1,456 research outputs found
Quantum mechanics of lattice gas automata. II. Boundary conditions and other inhomogeneities
We continue our analysis of the physics of quantum lattice gas automata
(QLGA). Previous work has been restricted to periodic or infinite lattices;
simulation of more realistic physical situations requires finite sizes and
non-periodic boundary conditions. Furthermore, envisioning a QLGA as a
nanoscale computer architecture motivates consideration of inhomogeneities in
the `substrate'; this translates into inhomogeneities in the local evolution
rules. Concentrating on the one particle sector of the model, we determine the
various boundary conditions and rule inhomogeneities which are consistent with
unitary global evolution. We analyze the reflection of plane waves from
boundaries, simulate wave packet refraction across inhomogeneities, and
conclude by discussing the extension of these results to multiple particles.Comment: 24 pages, plain TeX, 9 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages), 3 additional large figures
available upon request or from
http://math.ucsd.edu/~dmeyer/papers/papers.htm
Boolean Delay Equations: A simple way of looking at complex systems
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with
Boolean-valued variables that evolve in continuous time. Systems of BDEs can be
classified into conservative or dissipative, in a manner that parallels the
classification of ordinary or partial differential equations. Solutions to
certain conservative BDEs exhibit growth of complexity in time. They represent
therewith metaphors for biological evolution or human history. Dissipative BDEs
are structurally stable and exhibit multiple equilibria and limit cycles, as
well as more complex, fractal solution sets, such as Devil's staircases and
``fractal sunbursts``. All known solutions of dissipative BDEs have stationary
variance. BDE systems of this type, both free and forced, have been used as
highly idealized models of climate change on interannual, interdecadal and
paleoclimatic time scales. BDEs are also being used as flexible, highly
efficient models of colliding cascades in earthquake modeling and prediction,
as well as in genetics. In this paper we review the theory of systems of BDEs
and illustrate their applications to climatic and solid earth problems. The
former have used small systems of BDEs, while the latter have used large
networks of BDEs. We moreover introduce BDEs with an infinite number of
variables distributed in space (``partial BDEs``) and discuss connections with
other types of dynamical systems, including cellular automata and Boolean
networks. This research-and-review paper concludes with a set of open
questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular
the discussion on partial BDEs is updated and enlarge
Quantum lattice gases and their invariants
The one particle sector of the simplest one dimensional quantum lattice gas
automaton has been observed to simulate both the (relativistic) Dirac and
(nonrelativistic) Schroedinger equations, in different continuum limits. By
analyzing the discrete analogues of plane waves in this sector we find
conserved quantities corresponding to energy and momentum. We show that the
Klein paradox obtains so that in some regimes the model must be considered to
be relativistic and the negative energy modes interpreted as positive energy
modes of antiparticles. With a formally similar approach--the Bethe ansatz--we
find the evolution eigenfunctions in the two particle sector of the quantum
lattice gas automaton and conclude by discussing consequences of these
calculations and their extension to more particles, additional velocities, and
higher dimensions.Comment: 19 pages, plain TeX, 11 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages
Lattice Gas Automata for Reactive Systems
Reactive lattice gas automata provide a microscopic approachto the dynamics
of spatially-distributed reacting systems. After introducing the subject within
the wider framework of lattice gas automata (LGA) as a microscopic approach to
the phenomenology of macroscopic systems, we describe the reactive LGA in terms
of a simple physical picture to show how an automaton can be constructed to
capture the essentials of a reactive molecular dynamics scheme. The statistical
mechanical theory of the automaton is then developed for diffusive transport
and for reactive processes, and a general algorithm is presented for reactive
LGA. The method is illustrated by considering applications to bistable and
excitable media, oscillatory behavior in reactive systems, chemical chaos and
pattern formation triggered by Turing bifurcations. The reactive lattice gas
scheme is contrasted with related cellular automaton methods and the paper
concludes with a discussion of future perspectives.Comment: to appear in PHYSICS REPORTS, 81 revtex pages; uuencoded gziped
postscript file; figures available from [email protected] or
[email protected]
Traffic flow on realistic road networks with adaptive traffic lights
We present a model of traffic flow on generic urban road networks based on
cellular automata. We apply this model to an existing road network in the
Australian city of Melbourne, using empirical data as input. For comparison, we
also apply this model to a square-grid network using hypothetical input data.
On both networks we compare the effects of non-adaptive vs adaptive traffic
lights, in which instantaneous traffic state information feeds back into the
traffic signal schedule. We observe that not only do adaptive traffic lights
result in better averages of network observables, they also lead to
significantly smaller fluctuations in these observables. We furthermore compare
two different systems of adaptive traffic signals, one which is informed by the
traffic state on both upstream and downstream links, and one which is informed
by upstream links only. We find that, in general, both the mean and the
fluctuation of the travel time are smallest when using the joint
upstream-downstream control strategy.Comment: 41 pages, pdflate
Exact matrix product decay modes of a boundary driven cellular automaton
We study integrability properties of a reversible deterministic cellular
automaton (the rule 54 of [Bobenko et al., Commun. Math. Phys. 158, 127
(1993)]) and present a bulk algebraic relation and its inhomogeneous extension
which allow for an explicit construction of Liouvillian decay modes for two
distinct families of stochastic boundary driving. The spectrum of the many-body
stochastic matrix defining the time propagation is found to separate into sets,
which we call orbitals, and the eigenvalues in each orbital are found to obey a
distinct set of Bethe-like equations. We construct the decay modes in the first
orbital (containing the leading decay mode) in terms of an exact inhomogeneous
matrix product ansatz, study the thermodynamic properties of the spectrum and
the scaling of its gap, and provide a conjecture for the Bethe-like equations
for all the orbitals and their degeneracy.Comment: 25 pages, 3 figure
Self-organized patterns of coexistence out of a predator-prey cellular automaton
We present a stochastic approach to modeling the dynamics of coexistence of
prey and predator populations. It is assumed that the space of coexistence is
explicitly subdivided in a grid of cells. Each cell can be occupied by only one
individual of each species or can be empty. The system evolves in time
according to a probabilistic cellular automaton composed by a set of local
rules which describe interactions between species individuals and mimic the
process of birth, death and predation. By performing computational simulations,
we found that, depending on the values of the parameters of the model, the
following states can be reached: a prey absorbing state and active states of
two types. In one of them both species coexist in a stationary regime with
population densities constant in time. The other kind of active state is
characterized by local coupled time oscillations of prey and predator
populations. We focus on the self-organized structures arising from
spatio-temporal dynamics of the coexistence. We identify distinct spatial
patterns of prey and predators and verify that they are intimally connected to
the time coexistence behavior of the species. The occurrence of a prey
percolating cluster on the spatial patterns of the active states is also
examined.Comment: 19 pages, 11 figure
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