107 research outputs found
Nonlinear response of a linear chain to weak driving
We study the escape of a chain of coupled units over the barrier of a
metastable potential. It is demonstrated that a very weak external driving
field with suitably chosen frequency suffices to accomplish speedy escape. The
latter requires the passage through a transition state the formation of which
is triggered by permanent feeding of energy from a phonon background into humps
of localised energy and elastic interaction of the arising breather solutions.
In fact, cooperativity between the units of the chain entailing coordinated
energy transfer is shown to be crucial for enhancing the rate of escape in an
extremely effective and low-energy cost way where the effect of entropic
localisation and breather coalescence conspire
Cooperative surmounting of bottlenecks
The physics of activated escape of objects out of a metastable state plays a
key role in diverse scientific areas involving chemical kinetics, diffusion and
dislocation motion in solids, nucleation, electrical transport, motion of flux
lines superconductors, charge density waves, and transport processes of
macromolecules, to name but a few. The underlying activated processes present
the multidimensional extension of the Kramers problem of a single Brownian
particle. In comparison to the latter case, however, the dynamics ensuing from
the interactions of many coupled units can lead to intriguing novel phenomena
that are not present when only a single degree of freedom is involved. In this
review we report on a variety of such phenomena that are exhibited by systems
consisting of chains of interacting units in the presence of potential
barriers.
In the first part we consider recent developments in the case of a
deterministic dynamics driving cooperative escape processes of coupled
nonlinear units out of metastable states. The ability of chains of coupled
units to undergo spontaneous conformational transitions can lead to a
self-organised escape. The mechanism at work is that the energies of the units
become re-arranged, while keeping the total energy conserved, in forming
localised energy modes that in turn trigger the cooperative escape. We present
scenarios of significantly enhanced noise-free escape rates if compared to the
noise-assisted case.
The second part deals with the collective directed transport of systems of
interacting particles overcoming energetic barriers in periodic potential
landscapes. Escape processes in both time-homogeneous and time-dependent driven
systems are considered for the emergence of directed motion. It is shown that
ballistic channels immersed in the associated high-dimensional phase space are
the source for the directed long-range transport
Macroscopic limit cycle via pure noise-induced phase transition
Bistability generated via a pure noise-induced phase transition is reexamined
from the view of bifurcations in macroscopic cumulant dynamics. It allows an
analytical study of the phase diagram in more general cases than previous
methods. In addition using this approach we investigate patially-extended
systems with two degrees of freedom per site. For this system, the analytic
solution of the stationary Fokker-Planck equation is not available and a
standard mean field approach cannot be used to find noise induced phase
transitions. A new approach based on cumulant dynamics predicts a noise-induced
phase transition through a Hopf bifurcation leading to a macroscopic limit
cycle motion, which is confirmed by numerical simulation.Comment: 8 pages, 8 figure
Active Brownian particles with velocity-alignment and active fluctuations
We consider a model of active Brownian particles with velocity-alignment in
two spatial dimensions with passive and active fluctuations. Hereby, active
fluctuations refers to purely non-equilibrium stochastic forces correlated with
the heading of an individual active particle. In the simplest case studied
here, they are assumed as independent stochastic forces parallel (speed noise)
and perpendicular (angular noise) to the velocity of the particle. On the other
hand, passive fluctuations are defined by a noise vector independent of the
direction of motion of a particle, and may account for example for thermal
fluctuations.
We derive a macroscopic description of the active Brownian particle gas with
velocity-alignment interaction. Hereby, we start from the individual based
description in terms of stochastic differential equations (Langevin equations)
and derive equations of motion for the coarse grained kinetic variables
(density, velocity and temperature) via a moment expansion of the corresponding
probability density function.
