1,555 research outputs found
Field-theoretic methods
Many complex systems are characterized by intriguing spatio-temporal
structures. Their mathematical description relies on the analysis of
appropriate correlation functions. Functional integral techniques provide a
unifying formalism that facilitates the computation of such correlation
functions and moments, and furthermore allows a systematic development of
perturbation expansions and other useful approximative schemes. It is explained
how nonlinear stochastic processes may be mapped onto exponential probability
distributions, whose weights are determined by continuum field theory actions.
Such mappings are madeexplicit for (1) stochastic interacting particle systems
whose kinetics is defined through a microscopic master equation; and (2)
nonlinear Langevin stochastic differential equations which provide a mesoscopic
description wherein a separation of time scales between the relevant degrees of
freedom and background statistical noise is assumed. Several well-studied
examples are introduced to illustrate the general methodology.Comment: Article for the Encyclopedia of Complexity and System Science, B.
Meyers (Ed.), Springer-Verlag Berlin, 200
Space-resolved dynamics of a tracer in a disordered solid
The dynamics of a tracer particle in a glassy matrix of obstacles displays
slow complex transport as the free volume approaches a critical value and the
void space falls apart. We investigate the emerging subdiffusive motion of the
test particle by extensive molecular dynamics simulations and characterize the
spatio-temporal transport in terms of two-time correlation functions, including
the time-dependent diffusion coefficient as well as the wavenumber-dependent
intermediate scattering function. We rationalize our findings within the
framework of critical phenomena and compare our data to a dynamic scaling
theory.Comment: 10 pages, 7 figures, submitted to Journal of Non-Crystalline Solid
Spacetime Approach to Phase Transitions
In these notes, the application of Feynman's sum-over-paths approach to
thermal phase transitions is discussed. The paradigm of such a spacetime
approach to critical phenomena is provided by the high-temperature expansion of
spin models. This expansion, known as the hopping expansion in the context of
lattice field theory, yields a geometric description of the phase transition in
these models, with the thermal critical exponents being determined by the
fractal structure of the high-temperature graphs. The graphs percolate at the
thermal critical point and can be studied using purely geometrical observables
known from percolation theory. Besides the phase transition in spin models and
in the closely related theory, other transitions discussed from this
perspective include Bose-Einstein condensation, and the transitions in the
Higgs model and the pure U(1) gauge theory.Comment: 59 pages, 18 figures. Write-up of Ising Lectures presented at the
National Academy of Sciences, Lviv, Ukraine, 2004. 2nd version: corrected
typo
25 Years of Self-Organized Criticality: Solar and Astrophysics
Shortly after the seminal paper {\sl "Self-Organized Criticality: An
explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has
been applied to solar physics, in {\sl "Avalanches and the Distribution of
Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring
cross-fertilization from complexity theory to solar and astrophysics took
place, where the SOC concept was initially applied to solar flares, stellar
flares, and magnetospheric substorms, and later extended to the radiation belt,
the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar
glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and
boson clouds. The application of SOC concepts has been performed by numerical
cellular automaton simulations, by analytical calculations of statistical
(powerlaw-like) distributions based on physical scaling laws, and by
observational tests of theoretically predicted size distributions and waiting
time distributions. Attempts have been undertaken to import physical models
into the numerical SOC toy models, such as the discretization of
magneto-hydrodynamics (MHD) processes. The novel applications stimulated also
vigorous debates about the discrimination between SOC models, SOC-like, and
non-SOC processes, such as phase transitions, turbulence, random-walk
diffusion, percolation, branching processes, network theory, chaos theory,
fractality, multi-scale, and other complexity phenomena. We review SOC studies
from the last 25 years and highlight new trends, open questions, and future
challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized
Criticality and Turbulence" (2012, 2013, Bern, Switzerland
Statistical Physics of Rupture in Heterogeneous Media
The damage and fracture of materials are technologically of enormous interest
due to their economic and human cost. They cover a wide range of phenomena like
e.g. cracking of glass, aging of concrete, the failure of fiber networks in the
formation of paper and the breaking of a metal bar subject to an external load.
Failure of composite systems is of utmost importance in naval, aeronautics and
space industry. By the term composite, we refer to materials with heterogeneous
microscopic structures and also to assemblages of macroscopic elements forming
a super-structure. Chemical and nuclear plants suffer from cracking due to
corrosion either of chemical or radioactive origin, aided by thermal and/or
mechanical stress. Despite the large amount of experimental data and the
considerable effort that has been undertaken by material scientists, many
questions about fracture have not been answered yet. There is no comprehensive
understanding of rupture phenomena but only a partial classification in
restricted and relatively simple situations. This lack of fundamental
understanding is indeed reflected in the absence of reliable prediction methods
for rupture, based on a suitable monitoring of the stressed system. Not only is
there a lack of non-empirical understanding of the reliability of a system, but
also the empirical laws themselves have often limited value. The difficulties
stem from the complex interplay between heterogeneities and modes of damage and
the possible existence of a hierarchy of characteristic scales (static and
dynamic).
The paper presents a review of recent efforts from the statistical physics
community to address these points.Comment: Enlarged review and updated references, 21 pages with 2 figure
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