611 research outputs found

### 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

### 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

### 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 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

### Front propagation techniques to calculate the largest Lyapunov exponent of dilute hard disk gases

A kinetic approach is adopted to describe the exponential growth of a small
deviation of the initial phase space point, measured by the largest Lyapunov
exponent, for a dilute system of hard disks, both in equilibrium and in a
uniform shear flow. We derive a generalized Boltzmann equation for an extended
one-particle distribution that includes deviations from the reference phase
space point. The equation is valid for very low densities n, and requires an
unusual expansion in powers of 1/|ln n|. It reproduces and extends results from
the earlier, more heuristic clock model and may be interpreted as describing a
front propagating into an unstable state. The asymptotic speed of propagation
of the front is proportional to the largest Lyapunov exponent of the system.
Its value may be found by applying the standard front speed selection mechanism
for pulled fronts to the case at hand. For the equilibrium case, an explicit
expression for the largest Lyapunov exponent is given and for sheared systems
we give explicit expressions that may be evaluated numerically to obtain the
shear rate dependence of the largest Lyapunov exponent.Comment: 26 pages REVTeX, 1 eps figure. Added remarks, a reference and
corrected some typo

### 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

### Lyapunov exponent for a gas of soft scatterers

For a fast particle moving within a two-dimensional array of soft scatterers
- centers of weak and short-range potential - the dependence of the Lyapunov
exponent on the system parameters is studied. The use of the linearized
equations for variations of the propagation angles and impact parameters of
consequent collisions reduces the problem to that of calculation of the
Lyapunov exponent of an ensemble of strongly correlated random matrices with
given statistics of matrix elements. In the simplest approximation this
Lyapunov exponent is proportional to the interaction strength and inversely
proportional to the square root of the interaction range. The model
satisfactorily describes the intensity of chaos in a system of two weakly
interacting particles moving in a two-dimensional regular confining potential.Comment: 3 pages, 3 figure

### 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.

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