Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Evidence of dispersion relations for the nonlinear response of the Lorenz 63 system

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.

A statistical mechanical approach for the computation of the climatic response to general forcings

The climate belongs to the class of non-equilibrium forced and dissipative systems, for which most results of quasi-equilibrium statistical mechanics, including the fluctuation-dissipation theorem, do not apply. In this paper we show for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings. The crucial value of the Ruelle theory lies in the fact that it allows to compute the response of the system in terms of expectation values of explicit and computable functions of the phase space averaged over the invariant measure of the unperturbed state. We choose as test bed a classical version of the Lorenz 96 model, which, in spite of its simplicity, has a well-recognized prototypical value as it is a spatially extended one-dimensional model and presents the basic ingredients, such as dissipation, advection and the presence of an external forcing, of the actual atmosphere. We recapitulate the main aspects of the general response theory and propose some new general results. We then analyze the frequency dependence of the response of both local and global observables to perturbations having localized as well as global spatial patterns. We derive analytically several properties of the corresponding susceptibilities, such as asymptotic behavior, validity of Kramers-Kronig relations, and sum rules, whose main ingredient is the causality principle. We show that all the coefficients of the leading asymptotic expansions as well as the integral constraints can be written as linear function of parameters that describe the unperturbed properties of the system, such as its average energy. Some newly obtained empirical closure equations for such parameters allow to define such properties as an explicit function of the unperturbed forcing parameter alone for a general class of chaotic Lorenz 96 models. We then verify the theoretical predictions from the outputs of the simulations up to a high degree of precision. The theory is used to explain differences in the response of local and global observables, to define the intensive properties of the system, which do not depend on the spatial resolution of the Lorenz 96 model, and to generalize the concept of climate sensitivity to all time scales. We also show how to reconstruct the linear Green function, which maps perturbations of general time patterns into changes in the expectation value of the considered observable for finite as well as infinite time. Finally, we propose a simple yet general methodology to study general Climate Change problems on virtually any time scale by resorting to only well selected simulations, and by taking full advantage of ensemble methods. The specific case of globally averaged surface temperature response to a general pattern of change of the CO2 concentration is discussed. We believe that the proposed approach may constitute a mathematically rigorous and practically very effective way to approach the problem of climate sensitivity, climate prediction, and climate change from a radically new perspective.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

Remote sensing of tidal networks and their relation to vegetation

The study of the morphology of tidal networks and their relation to salt marsh vegetation is currently an active area of research, and a number of theories have been developed which require validation using extensive observations. Conventional methods of measuring networks and associated vegetation can be cumbersome and subjective. Recent advances in remote sensing techniques mean that these can now often reduce measurement effort whilst at the same time increasing measurement scale. The status of remote sensing of tidal networks and their relation to vegetation is reviewed. The measurement of network planforms and their associated variables is possible to sufficient resolution using digital aerial photography and airborne scanning laser altimetry (LiDAR), with LiDAR also being able to measure channel depths. A multi-level knowledge-based technique is described to extract networks from LiDAR in a semi-automated fashion. This allows objective and detailed geomorphological information on networks to be obtained over large areas of the inter-tidal zone. It is illustrated using LIDAR data of the River Ems, Germany, the Venice lagoon, and Carnforth Marsh, Morecambe Bay, UK. Examples of geomorphological variables of networks extracted from LiDAR data are given. Associated marsh vegetation can be classified into its component species using airborne hyperspectral and satellite multispectral data. Other potential applications of remote sensing for network studies include determining spatial relationships between networks and vegetation, measuring marsh platform vegetation roughness, in-channel velocities and sediment processes, studying salt pans, and for marsh restoration schemes.

The role of animal nutrition in improving the nutritive value of animal-derived foods in relation to chronic disease

