Sampling fractional Brownian motion in presence of absorption: a Markov chain method

Thermally activated particle motion in biased correlated Gaussian disorder potentials

Thermally activated particle motion in disorder potentials is controlled by the large-ΔV tail of the distribution of height ΔV of the potential barriers created by the disorder. We employ the optimal fluctuation method to evaluate this tail for correlated quenched Gaussian potentials in one dimension in the presence of a small bias of the potential. We focus on the mean escape time (MET) of overdamped particles averaged over the disorder. We show that the bias leads to a strong (exponential) reduction of the MET in the direction along the bias. The reduction depends both on the bias and on detailed properties of the covariance of the disorder, such as its derivatives and asymptotic behavior at large distances. We verify our theoretical predictions for the large-ΔV tail of the barrier height distribution, as well as earlier predictions of this tail for zero bias, by performing large-deviation simulations of the potential disorder. The simulations employ correlated random potential sampling based on the circulant embedding method and the Wang-Landau algorithm, which enable us to probe probability densities smaller than 10^{-1200}.

Probing the large deviations for the beta random walk in random medium

We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.

First-passage area distribution and optimal fluctuations of fractional Brownian motion

We study the probability distribution P(A) of the area A=∫_{0}^{T}x(t)dt swept under fractional Brownian motion (fBm) x(t) until its first passage time T to the origin. The process starts at t=0 from a specified point x=L. We show that P(A) obeys exact scaling relation P(A)=D^{1/2H}/L^{1+1/H}Φ_{H}(D^{1/2H}A/L^{1+1/H}), where 0 Collapse

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Affiliation(s)

Alexander K Hartmann Institut für Physik, Universitåt Oldenburg - 26111 Oldenburg, Germany
Baruch Meerson Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

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Observing symmetry-broken optimal paths of the stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media

Consider the short-time probability distribution P(H,t) of the one-point interface height difference h(x=0,τ=t)-h(x=0,τ=0)=H of the stationary interface h(x,τ) described by the Kardar-Parisi-Zhang (KPZ) equation. It was previously shown that the optimal path, the most probable history of the interface h(x,τ) which dominates the upper tail of P(H,t), is described by any of two ramplike structures of h(x,τ) traveling either to the left, or to the right. These two solutions emerge, at a critical value of H, via a spontaneous breaking of the mirror symmetry x↔-x of the optimal path, and this symmetry breaking is responsible for a second-order dynamical phase transition in the system. We simulate the interface configurations numerically by employing a large-deviation Monte Carlo sampling algorithm in conjunction with the mapping between the KPZ interface and the directed polymer in a random potential at high temperature. This allows us to observe the optimal paths, which determine each of the two tails of P(H,t), down to probability densities as small as 10^{-500}. At short times we observe mirror-symmetry-broken traveling optimal paths for the upper tail, and a single mirror-symmetric path for the lower tail, in good quantitative agreement with analytical predictions. At long times, even at moderate values of H, where the optimal fluctuation method is not supposed to apply, we still observe two well-defined dominating paths. Each of them violates the mirror symmetry x↔-x and is a mirror image of the other.

Probing large deviations of the Kardar-Parisi-Zhang equation at short times with an importance sampling of directed polymers in random media

The one-point distribution of the height for the continuum Kardar-Parisi-Zhang equation is determined numerically using the mapping to the directed polymer in a random potential at high temperature. Using an importance sampling approach, the distribution is obtained over a large range of values, down to a probability density as small as 10^{-1000} in the tails. The short-time behavior is investigated and compared with recent analytical predictions for the large-deviation forms of the probability of rare fluctuations, showing a spectacular agreement with the analytical expressions. The flat and stationary initial conditions are studied in the full space, together with the droplet initial condition in the half-space.

Large-deviations of the basin stability of power grids

Energy grids play an important role in modern society. In recent years, there was a shift from using few central power sources to using many small power sources, due to efforts to increase the percentage of renewable energies. Therefore, the properties of extremely stable and unstable networks are of interest. In this paper, distributions of the basin stability, a nonlinear measure to quantify the ability of a power grid to recover from perturbations, and its correlations with other measurable quantities, namely, diameter, flow backup capacity, power-sign ratio, universal order parameter, biconnected component, clustering coefficient, two core, and leafs, are studied. The energy grids are modeled by an Erdős-Rényi random graph ensemble and a small-world graph ensemble, where the latter is defined in such a way that it does not exhibit dead ends. Using large-deviation techniques, we reach very improbable power grids that are extremely stable as well as ones that are extremely unstable. The 1/t-algorithm, a variation of Wang-Landau, which does not suffer from error saturation, and additional entropic sampling are used to achieve good precision even for very small probabilities ranging over eight decades.

High-precision work distributions for extreme nonequilibrium processes in large systems

The distributions of work for strongly nonequilibrium processes are studied using a very general form of a large-deviation approach, which allows one to study distributions down to extremely small probabilities of almost arbitrary quantities of interest for equilibrium, nonequilibrium stationary, and even nonstationary processes. The method is applied to quickly vary the external field in a wide range B = 3 ↔ 0 for a critical (T = 2.269) two-dimensional Ising system of size L × L = 128 × 128. To obtain free-energy differences from the work distributions, they must be studied in ranges where the probabilities are as small as 10^{-240}, which is not possible using direct simulation approaches. By comparison with the exact free energies, which are available for this model for the zero-field case, one sees that the present approach allows one to obtain the free energy with a very high relative precision of 10^{-4}. This works well also for a nonzero field, i.e., for a case where standard umbrella-sampling methods are not efficient to calculate free energies. Furthermore, for the present case it is verified that the resulting distributions of work for forward and backward processes fulfill Crooks theorem with high precision. Finally, the free energy for the Ising magnet as a function of the field strength is obtained.

Dynamics of a tagged monomer: effects of elastic pinning and harmonic absorption

We study the dynamics of a tagged monomer of a Rouse polymer for different initial configurations. In the case of free evolution, the monomer displays subdiffusive behavior with strong memory of the initial state. In the presence of either elastic pinning or harmonic absorption, we show that the steady state is independent of the initial condition that, however, strongly affects the transient regime, resulting in nonmonotonic behavior and power-law relaxation with varying exponents.