Diffusion with stochastic resetting at power-law times

What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals $\tau$ distributed as a power-law $\sim \tau^{-(1+\alpha)};\alpha>0$? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain {\em 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 $\alpha < 1$, to one that is time independent for $\alpha > 1$. 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 $\alpha$ that minimizes the mean time to reach the target, thereby offering a step towards a viable strategy to locate targets in a crowded environment.Comment: 8 pages, 3 figures. v2: Version published in Phys. Rev. E as a rapid comm., includes Suppl. Ma

Absence of jamming in ant trails: Feedback control of self propulsion and noise

We present a model of ant traffic considering individual ants as self-propelled particles undergoing single file motion on a one-dimensional trail. Recent experiments on unidirectional ant traffic in well-formed natural trails showed that the collective velocity of ants remains approximately unchanged, leading to absence of jamming even at very high densities [ John et. al., Phys. Rev. Lett. 102, 108001 (2009) ]. Assuming a feedback control mechanism of self-propulsion force generated by each ant using information about the distance from the ant in front, our model captures all the main features observed in the experiment. The distance headway distribution shows a maximum corresponding to separations within clusters. The position of this maximum remains independent of average number density. We find a non-equilibrium first order transition, with the formation of an infinite cluster at a threshold density where all the ants in the system suddenly become part of a single cluster.Comment: 6 pages, 4 figure

Passive sliders and scaling: from cusps to divergences

The steady state reached by a system of particles sliding down a fluctuating surface has interesting properties. Particle clusters form and break rapidly, leading to a broad distribution of sizes and large fluctuations. The density-density correlation function is a singular scaling function of the separation and system size. A simple mapping is shown to take a configuration of sliding hard-core particles with mutual exclusion (a system which shows a cusp singularity) to a configuration with multiparticle occupancy. For the mapped system, a calculation of the correlation function shows that it is of the same scaling form again, but with a stronger singularity (a divergence) of the sort observed earlier for noninteracting passive particles

Passive Sliders on Fluctuating Surfaces: Strong-Clustering States

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

Condensation transition in a model with attractive particles and non-local hops

We study a one dimensional nonequilibrium lattice model with competing features of particle attraction and non-local hops. The system is similar to a zero range process (ZRP) with attractive particles but the particles can make both local and non-local hops. The length of the non-local hop is dependent on the occupancy of the chosen site and its probability is given by the parameter $p$. Our numerical results show that the system undergoes a phase transition from a condensate phase to a homogeneous density phase as $p$ is increased beyond a critical value $p_c$. A mean-field approximation does not predict a phase transition and describes only the condensate phase. We provide heuristic arguments for understanding the numerical results.Comment: 11 Pages, 6 Figures. Published in Journal of Statistical Mechanics: Theory and Experimen

Strong clustering of non-interacting, passive sliders driven by a Kardar-Parisi-Zhang surface

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. Monte-Carlo 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 density-density 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.Comment: 14 pages, 9 figure

Boundary-induced abrupt transition in the symmetric exclusion process

We investigate the role of the boundary in the symmetric simple exclusion process with competing nonlocal and local hopping events. With open boundaries, the system undergoes a first order phase transition from a finite density phase to an empty road phase as the nonlocal hopping rate increases. Using a cluster stability analysis, we determine the location of such an abrupt nonequilibrium phase transition, which agrees well with numerical results. Our cluster analysis provides a physical insight into the mechanism behind this transition. We also explain why the transition becomes discontinuous in contrast to the case with periodic boundary conditions, in which the continuous phase transition has been observed.Comment: 8 pages, 11 figures (12 eps files); revised as the publised versio

Exact phase diagram of quasispecies model with mutation rate modifier

We consider an infinite asexual population with a mutator allele which can elevate mutation rates. With probability $f$, a transition from nonmutator to mutator state occurs but the reverse transition is forbidden. We find that at $f=0$, the population is in the state with minimum mutation rate and at $f=f_c$, a phase transition occurs between a mixed phase with both nonmutators and mutators and a pure mutator phase. We calculate the critical probability $f_c$ and the total mutator fraction $Q$ in the mixed phase exactly. Our predictions for $Q$ are in agreement with those seen in microbial populations in static environments.Comment: Revised versio