49 research outputs found
Brussels-Austin Nonequilibrium Statistical Mechanics in the Early Years: Similarity Transformations between Deterministic and Probabilistic Descriptions
The fundamental problem on which Ilya Prigogine and the Brussels-Austin Group
have focused can be stated briefly as follows. Our observations indicate that
there is an arrow of time in our experience of the world (e.g., decay of
unstable radioactive atoms like Uranium, or the mixing of cream in coffee).
Most of the fundamental equations of physics are time reversible, however,
presenting an apparent conflict between our theoretical descriptions and
experimental observations. Many have thought that the observed arrow of time
was either an artifact of our observations or due to very special initial
conditions. An alternative approach, followed by the Brussels-Austin Group, is
to consider the observed direction of time to be a basics physical phenomenon
and to develop a mathematical formalism that can describe this direction as
being due to the dynamics of physical systems. In part I of this essay, I
review and assess an attempt to carry out an approach that received much of
their attention from the early 1970s to the mid 1980s. In part II, I will
discuss their more recent approach using rigged Hilbert spaces.Comment: 22 pages, Part I of two parts; updated institutional affiliatio
Noise induced dissipation in Lebesgue-measure preserving maps on dimensional torus
We consider dissipative systems resulting from the Gaussian and
-stable noise perturbations of measure-preserving maps on the
dimensional torus. We study the dissipation time scale and its physical
implications as the noise level \vep vanishes.
We show that nonergodic maps give rise to an O(1/\vep) dissipation time
whereas ergodic toral automorphisms, including cat maps and their
-dimensional generalizations, have an O(\ln{(1/\vep)}) dissipation time
with a constant related to the minimal, {\em dimensionally averaged entropy}
among the automorphism's irreducible blocks. Our approach reduces the
calculation of the dissipation time to a nonlinear, arithmetic optimization
problem which is solved asymptotically by means of some fundamental theorems in
theories of convexity, Diophantine approximation and arithmetic progression. We
show that the same asymptotic can be reproduced by degenerate noises as well as
mere coarse-graining. We also discuss the implication of the dissipation time
in kinematic dynamo.Comment: The research is supported in part by the grant from U.S. National
Science Foundation, DMS-9971322 and Lech Wolowsk
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
Discovering Causal Relations and Equations from Data
Physics is a field of science that has traditionally used the scientific
method to answer questions about why natural phenomena occur and to make
testable models that explain the phenomena. Discovering equations, laws and
principles that are invariant, robust and causal explanations of the world has
been fundamental in physical sciences throughout the centuries. Discoveries
emerge from observing the world and, when possible, performing interventional
studies in the system under study. With the advent of big data and the use of
data-driven methods, causal and equation discovery fields have grown and made
progress in computer science, physics, statistics, philosophy, and many applied
fields. All these domains are intertwined and can be used to discover causal
relations, physical laws, and equations from observational data. This paper
reviews the concepts, methods, and relevant works on causal and equation
discovery in the broad field of Physics and outlines the most important
challenges and promising future lines of research. We also provide a taxonomy
for observational causal and equation discovery, point out connections, and
showcase a complete set of case studies in Earth and climate sciences, fluid
dynamics and mechanics, and the neurosciences. This review demonstrates that
discovering fundamental laws and causal relations by observing natural
phenomena is being revolutionised with the efficient exploitation of
observational data, modern machine learning algorithms and the interaction with
domain knowledge. Exciting times are ahead with many challenges and
opportunities to improve our understanding of complex systems.Comment: 137 page