Application of importance sampling to the computation of large deviations in non-equilibrium processes

We present an algorithm for finding the probabilities of rare events in nonequilibrium processes. The algorithm consists of evolving the system with a modified dynamics for which the required event occurs more frequently. By keeping track of the relative weight of phase-space trajectories generated by the modified and the original dynamics one can obtain the required probabilities. The algorithm is tested on two model systems of steady-state particle and heat transport where we find a huge improvement from direct simulation methods.Comment: 5 pages, 4 figures; some modification

Dynamics of bootstrap percolation

Bootstrap percolation transition may be first order or second order, or it may have a mixed character where a first order drop in the order parameter is preceded by critical fluctuations. Recent studies have indicated that the mixed transition is characterized by power law avalanches, while the continuous transition is characterized by truncated avalanches in a related sequential bootstrap process. We explain this behavior on the basis of a through analytical and numerical study of the avalanche distributions on a Bethe lattice.Comment: Proceedings of the International Workshop and Conference on Statistical Physics Approaches to Multidisciplinary Problems, IIT Guwahati, India, 7-13 January 200

Accurate statistics of a flexible polymer chain in shear flow

We present exact and analytically accurate results for the problem of a flexible polymer chain in shear flow. Under such a flow the polymer tumbles, and the probability distribution of the tumbling times $\tau$ of the polymer decays exponentially as $\sim \exp(-\alpha \tau/\tau_0)$ (where $\tau_0$ is the longest relaxation time). We show that for a Rouse chain, this nontrivial constant $\alpha$ can be calculated in the limit of large Weissenberg number (high shear rate) and is in excellent agreement with our simulation result of $\alpha \simeq 0.324$. We also derive exactly the distribution functions for the length and the orientational angles of the end-to-end vector of the polymer.Comment: 4 pages, 2 figures. Minor changes. Texts differ slightly from the PRL published versio

Crowding at the Front of the Marathon Packs

We study the crowding of near-extreme events in the time gaps between successive finishers in major international marathons. Naively, one might expect these gaps to become progressively larger for better-placing finishers. While such an increase does indeed occur from the middle of the finishing pack down to approximately 20th place, the gaps saturate for the first 10-20 finishers. We give a probabilistic account of this feature. However, the data suggests that the gaps have a weak maximum around the 10th place, a feature that seems to have a sociological origin.Comment: 5 pages, 2 figures; version 2: published manuscript with various changes in response to referee comments and some additional improvement

Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verifications of our analytical results

Exact Expressions for Minor Hysteresis Loops in the Random Field Ising Model on a Bethe Lattice at Zero Temperature

We obtain exact expressions for the minor hysteresis loops in the ferromagnetic random field Ising model on a Bethe lattice at zero temperature in the case when the driving field is cycled infinitely slowly.Comment: Replaced with the published versio

Hysteresis in the Random Field Ising Model and Bootstrap Percolation

We study hysteresis in the random-field Ising model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and show that the characteristic length for self-averaging $L^*$ increases as $exp(exp (J/\Delta))$ in 2d, and as $exp(exp(exp(J/\Delta)))$ in 3d, for disorder strength $\Delta$ much less than the exchange coupling J. For system size $1 << L < L^*$, the coercive field $h_{coer}$ varies as $2J - \Delta \ln \ln L$ for the square lattice, and as $2J - \Delta \ln \ln \ln L$ on the cubic lattice. Its limiting value is 0 for L tending to infinity, both for square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and $h_{coer}$ tends to J.Comment: 4 pages, 4 figure

A Note on Limit Shapes of Minimal Difference Partitions

We provide a variational derivation of the limit shape of minimal difference partitions and discuss the link with exclusion statistic

Record statistics in random vectors and quantum chaos

The record statistics of complex random states are analytically calculated, and shown that the probability of a record intensity is a Bernoulli process. The correlation due to normalization leads to a probability distribution of the records that is non-universal but tends to the Gumbel distribution asymptotically. The quantum standard map is used to study these statistics for the effect of correlations apart from normalization. It is seen that in the mixed phase space regime the number of intensity records is a power law in the dimensionality of the state as opposed to the logarithmic growth for random states.Comment: figures redrawn, discussion adde

Performance Limitations of Flat Histogram Methods and Optimality of Wang-Landau Sampling

We determine the optimal scaling of local-update flat-histogram methods with system size by using a perfect flat-histogram scheme based on the exact density of states of 2D Ising models.The typical tunneling time needed to sample the entire bandwidth does not scale with the number of spins N as the minimal N^2 of an unbiased random walk in energy space. While the scaling is power law for the ferromagnetic and fully frustrated Ising model, for the +/- J nearest-neighbor spin glass the distribution of tunneling times is governed by a fat-tailed Frechet extremal value distribution that obeys exponential scaling. We find that the Wang-Landau algorithm shows the same scaling as the perfect scheme and is thus optimal.Comment: 5 pages, 6 figure