Heat Conduction and Entropy Production in a One-Dimensional Hard-Particle Gas

We present large scale simulations for a one-dimensional chain of hard-point particles with alternating masses. We correct several claims in the recent literature based on much smaller simulations. Both for boundary conditions with two heat baths at different temperatures at both ends and from heat current autocorrelations in equilibrium we find heat conductivities kappa to diverge with the number N of particles. These depended very strongly on the mass ratios, and extrapolation to N -> infty resp. t -> infty is difficult due to very large finite-size and finite-time corrections. Nevertheless, our data seem compatible with a universal power law kappa ~ N^alpha with alpha approx 0.33. This suggests a relation to the Kardar-Parisi-Zhang model. We finally show that the hard-point gas with periodic boundary conditions is not chaotic in the usual sense and discuss why the system, when kept out of equilibrium, leads nevertheless to energy dissipation and entropy production.Comment: 4 pages (incl. 5 figures), RevTe

The wave-vector power spectrum of the local tunnelling density of states: ripples in a d-wave sea

A weak scattering potential imposed on a $CuO_2$ layer of a cuprate superconductor modulates the local density of states $N(x,\omega)$. In recently reported experimental studies scanning-tunneling maps of $N(x,\omega)$ have been Fourier transformed to obtain a wave-vector power spectrum. Here, for the case of a weak scattering potential, we discuss the structure of this power spectrum and its relationship to the quasi-particle spectrum and the structure factor of the scattering potential. Examples of quasi-particle interferences in normal metals and $s$- and d-wave superconductors are discussed.Comment: 22 pages, 21 figures; enlarged discussion of the d-wave response, to be published in Physical Review

A Search for Instantons at HERA

A search for QCD instanton (I) induced events in deep-inelastic scattering (DIS) at HERA is presented in the kinematic range of low x and low Q^2. After cutting into three characteristic variables for I-induced events yielding a maximum suppression of standard DIS background to the 0.1% level while still preserving 10% of the I-induced events, 549 data events are found while 363^{+22}_{-26} (CDM) and 435^{+36}_{-22} (MEPS) standard DIS events are expected. More events than expected by the standard DIS Monte Carlo models are found in the data. However, the systematic uncertainty between the two different models is of the order of the expected signal, so that a discovery of instantons can not be claimed. An outlook is given on the prospect to search for QCD instanton events using a discriminant based on range searching in the kinematical region Q^2\gtrsim100\GeV^2 where the I-theory makes safer predictions and the QCD Monte Carlos are expected to better describe the inclusive data.Comment: Invited talk given at the Ringberg Workshop on HERA Physics on June 19th, 2001 on behalf of the H1 collaboratio

Phase Transition in the Aldous-Shields Model of Growing Trees

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to c^{-l} where c is a positive parameter and l is the distance of the perimeter site from the root. For c=1, this model corresponds to random binary search trees and for c=2 it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value c=sqrt{2}. While for c>sqrt{2} the variance is proportional to the mean and the distribution is normal, for c<sqrt{2} the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at c=sqrt{m}.Comment: Latex 17 pages, 6 figure

Phase transition for cutting-plane approach to vertex-cover problem

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from "easy" to "hard". In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an "easy-hard" transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.Comment: 4 pages, 3 figure

Efficient Monte Carlo algorithm and high-precision results for percolation

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at occupation probability 0.59274621(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.Comment: 8 pages, including 3 postscript figures, minor corrections in this version, plus updated figures for the position of the percolation transitio