We focus here in particular on the different impact of active and passive
fluctuations on the onset of collective motion and show how active fluctuations
in the active Brownian dynamics can change the phase-transition behaviour of
the system. In particular, we show that active angular fluctuation lead to an
earlier breakdown of collective motion and to emergence of a new bistable
regime in the mean-field case.Comment: 5 figures, 22 pages, submitted to New Journal of Physic
Self-organized escape of oscillator chains in nonlinear potentials
We present the noise free escape of a chain of linearly interacting units
from a metastable state over a cubic on-site potential barrier. The underlying
dynamics is conservative and purely deterministic. The mutual interplay between
nonlinearity and harmonic interactions causes an initially uniform lattice
state to become unstable, leading to an energy redistribution with strong
localization. As a result a spontaneously emerging localized mode grows into a
critical nucleus. By surpassing this transition state, the nonlinear chain
manages a self-organized, deterministic barrier crossing. Most strikingly,
these noise-free, collective nonlinear escape events proceed generally by far
faster than transitions assisted by thermal noise when the ratio between the
average energy supplied per unit in the chain and the potential barrier energy
assumes small values
Rectification of motion in nonlinear media with asymmetric random drive
We consider moving particles in media with nonlinear friction and drive them
by an asymmetric dichotomic Markov process. Due to different energy
dissipations, during the forward and backward stroke, we obtain a mean
non-vanishing directed flow of the particles. Starting with the stationary
velocity distribution, we calculate the stationary current of particles, the
variance and the relative variance in dependence on the degree of nonlinearity
of the friction, on the asymmetry and for different strengths of friction. In
two dimensions the particle performs diffusional motion, if in addition the
direction of the asymmetric drive changes stochastically.Comment: 17 pages, 6 figure
Noise Induced Complexity: From Subthreshold Oscillations to Spiking in Coupled Excitable Systems
We study stochastic dynamics of an ensemble of N globally coupled excitable
elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is
disturbed by independent Gaussian noise. In simulations of the Langevin
dynamics we characterize the collective behavior of the ensemble in terms of
its mean field and show that with the increase of noise the mean field displays
a transition from a steady equilibrium to global oscillations and then, for
sufficiently large noise, back to another equilibrium. Diverse regimes of
collective dynamics ranging from periodic subthreshold oscillations to
large-amplitude oscillations and chaos are observed in the course of this
transition. In order to understand details and mechanisms of noise-induced
dynamics we consider a thermodynamic limit of the ensemble, and
derive the cumulant expansion describing temporal evolution of the mean field
fluctuations. In the Gaussian approximation this allows us to perform the
bifurcation analysis; its results are in good agreement with dynamical
scenarios observed in the stochastic simulations of large ensembles
Canonical active Brownian motion
Active Brownian motion is the complex motion of active Brownian particles.
They are active in the sense that they can transform their internal energy into
energy of motion and thus create complex motion patterns. Theories of active
Brownian motion so far imposed couplings between the internal energy and the
kinetic energy of the system. We investigate how this idea can be naturally
taken further to include also couplings to the potential energy, which finally
leads to a general theory of canonical dissipative systems. Explicit analytical
and numerical studies are done for the motion of one particle in harmonic
external potentials. Apart from stationary solutions, we study non-equilibrium
dynamics and show the existence of various bifurcation phenomena.Comment: 11 pages, 6 figures, a few remarks and references adde
Asymptotic Scaling of the Diffusion Coefficient of Fluctuating "Pulled" Fronts
We present a (heuristic) theoretical derivation for the scaling of the
diffusion coefficient for fluctuating ``pulled'' fronts. In agreement
with earlier numerical simulations, we find that as ,
approaches zero as , where is the average number of particles per
correlation volume in the stable phase of the front. This behaviour of
stems from the shape fluctuations at the very tip of the front, and is
independent of the microscopic model.Comment: Some minor algebra corrected, to appear in Rapid Comm., Phys. Rev.
Surmounting collectively oscillating bottlenecks
We study the collective escape dynamics of a chain of coupled, weakly damped
nonlinear oscillators from a metastable state over a barrier when driven by a
thermal heat bath in combination with a weak, globally acting periodic
perturbation. Optimal parameter choices are identified that lead to a drastic
enhancement of escape rates as compared to a pure noise-assisted situation. We
elucidate the speed-up of escape in the driven Langevin dynamics by showing
that the time-periodic external field in combination with the thermal
fluctuations triggers an instability mechanism of the stationary homogeneous
lattice state of the system. Perturbations of the latter provided by incoherent
thermal fluctuations grow because of a parametric resonance, leading to the
formation of spatially localized modes (LMs). Remarkably, the LMs persist in
spite of continuously impacting thermal noise. The average escape time assumes
a distinct minimum by either tuning the coupling strength and/or the driving
frequency. This weak ac-driven assisted escape in turn implies a giant speed of
the activation rate of such thermally driven coupled nonlinear oscillator
chains
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