Large deviations of a tracer position in the dense and the dilute limits of single-file diffusion.

We apply the macroscopic fluctuation theory to analyze the long-time statistics of the position of a tracer in the dense and the dilute limits of diffusive single-file systems. Our explicit results are about the corresponding large deviation functions for an initial step density profile with the fluctuating (annealed) and the fixed (quenched) initial conditions. These hydrodynamic results are applicable for a general single-file system and they agree with recent exact results obtained by microscopic solutions for specific model systems.

Sequence of phase transitions in a model of interacting rods.

In a system of interacting thin rigid rods of equal length 2ℓ on a two-dimensional grid of lattice spacing a, we show that there are multiple phase transitions as the coupling strength κ=ℓ/a and the temperature are varied. There are essentially two classes of transitions. One corresponds to the Ising-type spontaneous symmetry-breaking transition and the second belongs to less-studied phase transitions of geometrical origin. The latter class of transitions appears at fixed values of κ irrespective of the temperature, whereas the critical coupling for the spontaneous symmetry-breaking transition depends on it. By varying the temperature, the phase boundaries may cross each other, leading to a rich phase behavior with infinitely many phases. Our results are based on Monte Carlo simulations on the square lattice and a fixed-point analysis of a functional flow equation on a Bethe lattice.

Functionals of fractional Brownian motion and the three arcsine laws.

Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent H∈(0,1), generalizing standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for nonequilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many nontrivial observables analytically: We generalize the celebrated three arcsine laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in ɛ=H-1/2, up to second order. We find that the three probabilities are different, except for H=1/2, where they coincide. Our results are confirmed to high precision by numerical simulations.

Generalized Arcsine Laws for Fractional Brownian Motion.

The three arcsine laws for Brownian motion are a cornerstone of extreme-value statistics. For a Brownian B_{t} starting from the origin, and evolving during time T, one considers the following three observables: (i) the duration t_{+} the process is positive, (ii) the time t_{last} the process last visits the origin, and (iii) the time t_{max} when it achieves its maximum (or minimum). All three observables have the same cumulative probability distribution expressed as an arcsine function, thus the name arcsine laws. We show how these laws change for fractional Brownian motion X_{t}, a non-Markovian Gaussian process indexed by the Hurst exponent H. It generalizes standard Brownian motion (i.e., H=1/2). We obtain the three probabilities using a perturbative expansion in ϵ=H-1/2. While all three probabilities are different, this distinction can only be made at second order in ϵ. Our results are confirmed to high precision by extensive numerical simulations.

Multiple Singularities of the Equilibrium Free Energy in a One-Dimensional Model of Soft Rods.

There is a misconception, widely shared among physicists, that the equilibrium free energy of a one-dimensional classical model with strictly finite-ranged interactions, and at nonzero temperatures, cannot show any singularities as a function of the coupling constants. In this Letter, we discuss an instructive counterexample. We consider thin rigid linear rods of equal length 2ℓ whose centers lie on a one-dimensional lattice, of lattice spacing a. The interaction between rods is a soft-core interaction, having a finite energy U per overlap of rods. We show that the equilibrium free energy per rod F[(ℓ/a),ß], at inverse temperature ß, has an infinite number of singularities, as a function of ℓ/a.

Long-range correlations in a locally driven exclusion process.

We show that the presence of a driven bond in an otherwise diffusive lattice gas with simple exclusion interaction results in long-range density-density correlation in its stationary state. In dimensions d > 1 we show that in the thermodynamic limit this correlation decays as C(r,s)∼(r(2)+s(2))(-d) at large distances r and s away from the drive with |r - s| ≫ 1. This is derived using an electrostatic analogy whereby C(r,s) is expressed as the potential due to a configuration of electrostatic charges distributed in 2d dimension. At bulk density ρ=1/2 we show that the potential is that of a localized quadrupolar charge. At other densities the same is correct in leading order in the strength of the drive and it is argued numerically to be valid at higher orders.

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.

Interface phase transition induced by a driven line in two dimensions.

The effect of a localized drive on the steady state of an interface separating two phases in coexistence is studied. This is done using a spin-conserving kinetic Ising model on a two-dimensional lattice with cylindrical boundary conditions, where a drive is applied along a single ring on which the interface separating the two phases is centered. The drive is found to induce an interface spontaneous symmetry breaking whereby the magnetization of the driven ring becomes nonzero. The width of the interface becomes finite and its fluctuations around the driven ring are nonsymmetric. The dynamical origin of these properties is analyzed in an adiabatic limit, which allows the evaluation of the large deviation function of the magnetization of the driven ring.

Pattern formation in fast-growing sandpiles.

We study the patterns formed by adding N sand grains at a single site on an initial periodic background in the Abelian sandpile models, and relaxing the configuration. When the heights at all sites in the initial background are low enough, one gets patterns showing proportionate growth, with the diameter of the pattern formed growing as N(1/d) for large N, in d dimensions. On the other hand, if sites with maximum stable height in the starting configuration form an infinite cluster, we get avalanches that do not stop. In this paper we describe our unexpected finding of an interesting class of backgrounds in two dimensions that show an intermediate behavior: For any N, the avalanches are finite, but the diameter of the pattern increases as N(α), for large N, with 1/2<α≤1. Different values of α can be realized on different backgrounds, and the patterns still show proportionate growth. The noncompact nature of growth simplifies their analysis significantly. We characterize the asymptotic pattern exactly for one illustrative example with α=1.

Long-range steady-state density profiles induced by localized drive.

We show that the presence of a localized drive in an otherwise diffusive system results in steady-state density and current profiles that decay algebraically to their global average value, away from the drive in two or higher dimensions. An analogy to an electrostatic problem is established, whereby the density profile induced by a driving bond maps onto the electrostatic potential due to an electric dipole located along the bond. The dipole strength is proportional to the drive, and is determined self-consistently by solving the electrostatic problem. The profile resulting from a localized configuration of more than one driving bond can be straightforwardly determined by the superposition principle of electrostatics. This picture is shown to hold even in the presence of exclusion interaction between particles.

Emergence of quasiunits in the one-dimensional Zhang model.

We study the Zhang model of sandpile on a one-dimensional chain of length L , where a random amount of energy is added at a randomly chosen site at each time step. We show that in spite of this randomness in the input energy, the probability distribution function of energy at a site in the steady state is sharply peaked, and the width of the peak decreases as L(-1/2) for large L . We discuss how the energy added at one time is distributed among different sites by topplings with time. We relate this distribution to the time-dependent probability distribution of the position of a marked grain in the one-dimensional Abelian model with discrete heights. We argue that in the large L limit, the variance of energy at site x has a scaling form L(-1)g(x/L) , where g(xi) varies as ln(1/xi) for small xi , which agrees very well with the results from numerical simulations.