593 research outputs found
Geometry of dynamics, Lyapunov exponents and phase transitions
The Hamiltonian dynamics of classical planar Heisenberg model is numerically
investigated in two and three dimensions. By considering the dynamics as a
geodesic flow on a suitable Riemannian manifold, it is possible to analytically
estimate the largest Lyapunov exponent in terms of some curvature fluctuations.
The agreement between numerical and analytical values for Lyapunov exponents is
very good in a wide range of temperatures. Moreover, in the three dimensional
case, in correspondence with the second order phase transition, the curvature
fluctuations exibit a singular behaviour which is reproduced in an abstract
geometric model suggesting that the phase transition might correspond to a
change in the topology of the manifold whose geodesics are the motions of the
system.Comment: REVTeX, 10 pages, 5 PostScript figures, published versio
Weyl's search for a difference between `physical' and `mathematical' automorphisms
During his whole scientific life Hermann Weyl was fascinated by the
interrelation of physical and mathematical theories. From the mid 1920s onward
he reflected also on the typical difference between the two epistemic fields
and tried to identify it by comparing their respective automorphism structures.
In a talk given at the end of the 1940s (ETH, Hs 91a:31) he gave the most
detailed and coherent discussion of his thoughts on this topic. This paper
presents his arguments in the talk and puts it in the context of the later
development of gauge theories.Comment: 30 p
Complex statistics in Hamiltonian barred galaxy models
We use probability density functions (pdfs) of sums of orbit coordinates,
over time intervals of the order of one Hubble time, to distinguish weakly from
strongly chaotic orbits in a barred galaxy model. We find that, in the weakly
chaotic case, quasi-stationary states arise, whose pdfs are well approximated
by -Gaussian functions (with ), while strong chaos is identified by
pdfs which quickly tend to Gaussians (). Typical examples of weakly
chaotic orbits are those that "stick" to islands of ordered motion. Their
presence in rotating galaxy models has been investigated thoroughly in recent
years due of their ability to support galaxy structures for relatively long
time scales. In this paper, we demonstrate, on specific orbits of 2 and 3
degree of freedom barred galaxy models, that the proposed statistical approach
can distinguish weakly from strongly chaotic motion accurately and efficiently,
especially in cases where Lyapunov exponents and other local dynamic indicators
appear to be inconclusive.Comment: 14 pages, 9 figures, submitted for publicatio
Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches
Micro-Electro-Mechanical-Systems are complex structures, often involving
nonlinearites of geometric and multiphysics nature, that are used as sensors
and actuators in countless applications. Starting from full-order
representations, we apply deep learning techniques to generate accurate,
efficient and real-time reduced order models to be used as virtual twin for the
simulation and optimization of higher-level complex systems. We extensively
test the reliability of the proposed procedures on micromirrors, arches and
gyroscopes, also displaying intricate dynamical evolutions like internal
resonances. In particular, we discuss the accuracy of the deep learning
technique and its ability to replicate and converge to the invariant manifolds
predicted using the recently developed direct parametrization approach that
allows extracting the nonlinear normal modes of large finite element models.
Finally, by addressing an electromechanical gyroscope, we show that the
non-intrusive deep learning approach generalizes easily to complex multiphysics
problem
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