121,883 research outputs found
A two-scale approach to the hydrodynamic limit, part II: local Gibbs behavior
This work is a follow-up on [GOVW]. In that previous work a two-scale
approach was used to prove the logarithmic Sobolev inequality for a system of
spins with fixed mean whose potential is a bounded perturbation of a Gaussian,
and to derive an abstract theorem for the convergence to the hydrodynamic
limit. This strategy was then successfully applied to Kawasaki dynamics. Here
we shall use again this two-scale approach to show that the microscopic
variable in such a model behaves according to a local Gibbs state. As a
consequence, we shall prove the convergence of the microscopic entropy to the
hydrodynamic entropy.Comment: 31 pages, 2nd version. The proof of Theorem 1.15 has been simplifie
Stabilization of the Witt group
Using an idea due to R.Thomason, we define a "homology theory" on the
category of rings which satisfies excision, exactness, homotopy (in the
algebraic sense) and periodicity of order 4. For regular noetherian rings, we
find P. Balmer's higher Witt groups. For more general rings, this homology
isomorphic to the KT-theory of J. Hornbostel, inspired by the work of B.
Williams. For real or complex C*-algebras, we recover - up to 2 torsion -
topological K-theory.Comment: 6 pages ; see also http://www.math.jussieu.fr/~karoubi
Consciousness as a State of Matter
We examine the hypothesis that consciousness can be understood as a state of
matter, "perceptronium", with distinctive information processing abilities. We
explore five basic principles that may distinguish conscious matter from other
physical systems such as solids, liquids and gases: the information,
integration, independence, dynamics and utility principles. If such principles
can identify conscious entities, then they can help solve the quantum
factorization problem: why do conscious observers like us perceive the
particular Hilbert space factorization corresponding to classical space (rather
than Fourier space, say), and more generally, why do we perceive the world
around us as a dynamic hierarchy of objects that are strongly integrated and
relatively independent? Tensor factorization of matrices is found to play a
central role, and our technical results include a theorem about Hamiltonian
separability (defined using Hilbert-Schmidt superoperators) being maximized in
the energy eigenbasis. Our approach generalizes Giulio Tononi's integrated
information framework for neural-network-based consciousness to arbitrary
quantum systems, and we find interesting links to error-correcting codes,
condensed matter criticality, and the Quantum Darwinism program, as well as an
interesting connection between the emergence of consciousness and the emergence
of time.Comment: Replaced to match accepted CSF version; discussion improved, typos
corrected. 36 pages, 15 fig
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