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 $\Omega^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order $\Omega^{-2}$ for systems obeying detailed balance, where $\Omega$ 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 $\Omega^{-1/2}$ and variance estimates accurate to order $\Omega^{-3/2}$. 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 $l_{\rm mean}$ is used to obtain the density field over the whole simulation box. Mesh cells having density contrast higher than a local cutoff threshold $\delta_{\rm LOC}$ 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 $\Lambda$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