Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization
of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.
We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagg
ed particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statistics of the tagged particle trajectory---the full single-time statistics, all multiple-time correlation functions, etc. are merely special cases.
Universal scaling of entanglement estimators of critical quantum systems has drawn a lot of attention in the past. Recent studies indicate that similar universal properties can be found for bipartite information estimators of classical systems near p
hase transitions, opening a new direction in the study of critical phenomena. We explore this subject by studying the information estimators of classical spin chains with general mean-field interactions. In our explicit analysis of two different bipartite information estimators in the canonical ensemble we find that, away from criticality both the estimators remain finite in the thermodynamic limit. On the other hand, along the critical line there is a logarithmic divergence with increasing system-size. The coefficient of the logarithm is fully determined by the mean-field interaction and it is the same for the class of models we consider. The scaling function, however, depends on the details of each model. In addition, we study the information estimators in the micro-canonical ensemble, where they are shown to exhibit a different universal behavior. We verify our results using numerical calculations of two specific cases of the general Hamiltonian.
We study the steady state of the abelian sandpile models with stochastic toppling rules. The particle addition operators commute with each other, but in general these operators need not be diagonalizable. We use their abelian algebra to determine the
ir eigenvalues, and the Jordan block structure. These are then used to determine the probability of different configurations in the steady state. We illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example, for systems of size $le12$, and also study the density profile in the steady state.
Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two
arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The size of the pattern formed scales with the number of grains added $N$ as $sqrt{N}$. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern.
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 distrib
ution 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 $log(1/xi)$ for small $xi$, which agrees very well with the results from numerical simulations.
