Current fluctuations in a partially asymmetric simple exclusion process with a defect particle.

We study an exclusion process on a ring comprising a free defect particle in a bath of normal particles. The model is one of the few integrable cases in which the bath particles are partially asymmetric. The presence of the free defect creates localized or shock phases according to parameter values. We use a functional approach to Bethe equations resulting from a nested Bethe ansatz to calculate exactly the mean currents and diffusion constants. The results agree very well with Monte Carlo simulations and reveal the main modes of fluctuation in the different phases of the steady state.

Dynamical fluctuations in the Riesz gas.

We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the |x-y|^{-s} Riesz potential, causing the overdamped motion of particles. We investigate fluctuations of the integrated current and the position of a tagged particle. We show that for 01, the interactions are effectively short-ranged, and the universal subdiffusive t^{1/4} growth emerges with only amplitude depending on the exponent s. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.

Exact Solution of the Macroscopic Fluctuation Theory for the Symmetric Exclusion Process.

We present the first exact solution for the time-dependent equations of the macroscopic fluctuation theory (MFT) for the symmetric simple exclusion process by combining a generalization of the canonical Cole-Hopf transformation with the inverse scattering method. For the step initial condition with two densities, we obtain exact and compact formulas for the optimal density profile and the response field that produce a required fluctuation, both at initial and final times. The large deviation function of the current is derived and coincides with the formula obtained previously by microscopic calculations. This provides the first analytic confirmation of the validity of the MFT for an interacting model in the time-dependent regime.

Random walk through a fertile site.

We study the dynamics of random walks hopping on homogeneous hypercubic lattices and multiplying at a fertile site. In one and two dimensions, the total number N(t) of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate µ at the fertile site. When d>d_{c}=2, the number of walkers may remain finite forever for any µ; it surely remains finite when µ≤µ_{d}. We determine µ_{d} and show that 〈N(t)〉 grows exponentially if µ>µ_{d}. The distribution of the total number of walkers remains broad when d≤2, and also when d>2 and µ>µ_{d}. We compute 〈N^{m}〉 explicitly for small m, and show how to determine higher moments. In the critical regime, 〈N〉 grows as sqrt[t] for d=3, t/lnt for d=4, and t for d>4. Higher moments grow anomalously, 〈N^{m}〉∼〈N〉^{2m-1}, in the critical regime; the growth is normal, 〈N^{m}〉∼〈N〉^{m}, in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz., it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if d>2, there is again a critical multiplication rate, above which 〈N(t)〉 grows linearly (not exponentially) in time; when d≤d_{c}=2, the leading behavior is independent on µ and 〈N(t)〉 exhibits a sublinear growth.

Large deviations and fluctuation theorem for the quantum heat current in the spin-boson model.

We study the heat current flowing between two baths consisting of harmonic oscillators interacting with a qubit through a spin-boson coupling. An explicit expression for the generating function of the total heat flowing between the right and left baths is derived by evaluating the corresponding Feynman-Vernon path integral by performing the noninteracting blip approximation (NIBA). We recover the known expression, obtained by using the polaron transform. This generating function satisfies the Gallavotti-Cohen fluctuation theorem, both before and after performing the NIBA. We also verify that the heat conductance is proportional to the variance of the heat current, retrieving the well-known fluctuation dissipation relation. Finally, we present numerical results for the heat current.

Large Deviations of a Tracer in the Symmetric Exclusion Process.

The one-dimensional symmetric exclusion process, the simplest interacting particle process, is a lattice gas made of particles that hop symmetrically on a discrete line respecting hard-core exclusion. The system is prepared on the infinite lattice with a step initial profile with average densities ρ_{+} and ρ_{-} on the right and on the left of the origin. When ρ_{+}=ρ_{-}, the gas is at equilibrium and undergoes stationary fluctuations. When these densities are unequal, the gas is out of equilibrium and will remain so forever. A tracer, or a tagged particle, is initially located at the boundary between the two domains; its position X_{t} is a random observable in time that carries information on the nonequilibrium dynamics of the whole system. We derive an exact formula for the cumulant generating function and the large deviation function of X_{t} in the long-time limit and deduce the full statistical properties of the tracer's position. The equilibrium fluctuations of the tracer's position, when the density is uniform, are obtained as an important special case.

