What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals τ distributed as a power law ∼τ^{-(1+α)};α>0? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain exact closed-form expressions for both static and dynamic quantities, while accounting for strong correlations implied by a power law. Our results show that the resulting dynamics exhibits a spectrum of rich long-time behavior, from an ever-spreading spatial distribution for α<1, to one that is time independent for α>1. The dynamics has strong consequences on the time to reach a distant target for the first time; we specifically show that there exists an optimal α that minimizes the mean time to reach the target, thereby offering a step towards a viable strategy to locate targets in a crowded environment.

Abstract. What happens when the time evolution of a fluctuating interface is interrupted with resetting to a given initial configuration after random time intervals τ distributed as a power-law ∼ τ −(1+α) ; α > 0? For an interface of length L in one dimension, and an initial flat configuration, we show that depending on α, the dynamics in the limit L → ∞ exhibits a spectrum of rich long-time behavior. It is known that without resetting, the interface width grows unbounded with time as t β in this limit, where β is the so-called growth exponent characteristic of the universality class for a given interface dynamics. We show that introducing resetting induces for α > 1 and at long times fluctuations that are bounded in time. Corresponding to such a resetinduced stationary state is a distribution of fluctuations that is strongly non-Gaussian, with tails decaying as a power-law. The distribution exhibits a distinctive cuspy behavior for small argument, implying that the stationary state is out of equilibrium. For α < 1, on the contrary, resetting to the flat configuration is unable to counter the otherwise unbounded growth of fluctuations in time, so that the distribution of fluctuations remains time dependent with an ever-increasing width even at long times. Although stationary for α > 1, the width of the interface grows forever with time as a power-law for 1 < α < α (w) , and converges to a finite constant only for larger α, thereby exhibiting a crossover at α (w) = 1 + 2β. The time-dependent distribution of fluctuations for α < 1 exhibits for small argument another interesting crossover behavior, from cusp to divergence, across α (d) = 1 − β. We demonstrate these results by exact analytical results for the paradigmatic Edwards-Wilkinson (EW) dynamical evolution of the interface, and further corroborate our findings by extensive numerical simulations of interface models in the EW and the Kardar-Parisi-Zhang universality class.

We study the clustering of passive, non-interacting particles moving under the influence of a fluctuating field and random noise, in one dimension. The fluctuating field in our case is provided by a surface governed by the Kardar-Parisi-Zhang (KPZ) equation and the sliding particles follow the local surface slope. As the KPZ equation can be mapped to the noisy Burgers equation, the problem translates to that of passive scalars in a Burgers fluid. We study the case of particles moving in the same direction as the surface, equivalent to advection in fluid language. MonteCarlo simulations on a discrete lattice model reveal extreme clustering of the passive particles. The resulting Strong Clustering State is defined using the scaling properties of the two point densitydensity correlation function. Our simulations show that the state is robust against changing the ratio of update speeds of the surface and particles. In the equilibrium limit of a stationary surface and finite noise, one obtains the Sinai model for random walkers on a random landscape. In this limit, we obtain analytic results which allow closed form expressions to be found for the quantities of interest. Surprisingly, these results for the equilibrium problem show good agreement with the results in the non-equilibrium regime.

We study the clustering properties of particles sliding downwards on a fluctuating surface evolving through the Kardar-Parisi-Zhang equation, a problem equivalent to passive scalars driven by a Burgers fluid. Monte Carlo simulations on a discrete version of the problem in one dimension reveal that particles cluster very strongly: the two point density correlation function scales with the system size with a scaling function which diverges at small argument. Analytic results are obtained for the Sinai problem of random walkers in a quenched random landscape. This equilibrium system too has a singular scaling function which agrees remarkably with that for advected particles.Comment: To be published in Physical Review Letter

A simple measure for the efficiency of protein synthesis by ribosomes is provided by the steady state amount of protein per messenger RNA (mRNA), the so-called translational ratio, which is proportional to the translation rate. Taking the degradation of mRNA into account, we show theoretically that both the translation rate and the translational ratio decrease with increasing mRNA length, in agreement with available experimental data for the prokaryote Escherichia coli. We also show that, compared to prokaryotes, mRNA degradation in eukaryotes leads to a less rapid decrease of the translational ratio. This finding is consistent with the fact that, compared to prokaryotes, eukaryotes tend to have longer proteins.

We study the one-dimensional totally asymmetric simple exclusion process (TASEP) with open boundaries having the additional dynamical feature of stochastic resetting to the initial, empty state. The system evolves according to the TASEP dynamics with particles entering the input side with rate α and leaving the other side with rate β. The system is brought back to its initial state of an empty lattice at random intervals τ . These intervals are drawn from probability distributions, for which we consider two possibilities-a power law ∼ τ −(1+γ) with γ > 0, and an exponential distribution λe −λτ . We use approximate expressions for the time evolution of density on the lattice for a normal TASEP to calculate the reset-averaged density as a function of time. We find that in the limit of large time, the system achieves a steady state when γ > 1, while for γ < 1 we see a time-dependent scaling function. The large time behaviour of the density distribution shows a power law decay at the input boundary in all the phases while it shows a non-monotonic behaviour in the high-density phase of the TASEP. One sees this monotonic behaviour also for the exponential resetting, with the system always achieving a steady state in the limit of long times. We also perform numerical simulations, results from which show good agreement with our analytic expressions.

The interplay between turnover or degradation and ribosome loading of messenger RNA (mRNA) is studied theoretically using a stochastic model that is motivated by recent experimental results. Random mRNA degradation affects the statistics of polysomes, i.e., the statistics of the number of ribosomes per mRNA as extracted from cells. Since ribosome loading of newly created mRNA chains requires some time to reach steady state, a fraction of the extracted mRNA/ribosome complexes does not represent steady state conditions. As a consequence, the mean ribosome density obtained from the extracted complexes is found to be inversely proportional to the mRNA length. On the other hand, the ribosome density profile shows an exponential decrease along the mRNA for prokaryotes and becomes uniform in eukaryotic cells.

We investigate the role of degradation of mRNA on protein synthesis using the totally asymmetric simple exclusion process (TASEP) as the underlying model for ribosome dynamics. mRNA degradation has a strong effect on the lifetime distribution of the mRNA, which in turn affects polysome statistics such as the number of ribosomes present on an mRNA strand of a given size. An average over mRNA of all ages is equivalent to an average over possible configurations of the corresponding TASEP-both before steady state and in steady state. To evaluate the relevant quantities for the translation problem, we first study the approach towards steady state of the TASEP, starting with an empty lattice representing an unloaded mRNA. When approaching the high density phase, the system shows two distinct phases with the entry and exit boundaries taking control of the density at their respective ends in the second phase. The approach towards the maximal current phase exhibits the surprising property that the ribosome entry flux can exceed the maximum possible steady state value. In all phases, the averaging over the mRNA age distribution shows a decrease in the average ribosome density profile as a function of distance from the entry boundary. For entry/exit parameters corresponding to the high density phase of TASEP, the average ribosome density profile also has a maximum near the exit end.