16 research outputs found
Stochastic Process Associated with Traveling Wave Solutions of the Sine-Gordon Equation
Stochastic processes associated with traveling wave solutions of the
sine-Gordon equation are presented. The structure of the forward Kolmogorov
equation as a conservation law is essential in the construction and so is the
traveling wave structure. The derived stochastic processes are analyzed
numerically. An interpretation of the behaviors of the stochastic processes is
given in terms of the equation of motion.Comment: 12 pages, 9 figures; corrected typo
An invariant in shock clustering and Burgers turbulence
1-D scalar conservation laws with convex flux and Markov initial data are now
known to yield a completely integrable Hamiltonian system. In this article, we
rederive the analogue of Loitsiansky's invariant in hydrodynamic turbulence
from the perspective of integrable systems. Other relevant physical notions
such as energy dissipation and spectrum are also discussed.Comment: 11 pages, no figures; v2: corrections mad
Dynamical Transition in the Open-boundary Totally Asymmetric Exclusion Process
We revisit the totally asymmetric simple exclusion process with open
boundaries (TASEP), focussing on the recent discovery by de Gier and Essler
that the model has a dynamical transition along a nontrivial line in the phase
diagram. This line coincides neither with any change in the steady-state
properties of the TASEP, nor the corresponding line predicted by domain wall
theory. We provide numerical evidence that the TASEP indeed has a dynamical
transition along the de Gier-Essler line, finding that the most convincing
evidence was obtained from Density Matrix Renormalisation Group (DMRG)
calculations. By contrast, we find that the dynamical transition is rather hard
to see in direct Monte Carlo simulations of the TASEP. We furthermore discuss
in general terms scenarios that admit a distinction between static and dynamic
phase behaviour.Comment: 27 pages, 18 figures. v2 to appear in J Phys A features minor
corrections and better-quality figure
Mixtures in non stable Levy processes
We analyze the Levy processes produced by means of two interconnected classes
of non stable, infinitely divisible distribution: the Variance Gamma and the
Student laws. While the Variance Gamma family is closed under convolution, the
Student one is not: this makes its time evolution more complicated. We prove
that -- at least for one particular type of Student processes suggested by
recent empirical results, and for integral times -- the distribution of the
process is a mixture of other types of Student distributions, randomized by
means of a new probability distribution. The mixture is such that along the
time the asymptotic behavior of the probability density functions always
coincide with that of the generating Student law. We put forward the conjecture
that this can be a general feature of the Student processes. We finally analyze
the Ornstein--Uhlenbeck process driven by our Levy noises and show a few
simulation of it.Comment: 28 pages, 3 figures, to be published in J. Phys. A: Math. Ge