11,899 research outputs found
Symmetry breaking, mixing, instability, and low frequency variability in a minimal Lorenz-like system
Starting from the classical Saltzman two-dimensional convection equations, we derive via a severe spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled generalized Lorenz system. The consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables, with the ensuing modifications to the structural properties of the attractor and of the spectral features. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of O(Ec−1) is introduced in the system, as shown with standard multiscale analysis and made clear by several numerical evidences. Moreover, the system is ergodic and hyperbolic, the slow variables feature long-term memory with 1/f3/2 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics, such as the interaction between slow and fast variables, the presence of long-term memory, and the associated extreme value statistics. This analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing ability in describing intrinsically multiscale processes
Nonlinear brain dynamics as macroscopic manifestation of underlying many-body field dynamics
Neural activity patterns related to behavior occur at many scales in time and
space from the atomic and molecular to the whole brain. Here we explore the
feasibility of interpreting neurophysiological data in the context of many-body
physics by using tools that physicists have devised to analyze comparable
hierarchies in other fields of science. We focus on a mesoscopic level that
offers a multi-step pathway between the microscopic functions of neurons and
the macroscopic functions of brain systems revealed by hemodynamic imaging. We
use electroencephalographic (EEG) records collected from high-density electrode
arrays fixed on the epidural surfaces of primary sensory and limbic areas in
rabbits and cats trained to discriminate conditioned stimuli (CS) in the
various modalities. High temporal resolution of EEG signals with the Hilbert
transform gives evidence for diverse intermittent spatial patterns of amplitude
(AM) and phase modulations (PM) of carrier waves that repeatedly re-synchronize
in the beta and gamma ranges at near zero time lags over long distances. The
dominant mechanism for neural interactions by axodendritic synaptic
transmission should impose distance-dependent delays on the EEG oscillations
owing to finite propagation velocities. It does not. EEGs instead show evidence
for anomalous dispersion: the existence in neural populations of a low velocity
range of information and energy transfers, and a high velocity range of the
spread of phase transitions. This distinction labels the phenomenon but does
not explain it. In this report we explore the analysis of these phenomena using
concepts of energy dissipation, the maintenance by cortex of multiple ground
states corresponding to AM patterns, and the exclusive selection by spontaneous
breakdown of symmetry (SBS) of single states in sequences.Comment: 31 page
Fast magnetization switching of Stoner particles: A nonlinear dynamics picture
The magnetization reversal of Stoner particles is investigated from the point
of view of nonlinear dynamics within the Landau-Lifshitz-Gilbert formulation.
The following results are obtained. 1) We clarify that the so-called
Stoner-Wohlfarth (SW) limit becomes exact when damping constant is infinitely
large. Under the limit, the magnetization moves along the steepest energy
descent path. The minimal switching field is the one at which there is only one
stable fixed point in the system. 2) For a given magnetic anisotropy, there is
a critical value for the damping constant, above which the minimal switching
field is the same as that of the SW-limit. 3) We illustrate how fixed points
and their basins change under a field along different directions. This change
explains well why a non-parallel field gives a smaller minimal switching field
and a short switching time. 4) The field of a ballistic magnetization reversal
should be along certain direction window in the presence of energy dissipation.
The width of the window depends on both of the damping constant and the
magnetic anisotropy. The upper and lower bounds of the direction window
increase with the damping constant. The window width oscillates with the
damping constant for a given magnetic anisotropy. It is zero for both zero and
infinite damping. Thus, the perpendicular field configuration widely employed
in the current experiments is not the best one since the damping constant in a
real system is far from zero.Comment: 10 pages, 9 figures. submitted to PR
Least Rattling Feedback from Strong Time-scale Separation
In most interacting many-body systems associated with some "emergent
phenomena," we can identify sub-groups of degrees of freedom that relax on
dramatically different time-scales. Time-scale separation of this kind is
particularly helpful in nonequilibrium systems where only the fast variables
are subjected to external driving; in such a case, it may be shown through
elimination of fast variables that the slow coordinates effectively experience
a thermal bath of spatially-varying temperature. In this work, we investigate
how such a temperature landscape arises according to how the slow variables
affect the character of the driven quasi-steady-state reached by the fast
variables. Brownian motion in the presence of spatial temperature gradients is
known to lead to the accumulation of probability density in low temperature
regions. Here, we focus on the implications of attraction to low effective
temperature for the long-term evolution of slow variables. After quantitatively
deriving the temperature landscape for a general class of overdamped systems
using a path integral technique, we then illustrate in a simple dynamical
system how the attraction to low effective temperature has a fine-tuning effect
on the slow variable, selecting configurations that bring about exceptionally
low force fluctuation in the fast-variable steady-state. We furthermore
demonstrate that a particularly strong effect of this kind can take place when
the slow variable is tuned to bring about orderly, integrable motion in the
fast dynamics that avoids thermalizing energy absorbed from the drive. We thus
point to a potentially general feedback mechanism in multi-time-scale active
systems, that leads to the exploration of slow variable space, as if in search
of fine-tuning for a "least rattling" response in the fast coordinates.Comment: 9 pages, 4 figures + 2 Appendices, RevTe
A comparison of local simulations and reduced models of MRI-induced turbulence
We run mean-field shearing-box numerical simulations with a
temperature-dependent resistivity and compare them to a reduced dynamical
model. Our simulations reveal the co-existence of two quasi-steady states, a
`quiet' state and an `active' turbulent state, confirming the predictions of
the reduced model. The initial conditions determine on which state the
simulation ultimately settles. The active state is strongly influenced by the
geometry of the computational box and the thermal properties of the gas. Cubic
domains support permanent channel flows, bar-shaped domains exhibit eruptive
behaviour, and horizontal slabs give rise to infrequent channels. Meanwhile,
longer cooling time-scales lead to higher saturation amplitudes.Comment: MNRAS accepted, 9 pages, 11 figure
Particles and fields in fluid turbulence
The understanding of fluid turbulence has considerably progressed in recent
years. The application of the methods of statistical mechanics to the
description of the motion of fluid particles, i.e. to the Lagrangian dynamics,
has led to a new quantitative theory of intermittency in turbulent transport.
The first analytical description of anomalous scaling laws in turbulence has
been obtained. The underlying physical mechanism reveals the role of
statistical integrals of motion in non-equilibrium systems. For turbulent
transport, the statistical conservation laws are hidden in the evolution of
groups of fluid particles and arise from the competition between the expansion
of a group and the change of its geometry. By breaking the scale-invariance
symmetry, the statistically conserved quantities lead to the observed anomalous
scaling of transported fields. Lagrangian methods also shed new light on some
practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy
Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics
In dissipative ordinary differential equation systems different time scales
cause anisotropic phase volume contraction along solution trajectories. Model
reduction methods exploit this for simplifying chemical kinetics via a time
scale separation into fast and slow modes. The aim is to approximate the system
dynamics with a dimension-reduced model after eliminating the fast modes by
enslaving them to the slow ones via computation of a slow attracting manifold.
We present a novel method for computing approximations of such manifolds using
trajectory-based optimization. We discuss Riemannian geometry concepts as a
basis for suitable optimization criteria characterizing trajectories near slow
attracting manifolds and thus provide insight into fundamental geometric
properties of multiple time scale chemical kinetics. The optimization criteria
correspond to a suitable mathematical formulation of "minimal relaxation" of
chemical forces along reaction trajectories under given constraints. We present
various geometrically motivated criteria and the results of their application
to three test case reaction mechanisms serving as examples. We demonstrate that
accurate numerical approximations of slow invariant manifolds can be obtained.Comment: 22 pages, 18 figure
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