328 research outputs found
(Quantum) Space-Time as a Statistical Geometry of Lumps in Random Networks
In the following we undertake to describe how macroscopic space-time (or
rather, a microscopic protoform of it) is supposed to emerge as a
superstructure of a web of lumps in a stochastic discrete network structure. As
in preceding work (mentioned below), our analysis is based on the working
philosophy that both physics and the corresponding mathematics have to be
genuinely discrete on the primordial (Planck scale) level. This strategy is
concretely implemented in the form of \tit{cellular networks} and \tit{random
graphs}. One of our main themes is the development of the concept of
\tit{physical (proto)points} or \tit{lumps} as densely entangled subcomplexes
of the network and their respective web, establishing something like
\tit{(proto)causality}. It may perhaps be said that certain parts of our
programme are realisations of some early ideas of Menger and more recent ones
sketched by Smolin a couple of years ago. We briefly indicate how this
\tit{two-story-concept} of \tit{quantum} space-time can be used to encode the
(at least in our view) existing non-local aspects of quantum theory without
violating macroscopic space-time causality.Comment: 35 pages, Latex, under consideration by CQ
Dimension Theory of Graphs and Networks
Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face in this enterprise is to
find the discrete protoforms of the building blocks of continuum physics and
mathematics. A core concept is the notion of dimension. In the following we
develop such a notion for irregular structures like (large) graphs and networks
and derive a number of its properties. Among other things we show its stability
under a wide class of perturbations which is important if one has 'dimensional
phase transitions' in mind. Furthermore we systematically construct graphs with
almost arbitrary 'fractal dimension' which may be of some use in the context of
'dimensional renormalization' or statistical mechanics on irregular sets.Comment: 20 pages, 7 figures, LaTex2e, uses amsmath, amsfonts, amssymb,
latexsym, epsfi
(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces
We develop a kind of pregeometry consisting of a web of overlapping fuzzy
lumps which interact with each other. The individual lumps are understood as
certain closely entangled subgraphs (cliques) in a dynamically evolving network
which, in a certain approximation, can be visualized as a time-dependent random
graph. This strand of ideas is merged with another one, deriving from ideas,
developed some time ago by Menger et al, that is, the concept of probabilistic-
or random metric spaces, representing a natural extension of the metrical
continuum into a more microscopic regime. It is our general goal to find a
better adapted geometric environment for the description of microphysics. In
this sense one may it also view as a dynamical randomisation of the causal-set
framework developed by e.g. Sorkin et al. In doing this we incorporate, as a
perhaps new aspect, various concepts from fuzzy set theory.Comment: 25 pages, Latex, no figures, some references added, some minor
changes added relating to previous wor
Convolution of multifractals and the local magnetization in a random field Ising chain
The local magnetization in the one-dimensional random-field Ising model is
essentially the sum of two effective fields with multifractal probability
measure. The probability measure of the local magnetization is thus the
convolution of two multifractals. In this paper we prove relations between the
multifractal properties of two measures and the multifractal properties of
their convolution. The pointwise dimension at the boundary of the support of
the convolution is the sum of the pointwise dimensions at the boundary of the
support of the convoluted measures and the generalized box dimensions of the
convolution are bounded from above by the sum of the generalized box dimensions
of the convoluted measures. The generalized box dimensions of the convolution
of Cantor sets with weights can be calculated analytically for certain
parameter ranges and illustrate effects we also encounter in the case of the
measure of the local magnetization. Returning to the study of this measure we
apply the general inequalities and present numerical approximations of the
D_q-spectrum. For the first time we are able to obtain results on multifractal
properties of a physical quantity in the one-dimensional random-field Ising
model which in principle could be measured experimentally. The numerically
generated probability densities for the local magnetization show impressively
the gradual transition from a monomodal to a bimodal distribution for growing
random field strength h.Comment: An error in figure 1 was corrected, small additions were made to the
introduction and the conclusions, some typos were corrected, 24 pages,
LaTeX2e, 9 figure
Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
As an outgrowth of our investigation of non-regular spaces within the context
of quantum gravity and non-commutative geometry, we develop a graph Hilbert
space framework on arbitrary (infinite) graphs and use it to study spectral
properties of graph-Laplacians and graph-Dirac-operators. We define a spectral
triplet sharing most of the properties of what Connes calls a spectral triple.
