2,144 research outputs found
How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?
The chemical Fokker-Planck equation and the corresponding chemical Langevin
equation are commonly used approximations of the chemical master equation.
These equations are derived from an uncontrolled, second-order truncation of
the Kramers-Moyal expansion of the chemical master equation and hence their
accuracy remains to be clarified. We use the system-size expansion to show that
chemical Fokker-Planck estimates of the mean concentrations and of the variance
of the concentration fluctuations about the mean are accurate to order
for reaction systems which do not obey detailed balance and at
least accurate to order for systems obeying detailed balance,
where is the characteristic size of the system. Hence the chemical
Fokker-Planck equation turns out to be more accurate than the linear-noise
approximation of the chemical master equation (the linear Fokker-Planck
equation) which leads to mean concentration estimates accurate to order
and variance estimates accurate to order . This
higher accuracy is particularly conspicuous for chemical systems realized in
small volumes such as biochemical reactions inside cells. A formula is also
obtained for the approximate size of the relative errors in the concentration
and variance predictions of the chemical Fokker-Planck equation, where the
relative error is defined as the difference between the predictions of the
chemical Fokker-Planck equation and the master equation divided by the
prediction of the master equation. For dimerization and enzyme-catalyzed
reactions, the errors are typically less than few percent even when the
steady-state is characterized by merely few tens of molecules.Comment: 39 pages, 3 figures, accepted for publication in J. Chem. Phy
The Omega Dependence of the Evolution of xi(r)
The evolution of the two-point correlation function, xi(r,z), and the
pairwise velocity dispersion, sigma(r,z), for both the matter and halo
population, in three different cosmological models:
(Omega_M,Omega_Lambda)=(1,0), (0.2,0) and (0.2,0.8) are described. If the
evolution of xi is parameterized by xi(r,z)=(1+z)^{-(3+eps)}xi(r,0), where
xi(r,0)=(r/r_0)^{-gamma}, then eps(mass) ranges from 1.04 +/- 0.09 for (1,0) to
0.18 +/- 0.12 for (0.2,0), as measured by the evolution of at 1 Mpc (from z ~ 5
to the present epoch). For halos, eps depends on their mean overdensity. Halos
with a mean overdensity of about 2000 were used to compute the halo two-point
correlation function tested with two different group finding algorithms: the
friends of friends and the spherical overdensity algorithm. It is certainly
believed that the rate of growth of this xihh will give a good estimate of the
evolution of the galaxy two-point correlation function, at least from z ~ 1 to
the present epoch. The values we get for eps(halos) range from 1.54 for (1,0)
to -0.36 for (0.2,0), as measured by the evolution of xi(halos) from z ~ 1.0 to
the present epoch. These values could be used to constrain the cosmological
scenario. The evolution of the pairwise velocity dispersion for the mass and
halo distribution is measured and compared with the evolution predicted by the
Cosmic Virial Theorem (CVT). According to the CVT, sigma(r,z)^2 ~ G Q rho(z)
r^2 xi(r,z) or sigma proportional to (1+z)^{-eps/2}. The values of eps measured
from our simulated velocities differ from those given by the evolution of xi
and the CVT, keeping gamma and Q constant: eps(CVT) = 1.78 +/- 0.13 for (1,0)
or 1.40 +/- 0.28 for (0.2,0).Comment: Accepted for publication in the ApJ. Also available at
http://manaslu.astro.utoronto.ca/~carlberg/cnoc/xiev/xi_evo.ps.g
Relaxation Phenomena in a System of Two Harmonic Oscillators
We study the process by which quantum correlations are created when an
interaction Hamiltonian is repeatedly applied to a system of two harmonic
oscillators for some characteristic time interval. We show that, for the case
where the oscillator frequencies are equal, the initial Maxwell-Boltzmann
distributions of the uncoupled parts evolve to a new equilibrium
Maxwell-Boltzmann distribution through a series of transient Maxwell-Boltzmann
distributions. Further, we discuss why the equilibrium reached when the two
oscillator frequencies are unequal, is not a thermal one. All the calculations
are exact and the results are obtained through an iterative process, without
using perturbation theory.Comment: 22 pages, 6 Figures, Added contents, to appear in PR
A New Halo Finding Method for N-Body Simulations
We have developed a new halo finding method, Physically Self-Bound (PSB)
group finding algorithm, which can efficiently identify halos located even at
crowded regions. This method combines two physical criteria such as the tidal
radius of a halo and the total energy of each particle to find member
particles. Two hierarchical meshes are used to increase the speed and the power
of halo identification in the parallel computing environments. First, a coarse
mesh with cell size equal to the mean particle separation is
used to obtain the density field over the whole simulation box. Mesh cells
having density contrast higher than a local cutoff threshold
are extracted and linked together for those adjacent to each other. This
produces local-cell groups. Second, a finer mesh is used to obtain density
field within each local-cell group and to identify halos. If a density shell
contains only one density peak, its particles are assigned to the density peak.
But in the case of a density shell surrounding at least two density peaks, we
use both the tidal radii of halo candidates enclosed by the shell and the total
energy criterion to find physically bound particles with respect to each halo.
