1,548 research outputs found
Relating Theories via Renormalization
The renormalization method is specifically aimed at connecting theories
describing physical processes at different length scales and thereby connecting
different theories in the physical sciences.
The renormalization method used today is the outgrowth of one hundred and
fifty years of scientific study of thermal physics and phase transitions.
Different phases of matter show qualitatively different behavior separated by
abrupt phase transitions. These qualitative differences seem to be present in
experimentally observed condensed-matter systems. However, the "extended
singularity theorem" in statistical mechanics shows that sharp changes can only
occur in infinitely large systems. Abrupt changes from one phase to another are
signaled by fluctuations that show correlation over infinitely long distances,
and are measured by correlation functions that show algebraic decay as well as
various kinds of singularities and infinities in thermodynamic derivatives and
in measured system parameters.
Renormalization methods were first developed in field theory to get around
difficulties caused by apparent divergences at both small and large scales.
The renormalization (semi-)group theory of phase transitions was put together
by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality
developed earlier in the context of phase transitions and of couplings
dependent upon spatial scale coming from field theory. Correlations among
regions with fluctuations in their order underlie renormalization ideas.
Wilson's theory is the first approach to phase transitions to agree with the
extended singularity theorem.
Some of the history of the study of these correlations and singularities is
recounted, along with the history of renormalization and related concepts of
scaling and universality. Applications are summarized.Comment: This note is partially a summary of a talk given at the workshop
"Part and Whole" in Leiden during the period March 22-26, 201
MLAPM - a C code for cosmological simulations
We present a computer code written in C that is designed to simulate
structure formation from collisionless matter. The code is purely grid-based
and uses a recursively refined Cartesian grid to solve Poisson's equation for
the potential, rather than obtaining the potential from a Green's function.
Refinements can have arbitrary shapes and in practice closely follow the
complex morphology of the density field that evolves. The timestep shortens by
a factor two with each successive refinement. It is argued that an appropriate
choice of softening length is of great importance and that the softening should
be at all points an appropriate multiple of the local inter-particle
separation. Unlike tree and P3M codes, multigrid codes automatically satisfy
this requirement. We show that at early times and low densities in cosmological
simulations, the softening needs to be significantly smaller relative to the
inter-particle separation than in virialized regions. Tests of the ability of
the code's Poisson solver to recover the gravitational fields of both
virialized halos and Zel'dovich waves are presented, as are tests of the code's
ability to reproduce analytic solutions for plane-wave evolution. The times
required to conduct a LCDM cosmological simulation for various configurations
are compared with the times required to complete the same simulation with the
ART, AP3M and GADGET codes. The power spectra, halo mass functions and
halo-halo correlation functions of simulations conducted with different codes
are compared.Comment: 20 pages, 20 figures, MNRAS in press, the code can be downloaded at
http://www-thphys.physics.ox.ac.uk/users/MLAPM
Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
A statistical mechanics argument relating partition functions to martingales
is used to get a condition under which random geometric processes can describe
interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs
to satisfy this condition leads to some natural processes, which we study in
this note. We give examples of such multiple SLEs and discuss how a choice of
conformal block is related to geometric configuration of the interfaces and
what is the physical meaning of mixed conformal blocks. We illustrate the
general ideas on concrete computations, with applications to percolation and
the Ising model.Comment: 40 pages, 6 figures. V2: well, it looks better with the addresse
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