We consider a system of non-interacting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run and tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomised. We study the effects of resetting on the distribution P (Q, t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution Pan(Q, t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution Pqu(Q, t) attains its stationary form for Q < Qcrit(t), while it remains time-dependent for Q > Qcrit(t). We show that Qcrit(t) increases linearly with t for large t. On the scale where Q ∼ Qcrit(t), we show that Pqu(Q, t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10 −14000 , we were able to compute numerically this non-analytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

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Stochastic resetting is a rapidly developing topic in the field of stochastic&#xD;processes and their applications. It denotes the occasional reset of a&#xD;diffusing particle to its starting point and effects, inter alia, optimal&#xD;first-passage times to a target. Recently the concept of partial resetting,&#xD;in which the particle is reset to a given fraction of the current value&#xD;of the process, has been established and the associated search behaviour&#xD;analysed. Here we go one step further and we develop a general technique to&#xD;determine the time-dependent probability density function (PDF) for Markov&#xD;processes with partial resetting. We obtain an exact representation of the&#xD;PDF in the case of general symmetric L{'e}vy flights with stable index&#xD;$0<\alpha\le2$. For Cauchy and Brownian motions (i.e., $\alpha=1,2$), this&#xD;PDF can be expressed in terms of elementary functions in position&#xD;space. We also determine the stationary PDF. Our numerical analysis of the&#xD;PDF demonstrates intricate crossover behaviours as function of time.