2,082 research outputs found

### Space and time from translation symmetry

We show that the notions of space and time in algebraic quantum field theory
arise from translation symmetry if we assume asymptotic commutativity. We argue
that this construction can be applied to string theory.Comment: 10 pages, Essential changes and additions, in particular, in the
discussion of string field theor

### Extending the definition of entropy to nonequilibrium steady states

We study the nonequilibrium statistical mechanics of a finite classical
system subjected to nongradient forces $\xi$ and maintained at fixed kinetic
energy (Hoover-Evans isokinetic thermostat). We assume that the microscopic
dynamics is sufficiently chaotic (Gallavotti-Cohen chaotic hypothesis) and that
there is a natural nonequilibrium steady state $\rho_\xi$. When $\xi$ is
replaced by $\xi+\delta\xi$ one can compute the change $\delta\rho$ of
$\rho_\xi$ (linear response) and define an entropy change $\delta S$ based on
energy considerations. When $\xi$ is varied around a loop, the total change of
$S$ need not vanish: outside of equilibrium the entropy has curvature. But at
equilibrium (i.e. if $\xi$ is a gradient) we show that the curvature is zero,
and that the entropy $S(\xi+\delta\xi)$ near equilibrium is well defined to
second order in $\delta\xi$.Comment: plain TeX, 10 pagesemacs ded

### Stable resonances and signal propagation in a chaotic network of coupled units

We apply the linear response theory developed in \cite{Ruelle} to analyze how
a periodic signal of weak amplitude, superimposed upon a chaotic background, is
transmitted in a network of non linearly interacting units. We numerically
compute the complex susceptibility and show the existence of specific poles
(stable resonances) corresponding to the response to perturbations transverse
to the attractor. Contrary to the poles of correlation functions they depend on
the pair emitting/receiving units. This dynamic differentiation, induced by non
linearities, exhibits the different ability that units have to transmit a
signal in this network.Comment: 10 pages, 3 figures, to appear in Phys. rev.

### On the susceptibility function of piecewise expanding interval maps

We study the susceptibility function Psi(z) associated to the perturbation
f_t=f+tX of a piecewise expanding interval map f. The analysis is based on a
spectral description of transfer operators. It gives in particular sufficient
conditions which guarantee that Psi(z) is holomorphic in a disc of larger than
one. Although Psi(1) is the formal derivative of the SRB measure of f_t with
respect to t, we present examples satisfying our conditions so that the SRB
measure is not Lipschitz.*We propose a new version of Ruelle's conjectures.* In
v2, we corrected a few minor mistakes and added Conjectures A-B and Remark 4.5.
In v3, we corrected the perturbation (X(f(x)) instead of X(x)), in particular
in the examples from Section 6. As a consequence, Psi(z) has a pole at z=1 for
these examples.Comment: To appear Comm. Math. Phy

### Topics in chaotic dynamics

Various kinematical quantities associated with the statistical properties of
dynamical systems are examined: statistics of the motion, dynamical bases and
Lyapunov exponents. Markov partitons for chaotic systems, without any attempt
at describing ``optimal results''. The Ruelle principle is illustrated via its
relation with the theory of gases. An example of an application predicts the
results of an experiment along the lines of Evans, Cohen, Morriss' work on
viscosity fluctuations. A sequence of mathematically oriented problems
discusses the details of the main abstract ergodic theorems guiding to a proof
of Oseledec's theorem for the Lyapunov exponents and products of random
matricesComment: Plain TeX; compile twice; 30 pages; 140K Keywords: chaos,
nonequilibrium ensembles, Markov partitions, Ruelle principle, Lyapunov
exponents, random matrices, gaussian thermostats, ergodic theory, billiards,
conductivity, gas.

### Characterizing dynamics with covariant Lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete-
and continuous-time dynamical systems is introduced. This allows to address
fundamental questions such as the degree of hyperbolicity, which can be
quantified in terms of the transversality of these intrinsic vectors. For
spatially extended systems, the covariant Lyapunov vectors have localization
properties and spatial Fourier spectra qualitatively different from those
composing the orthonormalized basis obtained in the standard procedure used to
calculate the Lyapunov exponents.Comment: 4 pages, 3 figures, submitted to Physical Review letter

### Phase transitions with four-spin interactions

Using an extended Lee-Yang theorem and GKS correlation inequalities, we
prove, for a class of ferromagnetic multi-spin interactions, that they will
have a phase transition(and spontaneous magnetization) if, and only if, the
external field $h=0$ (and the temperature is low enough). We also show the
absence of phase transitions for some nonferromagnetic interactions. The FKG
inequalities are shown to hold for a larger class of multi-spin interactions

### Linear response formula for piecewise expanding unimodal maps

The average R(t) of a smooth function with respect to the SRB measure of a
smooth one-parameter family f_t of piecewise expanding interval maps is not
always Lipschitz. We prove that if f_t is tangent to the topological class of
f_0, then R(t) is differentiable at zero, and the derivative coincides with the
resummation previously proposed by the first named author of the (a priori
divergent) series given by Ruelle's conjecture.Comment: We added Theorem 7.1 which shows that the horizontality condition is
necessary. The paper "Smooth deformations..." containing Thm 2.8 is now
available on the arxiv; see also Corrigendum arXiv:1205.5468 (to appear
Nonlinearity 2012

### Note on nonequilibrium stationary states and entropy

In transformations between nonequilibrium stationary states, entropy might be
a not well defined concept. It might be analogous to the ``heat content'' in
transformations in equilibrium which is not well defined either, if they are
not isochoric ({\it i.e.} do involve mechanical work). Hence we conjecture that
un a nonequilbrium stationary state the entropy is just a quantity that can be
transferred or created, like heat in equilibrium, but has no physical meaning
as ``entropy content'' as a property of the system.Comment: 4 page

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