Variational calculation of transport coefficients in diffusive lattice gases.

A diffusive lattice gas is characterized by the diffusion coefficient depending only on the density. The Green-Kubo formula for diffusivity can be represented as a variational formula, but even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient can be computed only in the exceptional situation when the lattice gas is gradient. In the general case, minimization over an infinite-dimensional space is required. We propose an approximation scheme based on minimizing over finite-dimensional subspaces of functions. The procedure is demonstrated for one-dimensional generalized exclusion processes in which each site can accommodate at most two particles. Our analytical predictions provide upper bounds for the diffusivity that are very close to simulation results throughout the entire density range. We also analyze nonequilibrium density profiles for finite chains coupled to reservoirs. The predictions for the profiles are in excellent agreement with simulations.

Coalescing colony model: Mean-field, scaling, and geometry.

We analyze the coalescing model where a 'primary' colony grows and randomly emits secondary colonies that spread and eventually coalesce with it. This model describes population proliferation in theoretical ecology, tumor growth, and is also of great interest for modeling urban sprawl. Assuming the primary colony to be always circular of radius r(t) and the emission rate proportional to r(t)^{θ}, where θ>0, we derive the mean-field equations governing the dynamics of the primary colony, calculate the scaling exponents versus θ, and compare our results with numerical simulations. We then critically test the validity of the circular approximation for the colony shape and show that it is sound for a constant emission rate (θ=0). However, when the emission rate is proportional to the perimeter, the circular approximation breaks down and the roughness of the primary colony cannot be discarded, thus modifying the scaling exponents.

Reply to "Comment on 'Generalized exclusion processes: Transport coefficients' ".

We reply to the Comment of Becker, Nelissen, Cleuren, Partoens, and Van den Broeck [Phys. Rev. E 93, 046101 (2016)1539-375510.1103/PhysRevE.93.046101] on our article [Arita, Krapivsky, and Mallick, Phys. Rev. E 90, 052108 (2014)PLEEE81539-375510.1103/PhysRevE.90.052108] about the transport properties of a class of generalized exclusion processes.

Generalized exclusion processes: Transport coefficients.

A class of generalized exclusion processes with symmetric nearest-neighbor hopping which are parametrized by the maximal occupancy, k≥1, is investigated. For these processes on hypercubic lattices we compute the diffusion coefficient in all spatial dimensions. In the extreme cases of k=1 (symmetric simple exclusion process) and k=∞ (noninteracting symmetric random walks) the diffusion coefficient is constant, while for 2≤k<∞ it depends on the density and k. We also study the evolution of the tagged particle, show that it exhibits a normal diffusive behavior in all dimensions, and probe numerically the coefficient of self-diffusion.

Large deviations in single-file diffusion.

We apply macroscopic fluctuation theory to study the diffusion of a tracer in a one-dimensional interacting particle system with excluded mutual passage, known as single-file diffusion. In the case of Brownian point particles with hard-core repulsion, we derive the cumulant generating function of the tracer position and its large deviation function. In the general case of arbitrary interparticle interactions, we express the variance of the tracer position in terms of the collective transport properties, viz., the diffusion coefficient and the mobility. Our analysis applies both for fluctuating (annealed) and fixed (quenched) initial configurations.

Asymmetric exclusion process with global hopping.

We study a one-dimensional totally asymmetric simple exclusion process with one special site from which particles fly to any empty site (not just to the neighboring site). The system attains a nontrivial stationary state with a density profile varying over the spatial extent of the system. The density profile undergoes a nonequilibrium phase transition when the average density passes through the critical value 1-[4(1-ln2)](-1)=0.185277..., viz., in addition to the discontinuity in the vicinity of the special site, a shock wave is formed in the bulk of the system when the density exceeds the critical density.

Exact current statistics of the asymmetric simple exclusion process with open boundaries.

Nonequilibrium systems are often characterized by the transport of some quantity at a macroscopic scale, such as, for instance, a current of particles through a wire. The asymmetric simple exclusion process (ASEP) is a paradigm for nonequilibrium transport that is amenable to exact analytical solution. In the present work, we determine the full statistics of the current in the finite size open ASEP for all values of the parameters. Our exact analytical results are checked against numerical calculations using density matrix renormalization group techniques.