With the help of this scheme we derive an explicit expression for the
Connes-distance function on general directed or undirected graphs. We derive a
series of apriori estimates and calculate it for a variety of examples of
graphs. As a possibly interesting aside, we show that the natural setting of
approaching such problems may be the framework of (non-)linear programming or
optimization. We compare our results (arrived at within our particular
framework) with the results of other authors and show that the seeming
differences depend on the use of different graph-geometries and/or Dirac
operators.Comment: 27 pages, Latex, comlementary to an earlier paper, general treatment
of directed and undirected graphs, in section 4 a series of general results
and estimates concerning the Connes Distance on graphs together with examples
and numerical estimate
Orbits and phase transitions in the multifractal spectrum
We consider the one dimensional classical Ising model in a symmetric
dichotomous random field. The problem is reduced to a random iterated function
system for an effective field. The D_q-spectrum of the invariant measure of
this effective field exhibits a sharp drop of all D_q with q < 0 at some
critical strength of the random field. We introduce the concept of orbits which
naturally group the points of the support of the invariant measure. We then
show that the pointwise dimension at all points of an orbit has the same value
and calculate it for a class of periodic orbits and their so-called offshoots
as well as for generic orbits in the non-overlapping case. The sharp drop in
the D_q-spectrum is analytically explained by a drastic change of the scaling
properties of the measure near the points of a certain periodic orbit at a
critical strength of the random field which is explicitly given. A similar
drastic change near the points of a special family of periodic orbits explains
a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a
decisive role in this mechanism is played by a specific offshoot. We
furthermore give rigorous upper and/or lower bounds on all D_q in a wide
parameter range. In most cases the numerically obtained D_q coincide with
either the upper or the lower bound. The results in this paper are relevant for
the understanding of random iterated function systems in the case of moderate
overlap in which periodic orbits with weak singularity can play a decisive
role.Comment: The article has been completely rewritten; the title has changed; a
section about the typical pointwise dimension as well as several references
and remarks about more general systems have been added; to appear in J. Phys.
A; 25 pages, 11 figures, LaTeX2
X Her and TX Psc: Two cases of ISM interaction with stellar winds observed by Herschel
The asymptotic giant branch (AGB) stars X Her and TX Psc have been imaged at
70 and 160 microns with the PACS instrument onboard the Herschel satellite, as
part of the large MESS (Mass loss of Evolved StarS) Guaranteed Time Key
Program. The images reveal an axisymmetric extended structure with its axis
oriented along the space motion of the stars. This extended structure is very
likely to be shaped by the interaction of the wind ejected by the AGB star with
the surrounding interstellar medium (ISM). As predicted by numerical
simulations, the detailed structure of the wind-ISM interface depends upon the
relative velocity between star+wind and the ISM, which is large for these two
stars (108 and 55 km/s for X Her and TX Psc, respectively). In both cases,
there is a compact blob upstream whose origin is not fully elucidated, but that
could be the signature of some instability in the wind-ISM shock. Deconvolved
images of X Her and TX Psc reveal several discrete structures along the
outermost filaments, which could be Kelvin-Helmholtz vortices. Finally, TX Psc
is surrounded by an almost circular ring (the signature of the termination
shock?) that contrasts with the outer, more structured filaments. A similar
inner circular structure seems to be present in X Her as well, albeit less
clearly.Comment: 11 pages, Astronomy & Astrophysics, in pres
Production of dust by massive stars at high redshift
The large amounts of dust detected in sub-millimeter galaxies and quasars at
high redshift pose a challenge to galaxy formation models and theories of
cosmic dust formation. At z > 6 only stars of relatively high mass (> 3 Msun)
are sufficiently short-lived to be potential stellar sources of dust. This
review is devoted to identifying and quantifying the most important stellar
channels of rapid dust formation. We ascertain the dust production efficiency
of stars in the mass range 3-40 Msun using both observed and theoretical dust
yields of evolved massive stars and supernovae (SNe) and provide analytical
expressions for the dust production efficiencies in various scenarios. We also
address the strong sensitivity of the total dust productivity to the initial
mass function. From simple considerations, we find that, in the early Universe,
high-mass (> 3 Msun) asymptotic giant branch stars can only be dominant dust
producers if SNe generate <~ 3 x 10^-3 Msun of dust whereas SNe prevail if they
are more efficient. We address the challenges in inferring dust masses and
star-formation rates from observations of high-redshift galaxies. We conclude
that significant SN dust production at high redshift is likely required to
reproduce current dust mass estimates, possibly coupled with rapid dust grain
growth in the interstellar medium.Comment: 72 pages, 9 figures, 5 tables; to be published in The Astronomy and
Astrophysics Revie
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