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Fluctuations of Current in Non-Stationary Diffusive Lattice Gases  [PDF]
P. L. Krapivsky,Baruch Meerson
Physics , 2012, DOI: 10.1103/PhysRevE.86.031106
Abstract: We employ the macroscopic fluctuation theory to study fluctuations of integrated current in one-dimensional lattice gases with a step-like initial density profile. We analytically determine the variance of the current fluctuations for a class of diffusive processes with a density-independent diffusion coefficient, but otherwise arbitrary. Our calculations rely on a perturbation theory around the noiseless hydrodynamic solution. We consider both quenched and annealed types of averaging (the initial condition is allowed to fluctuate in the latter situation). The general results for the variance are specialized to a few interesting models including the symmetric exclusion process and the Kipnis-Marchioro-Presutti model. We also probe large deviations of the current for the symmetric exclusion process. This is done by numerically solving the governing equations of the macroscopic fluctuation theory using an efficient iteration algorithm.
Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition  [PDF]
Bernard Derrida,Antoine Gerschenfeld
Physics , 2009, DOI: 10.1007/s10955-009-9772-7
Abstract: For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current $Q_t$ during time $t$ through the origin when, in the initial condition, the sites are occupied with density $\rho_a$ on the negative axis and with density $\rho_b$ on the positive axis. All the cumulants of $Q_t$ grow like $\sqrt{t}$. In the range where $Q_t \sim \sqrt{t}$, the decay $\exp [-Q_t^3/t]$ of the distribution of $Q_t$ is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities recently derived by Tracy and Widom for exclusion processes on the infinite line.
The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step like initial profile decaying on the right half line  [PDF]
Alexei Rybkin
Mathematics , 2010, DOI: 10.1088/0951-7715/24/10/015
Abstract: We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V_0 which is decaying sufficiently fast at +\infty and arbitrarily enough (i.e., no decay or pattern of behavior) at -\infty. We show that this system is completely integrable in a very strong sense. Namely, the solution V(x,t) admits the Hirota {\tau}-function representation V(x,t)=-2\partial_{x}^2 logdet(I+M_{x,t}) where M_{x,t} is a Hankel integral operator constucted from certain scattering and spectral data suitably defined in terms of the Titchmarsh-Weyl m-functions associated with the two half-line Schr\"odinger operators corresponding to V_0. We show that V(x,t) is real meromorphic with respect to x for any t>0. We also show that under a very mild additional condition on V_0 representation implies a strong well-posedness of the KdV equation with such V_0's. Among others, our approach yields some relevant results due to Cohen, Kappeler, Khruslov, Kotlyarov, Venakides, Zhang and others.
On ASEP with Step Bernoulli Initial Condition  [PDF]
Craig A. Tracy,Harold Widom
Mathematics , 2009, DOI: 10.1007/s10955-009-9867-1
Abstract: This paper extends results of earlier work on ASEP to the case of step Bernoulli initial condition. The main results are a representation in terms of a Fredholm determinant for the probability distribution of a fixed particle, and asymptotic results which in particular establish KPZ universality for this probability in one regime. (And, as a corollary, for the current fluctuations.)
Diffusive and subdiffusive step dynamics  [PDF]
W. Selke,M. Bisani
Physics , 1998, DOI: 10.1007/BFb0106851
Abstract: The dynamics of steps on crystal surfaces is considered. In general, the meandering of the steps obeys a subdiffusive behaviour. The characteristic asymptotic time laws depend on the microscopic mechanism for detachment and attachment of the atoms at the steps. The three limiting cases of step-edge diffusion, evaporation-condensation and terrace diffusion are studied in the framework of Langevin descriptions and by Monte Carlo simulations.
Step fluctuations and random walks  [PDF]
M. Bisani,W. Selke
Physics , 1999, DOI: 10.1016/S0039-6028(99)00709-8
Abstract: The probability distribution p(l) of an atom to return to a step at distance l from the detachment site, with a random walk in between, is exactly enumerated. In particular, we study the dependence of p(l) on step roughness, presence of other reflecting or absorbing steps, interaction between steps and diffusing atom, as well as concentration of defects on the terrace neighbouring the step. Applying Monte Carlo techniques, the time evolution of equilibrium step fluctuations is computed for specific forms of return probabilities. Results are compared to previous theoretical and experimental findings.
Survival in equilibrium step fluctuations  [PDF]
C. Dasgupta,M. Constantin,S. Das Sarma,Satya N. Majumdar
Physics , 2003, DOI: 10.1103/PhysRevE.69.022101
Abstract: We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability $S(t)$ in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. $S(t)$ is shown to exhibit simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical STM data on step fluctuations on Al/Si(111) and Ag(111) surfaces.
Diffusive fluctuations for one-dimensional totally asymmetric interacting random dynamics  [PDF]
Timo Seppalainen
Physics , 2001, DOI: 10.1007/s002200200660
Abstract: We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working.
Mesoscopic fluctuations of the supercurrent in diffusive Josephson junctions  [PDF]
M. Houzet,M. A. Skvortsov
Physics , 2007, DOI: 10.1103/PhysRevB.77.024525
Abstract: We study mesoscopic fluctuations and weak localization correction to the supercurrent in Josephson junctions with coherent diffusive electron dynamics in the normal part. Two kinds of junctions are considered: a chaotic dot coupled to superconductors by tunnel barriers and a diffusive junction with transparent normal--superconducting interfaces. The amplitude of current fluctuations and the weak localization correction to the average current are calculated as functions of the ratio between the superconducting gap and the electron dwell energy, temperature, and superconducting phase difference across the junction. Technically, fluctuations on top of the spatially inhomogeneous proximity effect in the normal region are described by the replicated version of the \sigma-model. For the case of diffusive junctions with transparent interfaces, the magnitude of mesoscopic fluctuations of the critical current appears to be nearly 3 times larger than the prediction of the previous theory which did not take the proximity effect into account.
Influence of initial state fluctuations on the production of thermal photons  [PDF]
Rupa Chatterjee,Hannu Holopainen,Thorsten Renk,Kari J. Eskola
Physics , 2012, DOI: 10.1016/j.nuclphysa.2013.02.149
Abstract: Inhomogeneities in the initial QCD matter density distribution increase the production of thermal photons significantly compared to a smooth initial-state-averaged profile in the region $p_T > 1$ GeV/$c$ in an ideal hydrodynamic calculation. This relative enhancement is more pronounced for peripheral collisions, for smaller size systems as well as for lower beam energies. A suitably normalized ratio of central-to-peripheral yield of thermal photons reduce the uncertainties in the hydrodynamical initial conditions and can be a useful parameter to study the density fluctuations and their size. The fluctuations in the initial density distribution also lead to a larger elliptic flow of thermal photons for $p_T >$ 2.0 GeV/$c$ compared to the flow from a smooth profile.
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