37 research outputs found
Expanding direction of the period doubling operator
We prove that the period doubling operator has an expanding direction at the
fixed point. We use the induced operator, a ``Perron-Frobenius type operator'',
to study the linearization of the period doubling operator at its fixed point.
We then use a sequence of linear operators with finite ranks to study this
induced operator. The proof is constructive. One can calculate the expanding
direction and the rate of expansion of the period doubling operator at the
fixed point
The Wigner-Yanase entropy is not subadditive
Wigner and Yanase introduced in 1963 the Wigner-Yanase entropy defined as
minus the skew information of a state with respect to a conserved observable.
They proved that the Wigner-Yanase entropy is a concave function in the state
and conjectured that it is subadditive with respect to the aggregation of
possibly interacting subsystems. While this turned out to be true for the
quantum-mechanical entropy, we negate the conjecture for the Wigner-Yanase
entropy by providing a counter example.Comment: To appear in Journal of Statistical Physic
Theory of Circle Maps and the Problem of One-Dimensional Optical Resonator with a Periodically Moving Wall
We consider the electromagnetic field in a cavity with a periodically
oscillating perfectly reflecting boundary and show that the mathematical theory
of circle maps leads to several physical predictions. Notably, well-known
results in the theory of circle maps (which we review briefly) imply that there
are intervals of parameters where the waves in the cavity get concentrated in
wave packets whose energy grows exponentially. Even if these intervals are
dense for typical motions of the reflecting boundary, in the complement there
is a positive measure set of parameters where the energy remains bounded.Comment: 34 pages LaTeX (revtex) with eps figures, PACS: 02.30.Jr, 42.15.-i,
42.60.Da, 42.65.Y
Quantum Equilibrium and the Origin of Absolute Uncertainty
The quantum formalism is a ``measurement'' formalism--a phenomenological
formalism describing certain macroscopic regularities. We argue that it can be
regarded, and best be understood, as arising from Bohmian mechanics, which is
what emerges from Schr\"odinger's equation for a system of particles when we
merely insist that ``particles'' means particles. While distinctly
non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles
in motion, a motion choreographed by the wave function. We find that a Bohmian
universe, though deterministic, evolves in such a manner that an {\it
appearance} of randomness emerges, precisely as described by the quantum
formalism and given, for example, by ``\rho=|\psis|^2.'' A crucial ingredient
in our analysis of the origin of this randomness is the notion of the effective
wave function of a subsystem, a notion of interest in its own right and of
relevance to any discussion of quantum theory. When the quantum formalism is
regarded as arising in this way, the paradoxes and perplexities so often
associated with (nonrelativistic) quantum theory simply evaporate.Comment: 75 pages. This paper was published a long time ago, but was never
archived. We do so now because it is basic for our recent article
quant-ph/0308038, which can in fact be regarded as an appendix of the earlier
on
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
with Observables at Infinity and States Short Range Correlations in Statistical Mechanics
Abstract. We say that a representation of an algebra, of local observables has short-range correlations if any observable which can be measured outside all bounded sets is a multiple of the identity, and that a state has finite range correlations if the corresponding cyclic representation does. We characterize states with short-range correlations by a cluster property. For classical lattice systems and continuous systems with hard cores, we give a definition of equilibrium state for a specific interaction, based on a local version of the grand canonical prescription; an equilibrium state need not be translation invariant. We show that every equi. librium state has a unique decomposition into equilibrium states with short-range correlations. We use the properties of equilibrium states to prove some negative results about the existence of metastable states. We show that the correlation functions for an equilibrium state satisfy the Kirkwood-Salsburg equations; thus, at low activity, there is only one equilibrium state for a given interaction, temperature, and chemical potential. Finally, we argue heuristically that equilibrium states are invariant under time-evolution. 1
Communications in Physics © by Springer-Verlag 1980 Universal Properties of Maps on an Interval
Abstract. We consider iterates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points. 1