Large deviations of Rouse polymer chain: first passage problem

The purpose of this paper is to investigate several analytical methods of solving first passage (FP) problem for the Rouse model, a simplest model of a polymer chain. We show that this problem has to be treated as a multi-dimensional Kramers' problem, which presents rich and unexpected behavior. We first perform direct and forward-flux sampling (FFS) simulations, and measure the mean first-passage time $\tau(z)$ for the free end to reach a certain distance $z$ away from the origin. The results show that the mean FP time is getting faster if the Rouse chain is represented by more beads. Two scaling regimes of $\tau(z)$ are observed, with transition between them varying as a function of chain length. We use these simulations results to test two theoretical approaches. One is a well known asymptotic theory valid in the limit of zero temperature. We show that this limit corresponds to fully extended chain when each chain segment is stretched, which is not particularly realistic. A new theory based on the well known Freidlin-Wentzell theory is proposed, where dynamics is projected onto the minimal action path. The new theory predicts both scaling regimes correctly, but fails to get the correct numerical prefactor in the first regime. Combining our theory with the FFS simulations lead us to a simple analytical expression valid for all extensions and chain lengths. One of the applications of polymer FP problem occurs in the context of branched polymer rheology. In this paper, we consider the arm-retraction mechanism in the tube model, which maps exactly on the model we have solved. The results are compared to the Milner-McLeish theory without constraint release, which is found to overestimate FP time by a factor of 10 or more.

Non-competitive markets

Among all the paradigms in economic theory, the theoretical predictions of oligopoly were the first to be examined in the laboratory. In this chapter, instead of surveying all the experiments with few sellers, we adopt a narrower definition of the term “oligopoly”, and focus on the experiments that were directly inspired by the basic oligopolistic models of Cournot, Bertrand, Hotelling, Stackelberg, and some extensions. Most of the experiments we consider in this chapter have been run in the last three decades. This literature can be considered as a new wave of experimental works aiming at representing basic oligopolistic markets and testing their properties. The chapter is divided into independent sections referring to different parts of the oligopolistic theory, including both monopoly as well as a number of extensions of the basic models, which have been chosen with the aim of providing a representative list of the relevant experimental findings.

Toeplitz operators on Dirichlet-Besov spaces

We study Toeplitz operators on the Besov spaces in the case of the open unit disk. We prove that a symbol satisfying a weak Lipschitz type condition induces a bounded Toeplitz operator. Such symbols do not need to be bounded functions or have continuous extensions to the boundary of the open unit disk. We discuss the problem of the existence of nontrivial compact Toeplitz operators, and also consider Fredholm properties and prove an index formula.

First-order turbulence closure for modelling complex canopy flows

Simple first-order closure remains an attractive way of formulating equations for complex canopy flows when the aim is to find analytic or simple numerical solutions to illustrate fundamental physical processes. Nevertheless, the limitations of such closures must be understood if the resulting models are to illuminate rather than mislead. We propose five conditions that first-order closures must satisfy then test two widely used closures against them. The first is the eddy diffusivity based on a mixing length. We discuss the origins of this approach, its use in simple canopy flows and extensions to more complex flows. We find that it satisfies most of the conditions and, because the reasons for its failures are well understood, it is a reliable methodology. The second is the velocity-squared closure that relates shear stress to the square of mean velocity. Again we discuss the origins of this closure and show that it is based on incorrect physical principles and fails to satisfy any of the five conditions in complex canopy flows; consequently its use can lead to actively misleading conclusions.

A cautionary note on the use of Ornstein Uhlenbeck models in macroevolutionary studies

Phylogenetic comparative methods are increasingly used to give new insights into the dynamics of trait evolution in deep time. For continuous traits the core of these methods is a suite of models that attempt to capture evolutionary patterns by extending the Brownian constant variance model. However, the properties of these models are often poorly understood, which can lead to the misinterpretation of results. Here we focus on one of these models – the Ornstein Uhlenbeck (OU) model. We show that the OU model is frequently incorrectly favoured over simpler models when using Likelihood ratio tests, and that many studies fitting this model use datasets that are small and prone to this problem. We also show that very small amounts of error in datasets can have profound effects on the inferences derived from OU models. Our results suggest that simulating fitted models and comparing with empirical results is critical when fitting OU and other extensions of the Brownian model. We conclude by making recommendations for best practice in fitting OU models in phylogenetic comparative analyses, and for interpreting the parameters of the OU model.

The water vapour continuum in near-infrared windows – current understanding and prospects for its inclusion in spectroscopic databases

Spectroscopic catalogues, such as GEISA and HITRAN, do not yet include information on the water vapour continuum that pervades visible, infrared and microwave spectral regions. This is partly because, in some spectral regions, there are rather few laboratory measurements in conditions close to those in the Earth’s atmosphere; hence understanding of the characteristics of the continuum absorption is still emerging. This is particularly so in the near-infrared and visible, where there has been renewed interest and activity in recent years. In this paper we present a critical review focusing on recent laboratory measurements in two near-infrared window regions (centred on 4700 and 6300 cm−1) and include reference to the window centred on 2600 cm−1 where more measurements have been reported. The rather few available measurements, have used Fourier transform spectroscopy (FTS), cavity ring down spectroscopy, optical-feedback – cavity enhanced laser spectroscopy and, in very narrow regions, calorimetric interferometry. These systems have different advantages and disadvantages. Fourier Transform Spectroscopy can measure the continuum across both these and neighbouring windows; by contrast, the cavity laser techniques are limited to fewer wavenumbers, but have a much higher inherent sensitivity. The available results present a diverse view of the characteristics of continuum absorption, with differences in continuum strength exceeding a factor of 10 in the cores of these windows. In individual windows, the temperature dependence of the water vapour self-continuum differs significantly in the few sets of measurements that allow an analysis. The available data also indicate that the temperature dependence differs significantly between different near-infrared windows. These pioneering measurements provide an impetus for further measurements. Improvements and/or extensions in existing techniques would aid progress to a full characterisation of the continuum – as an example, we report pilot measurements of the water vapour self-continuum using a supercontinuum laser source coupled to an FTS. Such improvements, as well as additional measurements and analyses in other laboratories, would enable the inclusion of the water vapour continuum in future spectroscopic databases, and therefore allow for a more reliable forward modelling of the radiative properties of the atmosphere. It would also allow a more confident assessment of different theoretical descriptions of the underlying cause or causes of continuum absorption.