15,922 research outputs found
An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns
This paper presents an introduction to phase transitions and critical
phenomena on the one hand, and nonequilibrium patterns on the other, using the
Ginzburg-Landau theory as a unified language. In the first part, mean-field
theory is presented, for both statics and dynamics, and its validity tested
self-consistently. As is well known, the mean-field approximation breaks down
below four spatial dimensions, where it can be replaced by a scaling
phenomenology. The Ginzburg-Landau formalism can then be used to justify the
phenomenological theory using the renormalization group, which elucidates the
physical and mathematical mechanism for universality. In the second part of the
paper it is shown how near pattern forming linear instabilities of dynamical
systems, a formally similar Ginzburg-Landau theory can be derived for
nonequilibrium macroscopic phenomena. The real and complex Ginzburg-Landau
equations thus obtained yield nontrivial solutions of the original dynamical
system, valid near the linear instability. Examples of such solutions are plane
waves, defects such as dislocations or spirals, and states of temporal or
spatiotemporal (extensive) chaos
Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High-Resolution Simulations
Climate projections continue to be marred by large uncertainties, which
originate in processes that need to be parameterized, such as clouds,
convection, and ecosystems. But rapid progress is now within reach. New
computational tools and methods from data assimilation and machine learning
make it possible to integrate global observations and local high-resolution
simulations in an Earth system model (ESM) that systematically learns from
both. Here we propose a blueprint for such an ESM. We outline how
parameterization schemes can learn from global observations and targeted
high-resolution simulations, for example, of clouds and convection, through
matching low-order statistics between ESMs, observations, and high-resolution
simulations. We illustrate learning algorithms for ESMs with a simple dynamical
system that shares characteristics of the climate system; and we discuss the
opportunities the proposed framework presents and the challenges that remain to
realize it.Comment: 32 pages, 3 figure
Product Reservoir Computing: Time-Series Computation with Multiplicative Neurons
Echo state networks (ESN), a type of reservoir computing (RC) architecture,
are efficient and accurate artificial neural systems for time series processing
and learning. An ESN consists of a core of recurrent neural networks, called a
reservoir, with a small number of tunable parameters to generate a
high-dimensional representation of an input, and a readout layer which is
easily trained using regression to produce a desired output from the reservoir
states. Certain computational tasks involve real-time calculation of high-order
time correlations, which requires nonlinear transformation either in the
reservoir or the readout layer. Traditional ESN employs a reservoir with
sigmoid or tanh function neurons. In contrast, some types of biological neurons
obey response curves that can be described as a product unit rather than a sum
and threshold. Inspired by this class of neurons, we introduce a RC
architecture with a reservoir of product nodes for time series computation. We
find that the product RC shows many properties of standard ESN such as
short-term memory and nonlinear capacity. On standard benchmarks for chaotic
prediction tasks, the product RC maintains the performance of a standard
nonlinear ESN while being more amenable to mathematical analysis. Our study
provides evidence that such networks are powerful in highly nonlinear tasks
owing to high-order statistics generated by the recurrent product node
reservoir
Chaotic dynamics of three-dimensional H\'enon maps that originate from a homoclinic bifurcation
We study bifurcations of a three-dimensional diffeomorphism, , that has
a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers
(\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma), where
and . We show that in a
three-parameter family, g_{\eps}, of diffeomorphisms close to , there
exist infinitely many open regions near \eps =0 where the corresponding
normal form of the first return map to a neighborhood of a homoclinic point is
a three-dimensional H\'enon-like map. This map possesses, in some parameter
regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that
this homoclinic bifurcation leads to a strange attractor. We also discuss the
place that these three-dimensional H\'enon maps occupy in the class of
quadratic volume-preserving diffeomorphisms.Comment: laTeX, 25 pages, 6 eps figure
Lorenz, G\"{o}del and Penrose: New perspectives on determinism and causality in fundamental physics
Despite being known for his pioneering work on chaotic unpredictability, the
key discovery at the core of meteorologist Ed Lorenz's work is the link between
space-time calculus and state-space fractal geometry. Indeed, properties of
Lorenz's fractal invariant set relate space-time calculus to deep areas of
mathematics such as G\"{o}del's Incompleteness Theorem. These properties,
combined with some recent developments in theoretical and observational
cosmology, motivate what is referred to as the `cosmological invariant set
postulate': that the universe can be considered a deterministic dynamical
system evolving on a causal measure-zero fractal invariant set in its
state space. Symbolic representations of are constructed explicitly based
on permutation representations of quaternions. The resulting `invariant set
theory' provides some new perspectives on determinism and causality in
fundamental physics. For example, whilst the cosmological invariant set appears
to have a rich enough structure to allow a description of quantum probability,
its measure-zero character ensures it is sparse enough to prevent invariant set
theory being constrained by the Bell inequality (consistent with a partial
violation of the so-called measurement independence postulate). The primacy of
geometry as embodied in the proposed theory extends the principles underpinning
general relativity. As a result, the physical basis for contemporary programmes
which apply standard field quantisation to some putative gravitational
lagrangian is questioned. Consistent with Penrose's suggestion of a
deterministic but non-computable theory of fundamental physics, a
`gravitational theory of the quantum' is proposed based on the geometry of
, with potential observational consequences for the dark universe.Comment: This manuscript has been accepted for publication in Contemporary
Physics and is based on the author's 9th Dennis Sciama Lecture, given in
Oxford and Triest
- …