Similar to DENMAX and HOP, the \hfind method can efficiently identify small
halos embedded in a large halo, while the FoF and the SO do not resolve such
small halos. We apply our new halo finding method to a 1-Giga particle
simulation of the CDM model and compare the resulting mass function
with those of previous studies. The abundance of physically self-bound halos is
larger at the low mass scale and smaller at the high mass scale than proposed
by the Jenkins et al. (2001) who used the FoF and SO methods. (abridged)Comment: 10 pages, 8 figs, submitted to Ap
Holomorphic transforms with application to affine processes
In a rather general setting of It\^o-L\'evy processes we study a class of
transforms (Fourier for example) of the state variable of a process which are
holomorphic in some disc around time zero in the complex plane. We show that
such transforms are related to a system of analytic vectors for the generator
of the process, and we state conditions which allow for holomorphic extension
of these transforms into a strip which contains the positive real axis. Based
on these extensions we develop a functional series expansion of these
transforms in terms of the constituents of the generator. As application, we
show that for multidimensional affine It\^o-L\'evy processes with state
dependent jump part the Fourier transform is holomorphic in a time strip under
some stationarity conditions, and give log-affine series representations for
the transform.Comment: 30 page
Field theoretic formulation of a mode-coupling equation for colloids
The only available quantitative description of the slowing down of the
dynamics upon approaching the glass transition has been, so far, the
mode-coupling theory, developed in the 80's by G\"otze and collaborators. The
standard derivation of this theory does not result from a systematic expansion.
We present a field theoretic formulation that arrives at very similar
mode-coupling equation but which is based on a variational principle and on a
controlled expansion in a small dimensioneless parameter. Our approach applies
to such physical systems as colloids interacting via a mildly repulsive
potential. It can in principle, with moderate efforts, be extended to higher
orders and to multipoint correlation functions
Lens magnification by CL0024+1654 in the U and R band
[ABRIDGED] We estimate the total mass distribution of the galaxy cluster
CL0024+1654 from the measured source depletion due to lens magnification in the
R band. Within a radius of 0.54Mpc/h, a total projected mass of
(8.1+/-3.2)*10^14 M_sol/h (EdS) is measured, which corresponds to a mass-
to-light ratio of M/L(B)=470+/-180. We compute the luminosity function of
CL0024+1654 in order to estimate contamination of the background source counts
from cluster galaxies. Three different magnification-based reconstruction
methods are employed using both local and non-local techniques. We have
modified the standard single power-law slope number count theory to incorporate
a break and applied this to our observations. Fitting analytical magnification
profiles of different cluster models to the observed number counts, we find
that the cluster is best described either by a NFW model with scale radius
r_s=334+/-191 kpc/h and normalisation kappa_s=0.23+/-0.08 or a power-law
profile with slope xi=0.61+/-0.11, central surface mass density
kappa_0=1.52+/-0.20 and assuming a core radius of r_core=35 kpc/h. The NFW
model predicts that the cumulative projected mass contained within a radius R
scales as M(<R)=2.9*10^14*(R/1')^[1.3-0.5lg (R/1')] M_sol/h. Finally, we have
exploited the fact that flux magnification effectively enables us to probe
deeper than the physical limiting magnitude of our observations in searching
for a change of slope in the U band number counts. We rule out both a total
flattening of the counts with a break up to U_AB<=26.6 and a change of slope,
reported by some studies, from dlog N/dm=0.4->0.15 up to U_AB<=26.4 with 95%
confidence.Comment: 19 pages, 12 figures, submitted to A&A. New version includes more
robust U band break analysis and contamination estimates, plus new plot
Coarse graining of master equations with fast and slow states
We propose a general method for simplifying master equations by eliminating
from the description rapidly evolving states. The physical recipe we impose is
the suppression of these states and a renormalization of the rates of all the
surviving states. In some cases, this decimation procedure can be analytically
carried out and is consistent with other analytical approaches, like in the
problem of the random walk in a double-well potential. We discuss the
application of our method to nontrivial examples: diffusion in a lattice with
defects and a model of an enzymatic reaction outside the steady state regime.Comment: 9 pages, 9 figures, final version (new subsection and many minor
improvements
Local in time master equations with memory effects: Applicability and interpretation
Non-Markovian local in time master equations give a relatively simple way to
describe the dynamics of open quantum systems with memory effects. Despite
their simple form, there are still many misunderstandings related to the
physical applicability and interpretation of these equations. Here we clarify
these issues both in the case of quantum and classical master equations. We
further introduce the concept of a classical non-Markov chain signified through
negative jump rates in the chain configuration.Comment: Special issue on loss of coherence and memory effects in quantum
dynamics, J. Phys. B., to appea
Information Flow through a Chaotic Channel: Prediction and Postdiction at Finite Resolution
We reconsider the persistence of information under the dynamics of the
logistic map in order to discuss communication through a nonlinear channel
where the sender can set the initial state of the system with finite
resolution, and the recipient measures it with the same accuracy. We separate
out the contributions of global phase space shrinkage and local phase space
contraction and expansion to the uncertainty in predicting and postdicting the
state of the system. Thus, we determine how the amplification parameter, the
time lag, and the resolution influence the possibility for communication. A
novel representation for real numbers is introduced that allows for a
visualization of the flow of information between scales.Comment: 14 pages, 13 figure
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