile is obtained by taking inverse Fourier transform of Eq. (<ref>). In particular, for the step-like initial profile ϕ(x)=1- Θ(x+1-L/2) we have,
    ρ_0(x,t) = 1/2 + 1/L∑_n=1,3^L-1 e^-i 2 π n x/L(1+i π n/L) e^-λ_n t    
    
    § BEHAVIOUR OF CURRENT IN ABSENCE OF RESETTING
     
    
    
    In the absence of resetting the only source of current in SEP is the hopping dynamics of the particles. The average instantaneous current across the initial step, , across the central bond (L/2 -1, L/2) is given by,
     j_0(t)    =   ρ_0(L/2-1,t) - ρ_0(L/2,t)    =   2/L∑_n=1,3^L-1 e^-λ_n t
    where we have used Eq. (<ref>) to calculate the average densities at the sites x=L/2-1 and x=L/2. Clearly, in the long-time limit t →∞, the instantaneous current vanishes as the density profile becomes flat.
    
    We are interested in the time-integrated current J_0(t) = ∫_0^t 𝕀 s  j_0(s) which measures the net number of particles crossing the central bond towards right. The average time-integrated current is obtained by integrating Eq. (<ref>),
     J_0(t)  =  2/L∑_n=1,3^L-11/λ_n(1-e^-λ_n t).
    
    For any finite L, the average time-integrated current J_0(t) saturates to an L-dependent constant value in the long-time limit. 
    
    To understand the behaviour of a thermodynamically large system, one has to take the limit L →∞ first. In this case, the sum in Eq. (<ref>) can be can be converted to an integral by denoting q= 2 π n/L, and we have the mean instantaneous current,
     j_0(t)    =   ∫_0^2 π𝕀 q/2 π  e^-2(1-cos q)t
     [0.25em]    =    e^-2 t I_0(2 t).
    
    Here I_0(x) is the Modified Bessel function of the first kind <cit.> (see Eq. 10.25.2 therein). In this limit, the average time-integrated current becomes,
     J_0(t)  = e^-2t t [I_0(2t)+I_1(2t)].
    For large values of the argument x, both I_0(x) and I_1(x) have the same asymptotic behaviour (see <cit.>, Eq. 10.40.1 	),
    lim_x→∞I_0,1(2x) ∼e^2x/2 √(π x)
    
    which yields, in the long-time regime,
     J_0(t) ≃√(t/π). 
    This result has been obtained in Ref. <cit.>, albeit using a different method.  In fact, it has also been shown <cit.> that, in the long-time regime, all the higher moments of J_0 show a similar behaviour. In particular, the variance is given by,
     J_0^2(t)  -  J_0(t) ^2 ≃√(t/π)(1- 1/√(2)). 
    The above equation is used in Eq. (<ref>) in the main text to calculate J_r^2.
    
     §.§ Probability Distribution of J_0
    
    
    For ordinary SEP, the probability distribution of the time-integrated current J_0  was explored in Ref. <cit.>. There the authors considered a scenario where, initially, each site to the left (respectively, right) of the origin (x ≤ 0 and x>0 respectively) is occupied with probability ρ_a (respectively ρ_b). It was shown that, for large t, the moment generating function of the total particle flux J_0(t) through the origin is given by,
     e^λ J_0(t)∼ e^√(t) F(ω)
    where ω = ρ_a(e^λ -1) + ρ_b(e^-λ-1) + ρ_a ρ_b(e^λ -1) and 
    
    F(ω)    =   1/√(π)∑_n=1^∞(-1)^n+1ω^n/n^3/2   ≡    -1/√(π)PolyLog_3/2(-ω).
    
    
    In our case, we have ρ_a=1 and ρ_b=0 which simplifies ω and in turn F(ω); we get ω = e^λ-1 and
    
    F(λ) = -1/√(π)PolyLog_3/2(1-e^λ)
    
    which is quoted in Eq. (<ref>) in the main text. 
    
    
    
    It has been shown in Ref. <cit.>  that the corresponding probability distribution P_0(J_0,t), in the long-time limit, is of the form,
    
    P_0(J_0,t) ∼ e^√(t)G(J_0/√(t)). 
    The large deviation function G(q=J_0/√(t)) is related to F(λ) through a Legendre transform,
    
    G(q)    =   min_λ[F(λ) - λ q]    =    F(λ^*) - λ^* q, 
    where λ^* corresponds to the minimum of the function F(λ)- λ q and is obtained by solving 𝕀 F(λ)/𝕀λ = q. It is easy to see that for small values of q,λ^* is also small. Hence, it is convenient to
    use the series expansion of  F(λ) near λ=0
    F(λ)= λ/√(π) + λ^2/2√(π)(1- 1/√(2)) +  O(λ^3),
    
    to find λ^* for small values of q. Restricting ourselves to the quadratic order in λ, we get 
    λ^*=(q √(π)- 1)√(2)/(√(2)-1). Substitution of this λ^* in Eq. (<ref>) yields,
    
    G(q) = (q - 1/√(π))^2/2/√(π)(1- 1/√(2))
    Using the above G(q) in Eq. (<ref>) results in a Gaussian form for the current distribution,
    
    P_0(J_0,t) = 1/√(2 πσ_0^2(t))exp(- [J_0 - μ_0(t)]^2/2 σ^2_0(t))
    where the prefactor is just a normalization constant. Here, μ_0(t)=√(t/π) is nothing but the average hopping current J_0(t) and σ_0^2(t)=√(t/π)(1- 1/√(2)) is the variance [see Eq. (<ref>)]. Note that, this Gaussian distribution is expected only in the long-time limit, as Eq. (<ref>) holds true in this limit only.  
    
    
    
    
    
    Figure <ref> (a) shows a comparison of P_0(J_0,t) obtained from numerical simulations (symbols) with the that predicted from Eq. (<ref>) (solid lines) for different (large) values of t. 
    Fig. <ref> (b) shows the same data but plotted against the scaled variable  y=J_0 - μ_0(t)/σ_0(t); the solid line corresponds to the standard normal distribution 1/√(2 π)e^-y^2/2. The numerical data shows a very good match with the predicted Gaussian curve for typical values of J_0, there are deviations only at the regime |y| ≫ 1 which are visible only at a logarithmic scale. The large deviation function calculated in Ref <cit.> describes the distribution for these atypical values. However, as shown in Sec. <ref>, for our purposes it suffices to consider the typical fluctuations and we use the Gaussian distribution (<ref>) to calculate the distribution of the diffusive current  in presence of resetting. 
    
    
    
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