The uphill turtle race: on short time nucleation probabilities

The short time behavior of nucleation probabilities is studied by representing nucleation as diffusion in a potential well with escape over a barrier. If initially all growing nuclei start at the bottom of the well, the first nucleation time on average is larger than the inverse nucleation frequency. Explicit expressions are obtained for the short time probability of first nucleation. For very short times these become independent of the shape of the potential well. They agree well with numerical results from an exact enumeration scheme. For a large number N of growing nuclei the average first nucleation time scales as 1/\log N in contrast to the long-time nucleation frequency, which scales as 1/N. For linear potential wells closed form expressions are obtained for all times.Comment: 8 pages, submitted to J. Stat. Phy

Exact results for anomalous transport in one dimensional Hamiltonian systems

Anomalous transport in one dimensional translation invariant Hamiltonian systems with short range interactions, is shown to belong in general to the KPZ universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Pr\"ahofer-Spohn scaling functions, obtained from their exact solution for the Polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains, but in need of corrections for strongly anharmonic interparticle potentials.Comment: Further corrections to equations have been made. A few comments have been added, e.g. on the non-applicability to exactly solved model

Systematic Density Expansion of the Lyapunov Exponents for a Two-dimensional Random Lorentz Gas

We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form $A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}$, where $A_{0}$ and $B_{0}$ are known constants and $\tilde{n}$ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order $(\tilde{n}^{2})$, the positive Lyapunov exponent is of the form $A_{0}\tilde{n}\ln\tilde{n}+B_{0}\tilde{n}+A_{1}\tilde{n}^{2}\ln\tilde{n} +B_{1}\tilde{n}^{2}$. Explicit numerical values of the new constants $A_{1}$ and $B_{1}$ are obtained by means of a systematic analysis. This takes into account, up to $O(\tilde{n}^{2})$, the effects of {\it all\/} possible trajectories in two versions of the model; in one version overlapping scatterer configurations are allowed and in the other they are not.Comment: 12 pages, 9 figures, minor changes in this version, to appear in J. Stat. Phy

A Transfer Matrix study of the staggered BCSOS model

The phase diagram of the staggered six vertex, or body centered solid on solid model, is investigated by transfer matrix and finite size scaling techniques. The phase diagram contains a critical region, bounded by a Kosterlitz-Thouless line, and a second order line describing a deconstruction transition. In part of the phase diagram the deconstruction line and the Kosterlitz-Thouless line approach each other without merging, while the deconstruction changes its critical behaviour from Ising-like to a different universality class. Our model has the same type of symmetries as some other two-dimensional models, such as the fully frustrated XY model, and may be important for understanding their phase behaviour. The thermal behaviour for weak staggering is intricate. It may be relevant for the description of surfaces of ionic crystals of CsCl structure.Comment: 13 pages, RevTex, 1 Postscript file with all figures, to be published in Phys. Rev.

Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases II: Open Systems

We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e. the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is finite in at least some directions, and with absorbing boundary conditions, the moving particle escapes the system with probability one. However, there is a set of zero Lebesgue measure of initial phase points for the moving particle, such that escape never occurs. Typically, this set of points forms a fractal repeller, and the Lyapunov spectrum is calculated here for trajectories on this repeller. For this calculation, we need the solution of the recently introduced extended Boltzmann equation for the nonequilibrium distribution of the radius of curvature matrix and the solution of the standard Boltzmann equation. The escape-rate formalism then gives an explicit result for the Kolmogorov Sinai entropy on the repeller.Comment: submitted to Phys Rev

Long-time-tail Effects on Lyapunov Exponents of a Random, Two-dimensional Field-driven Lorentz Gas

We study the Lyapunov exponents for a moving, charged particle in a two-dimensional Lorentz gas with randomly placed, non-overlapping hard disk scatterers placed in a thermostatted electric field, $\vec{E}$. The low density values of the Lyapunov exponents have been calculated with the use of an extended Lorentz-Boltzmann equation. In this paper we develop a method to extend these results to higher density, using the BBGKY hierarchy equations and extending them to include the additional variables needed for calculation of Lyapunov exponents. We then consider the effects of correlated collision sequences, due to the so-called ring events, on the Lyapunov exponents. For small values of the applied electric field, the ring terms lead to non-analytic, field dependent, contributions to both the positive and negative Lyapunov exponents which are of the form ${\tilde{\epsilon}}^{2} \ln\tilde{\epsilon}$, where $\tilde{\epsilon}$ is a dimensionless parameter proportional to the strength of the applied field. We show that these non-analytic terms can be understood as resulting from the change in the collision frequency from its equilibrium value, due to the presence of the thermostatted field, and that the collision frequency also contains such non-analytic terms.Comment: 45 pages, 4 figures, to appear in J. Stat. Phy

A Note on the Ruelle Pressure for a Dilute Disordered Sinai Billiard

The topological pressure is evaluated for a dilute random Lorentz gas, in the approximation that takes into account only uncorrelated collisions between the moving particle and fixed, hard sphere scatterers. The pressure is obtained analytically as a function of a temperature-like parameter, beta, and of the density of scatterers. The effects of correlated collisions on the topological pressure can be described qualitatively, at least, and they significantly modify the results obtained by considering only uncorrelated collision sequences. As a consequence, for large systems, the range of beta-values over which our expressions for the topological pressure are valid becomes very small, approaching zero, in most cases, as the inverse of the logarithm of system size.Comment: 15 pages RevTeX with 2 figures. Final version with some typo's correcte

On thermostats and entropy production

The connection between the rate of entropy production and the rate of phase space contraction for thermostatted systems in nonequilibrium steady states is discussed for a simple model of heat flow in a Lorentz gas, previously described by Spohn and Lebowitz. It is easy to show that for the model discussed here the two rates are not connected, since the rate of entropy production is non-zero and positive, while the overall rate of phase space contraction is zero. This is consistent with conclusions reached by other workers. Fractal structures appear in the phase space for this model and their properties are discussed. We conclude with a discussion of the implications of this and related work for understanding the role of chaotic dynamics and special initial conditions for an explanation of the Second Law of Thermodynamics.Comment: 14 pages, 1 figur