336 research outputs found
Depinning transitions in discrete reaction-diffusion equations
We consider spatially discrete bistable reaction-diffusion equations that
admit wave front solutions. Depending on the parameters involved, such wave
fronts appear to be pinned or to glide at a certain speed. We study the
transition of traveling waves to steady solutions near threshold and give
conditions for front pinning (propagation failure). The critical parameter
values are characterized at the depinning transition and an approximation for
the front speed just beyond threshold is given.Comment: 27 pages, 12 figures, to appear in SIAM J. Appl. Mat
Interface depinning versus absorbing-state phase transitions
According to recent numerical results from lattice models, the critical
exponents of systems with many absorbing states and an order parameter coupled
to a non-diffusive conserved field coincide with those of the linear interface
depinning model within computational accuracy. In this paper the connection
between absorbing state phase transitions and interface pinning in quenched
disordered media is investigated. For that, we present a mapping of the
interface dynamics in a disordered medium into a Langevin equation for the
active-site density and show that a Reggeon-field-theory like description,
coupled to an additional non-diffusive conserved field, appears rather
naturally. Reciprocally, we construct a mapping from a discrete model belonging
in the absorbing state with-a-conserved-field class to a discrete interface
equation, and show how a quenched disorder is originated.
We discuss the character of the possible noise terms in both representations,
and overview the critical exponent relations. Evidence is provided that, at
least for dimensions larger that one, both universality classes are just two
different representations of the same underlying physics.Comment: 8 page
Wave trains, self-oscillations and synchronization in discrete media
We study wave propagation in networks of coupled cells which can behave as
excitable or self-oscillatory media. For excitable media, an asymptotic
construction of wave trains is presented. This construction predicts their
shape and speed, as well as the critical coupling and the critical separation
of time scales for propagation failure. It describes stable wave train
generation by repeated firing at a boundary. In self-oscillatory media, wave
trains persist but synchronization phenomena arise. An equation describing the
evolution of the oscillator phases is derived.Comment: to appear in Physica D: Nonlinear Phenomen
Wavefront depinning transition in discrete one-dimensional reaction-diffusion systems
Pinning and depinning of wavefronts are ubiquitous features of spatially
discrete systems describing a host of phenomena in physics, biology, etc. A
large class of discrete systems is described by overdamped chains of nonlinear
oscillators with nearest-neighbor coupling and controlled by constant external
forces. A theory of the depinning transition for these systems, including
scaling laws and asymptotics of wavefronts, is presented and confirmed by
numerical calculations.Comment: 4 pages, 4 figure
On possible experimental realizations of directed percolation
Directed percolation is one of the most prominent universality classes of
nonequilibrium phase transitions and can be found in a large variety of models.
Despite its theoretical success, no experiment is known which clearly
reproduces the critical exponents of directed percolation. The present work
compares suggested experiments and discusses possible reasons why the
observation of the critical exponents of directed percolation is obscured or
even impossible.Comment: RevTeX, 13 pages, 11 eps figure
Pulse propagation in discrete systems of coupled excitable cells
Propagation of pulses in myelinated fibers may be described by appropriate
solutions of spatially discrete FitzHugh-Nagumo systems. In these systems,
propagation failure may occur if either the coupling between nodes is not
strong enough or the recovery is too fast. We give an asymptotic construction
of pulses for spatially discrete FitzHugh-Nagumo systems which agrees well with
numerical simulations and discuss evolution of initial data into pulses and
pulse generation at a boundary. Formulas for the speed and length of pulses are
also obtained.Comment: 16 pages, 10 figures, to appear in SIAM J. Appl. Mat
Self-organized criticality in the Kardar-Parisi-Zhang-equation
Kardar-Parisi-Zhang interface depinning with quenched noise is studied in an
ensemble that leads to self-organized criticality in the quenched
Edwards-Wilkinson (QEW) universality class and related sandpile models. An
interface is pinned at the boundaries, and a slowly increasing external drive
is added to compensate for the pinning. The ensuing interface behavior
describes the integrated toppling activity history of a QKPZ cellular
automaton. The avalanche picture consists of several phases depending on the
relative importance of the terms in the interface equation. The SOC state is
more complicated than in the QEW case and it is not related to the properties
of the bulk depinning transition.Comment: 5 pages, 3 figures; accepted for publication in Europhysics Letter
Regulating wave front dynamics from the strongly discrete to the continuum limit in magnetically driven colloidal systems
The emergence of wave fronts in dissipative driven systems is a fascinating phenomenon which can be found in a broad range of physical and biological disciplines. Here we report the direct experimental observation of discrete fronts propagating along chains of paramagnetic colloidal particles, the latter propelled above a traveling wave potential generated by a structured magnetic substrate. We develop a rigorously reduced theoretical framework and describe the dynamics of the system in terms of a generalized one-dimensional dissipative Frenkel-Kontorova model. The front dynamics is explored in a wide range of field parameters close to and far from depinning, where the discrete and continuum limits apply. We show how symmetry breaking and finite size of chains are used to control the direction of front propagation, a universal feature relevant to different systems and important for real applications
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