Foods derived from animals are an important source of nutrients in the diet; for example, milk and meat together provide about 60 and 55% of the dietary intake of Ca and protein respectively in the UK. However, certain aspects of some animal-derived foods, particularly their fat and saturated fatty acid (SFA) contents, have led to concerns that these foods substantially contribute to the risk of CVD, the metabolic syndrome and other chronic diseases. In most parts of Europe dairy products are the greatest single dietary source of SFA. The fatty acid composition of various animal-derived foods is, however, not constant and can, in many cases, be enhanced by animal nutrition. In particular, milk fat with reduced concentrations of the C12-16 SFA and an increased concentration of 18:1 MUFA is achievable, although enrichment with very-long-chain n-3 PUFA is much less efficient. However, there is now evidence that some animal-derived foods (notably milk products) contain compounds that may actively promote long-term health, and research is urgently required to fully characterise the benefits associated with the consumption of these compounds and to understand how the levels in natural foods can be enhanced. It is also vital that the beneficial effects are not inadvertently destroyed in the process of reducing the concentrations of SFA. In the future the role of animal nutrition in creating foods closer to the optimum composition for long-term human health is likely to become increasingly important, but production of such foods on a scale that will substantially affect national diets will require political and financial incentives and great changes in the animal production industry.

Results of a mechanistic model estimating methane in relation to methane emissions measured in dairy cows

This investigation determines the accuracy of estimation of methanogenesis by a dynamic mechanistic model with real data determined in a respiration trial, where cows were fed a wide range of different carbohydrates included in the concentrates. The model was able to predict ECM (Energy corrected milk) very well, while the NDF digestibility of fibrous feed was less well predicted. Methane emissions were predicted quite well, with the exception of one diet containing wheat. The mechanistic model is therefore a helpful tool to estimate methanogenesis based on chemical analysis and dry matter intake, but the prediction can still be improved.

The changing structure of diets in the European Union in relation to healthy eating guidelines

Objective: Our objective in this paper is to assess diets in the European Union (EU) in relation to the recommendations of the recent World Health Organization/Food and Agriculture Organization expert consultation and to show how diets have changed between 1961 and 2001. Data and methods: Computations make use of FAOSTAT data on food availability at country level linked to a food composition database to convert foods to nutrients. We further explore the growing similarity of diets in the EU by making use of a consumption similarity index. The index provides a single number measure of dietary overlap between countries. Results: The data confirm the excessive consumption by almost all countries of saturated fats, cholesterol and sugars, and the convergence of nutrient intakes across the EU. Whereas in 1961 diets in several European countries were more similar to US diets than to those of other European countries, this is no longer the case; moreover, while EU diets have become more homogeneous, the EU as a whole and the USA have become less similar over time. Conclusions: Although the dominant cause of greater similarity in EU diets over the period studied is increased intakes in Mediterranean countries of saturated fats, cholesterol and sugar, also important are reductions in saturated fat and sugar in some Northern European countries. This suggests that healthy eating messages are finally having an impact on diets; a distinctly European diet may also be emerging.

First lactation ovarian function in dairy heifers in relation to prepubertal metabolic profiles

The aim of this study was to determine whether any differences in the GH-IGF-I axis in juvenile calves were predictive of fertility problems as adult cows. Endogenous metabolic hormone profiles before and after feeding and the response to a GH-releasing factor (GRF) challenge were measured in prepubertal (6 month) dairy calves. These metabolic parameters were subsequently related to physical characteristics at puberty and to ovarian function during the first lactation. Milk progesterone analysis was used to categorize the animals into those with normal progesterone profiles following calving (n = 17) and those that developed delayed ovulation (DOV1, n = 9) or persistent corpus luteum (PCL1, n = 6) profiles. There were associations between prepubertal GH parameters, glucose and non-esterified fatty acid (NEFA) concentrations and the body condition score at which the animals attained puberty. The calves which subsequently developed DOV1 profiles as cows tended to have a higher GH pulse amplitude during fasting than normal profile animals, they did not show the anticipated decrease in circulating glucose concentrations following a post-prandial rise in insulin and they also had the lowest IGF-I concentrations. The calves that later developed PCL1 had a significantly larger GH pulse amplitude and pulse area than normal profile animals in the fed period and had the highest IGF-I concentrations. There were no differences in prepubertal insulin or NEFA concentrations or in the GH response to a GRF challenge between the different progesterone profile categories. Plasma IGF-I concentrations in prepubertal animals were positively correlated with their post-calving concentrations, whereas glucose concentrations had a negative correlation between these time-periods. These results suggested that the different juvenile endocrine profiles of the DOV1 cows may predispose them to a higher rate of tissue mobilization during lactation and a consequent reduction in fertility, while altered GH and IGF-I levels in PCL1 cows may later contribute to the maintenance of the persistent corpus luteum. Therefore metabolic differences in prepubertal calves were later reflected by altered reproductive function during the first lactation.