Role of ATP-hydrolysis in the dynamics of a single actin filament.

We study the stochastic dynamics of growth and shrinkage of single actin filaments taking into account insertion, removal, and ATP hydrolysis of subunits either according to the vectorial mechanism or to the random mechanism. In a previous work, we developed a model for a single actin or microtubule filament where hydrolysis occurred according to the vectorial mechanism: the filament could grow only from one end, and was in contact with a reservoir of monomers. Here we extend this approach in two ways--by including the dynamics of both ends and by comparing two possible mechanisms of ATP hydrolysis. Our emphasis is mainly on two possible limiting models for the mechanism of hydrolysis within a single filament, namely the vectorial or the random model. We propose a set of experiments to test the nature of the precise mechanism of hydrolysis within actin filaments.

Statistical properties of the final state in one-dimensional ballistic aggregation.

We investigate the long time behavior of the one-dimensional ballistic aggregation model that represents a sticky gas of N particles with random initial positions and velocities, moving deterministically, and forming aggregates when they collide. We obtain a closed formula for the stationary measure of the system which allows us to analyze some remarkable features of the final "fan" state. In particular, we identify universal properties which are independent of the initial position and velocity distributions of the particles. We study cluster distributions and derive exact results for extreme value statistics (because of correlations these distributions do not belong to the Gumbel-Fréchet-Weibull universality classes). We also derive the energy distribution in the final state. This model generates dynamically many different scales and can be viewed as one of the simplest exactly solvable model of N -body dissipative dynamics.

Nonequilibrium self-assembly of a filament coupled to ATP/GTP hydrolysis.

We study the stochastic dynamics of growth and shrinkage of single actin filaments or microtubules taking into account insertion, removal, and ATP/GTP hydrolysis of subunits. The resulting phase diagram contains three different phases: two phases of unbounded growth: a rapidly growing phase and an intermediate phase, and one bounded growth phase. We analyze all these phases, with an emphasis on the bounded growth phase. We also discuss how hydrolysis affects force-velocity curves. The bounded growth phase shows features of dynamic instability, which we characterize in terms of the time needed for the ATP/GTP cap to disappear as well as the time needed for the filament to reach a length of zero (i.e., to collapse) for the first time. We obtain exact expressions for all these quantities, which we test using Monte Carlo simulations.

Bethe Ansatz in the Bernoulli matching model of random sequence alignment.

For the Bernoulli matching model of the sequence alignment problem we apply the Bethe Ansatz technique via an exact mapping to the five-vertex model on a square lattice. Considering the terracelike representation of the sequence alignment problem, we reproduce by the Bethe Ansatz the results for the averaged length of the longest common subsequence in the Bernoulli approximation. In addition, we compute the average number of nucleation centers of the terraces.

Asymptotic localization of stationary states in the nonlinear Schrödinger equation.

The mapping of the nonlinear Schrödinger equation with a random potential on the Fokker-Planck equation is used to calculate the localization length of its stationary states. The asymptotic growth rates of the moments of the wave function and its derivative for the linear Schrödinger equation in a random potential are computed analytically, and resummation is used to obtain the corresponding growth rate for the nonlinear Schrödinger equation and the localization length of the stationary states.

Molecular Spiders in One Dimension.

Molecular spiders are synthetic bio-molecular systems which have "legs" made of short single-stranded segments of DNA. Spiders move on a surface covered with single-stranded DNA segments complementary to legs. Different mappings are established between various models of spiders and simple exclusion processes. For spiders with simple gait and varying number of legs we compute the diffusion coefficient; when the hopping is biased we also compute their velocity.

Low-frequency noise controls on-off intermittency of bifurcating systems.

A bifurcating system subject to multiplicative noise can display on-off intermittency. Using a canonical example, we investigate the extreme sensitivity of the intermittent behavior to the nature of the noise. Through a perturbative expansion and numerical studies of the probability density function of the unstable mode, we show that intermittency is controlled by the ratio between the departure from onset and the value of the noise spectrum at zero frequency. Reducing the noise spectrum at zero frequency shrinks the intermittency regime drastically. This effect also modifies the distribution of the duration that the system spends in the off phase. Mechanisms and applications to more complex bifurcating systems are discussed.