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.

Precise calculations of the existence of multiple AMOC equilibria in coupled climate models Part I: equilibrium states

This study examines criteria for the existence of two stable states of the Atlantic Meridional Overturning Circulation (AMOC) using a combination of theory and simulations from a numerical coupled atmosphere–ocean climate model. By formulating a simple collection of state parameters and their relationships, the authors reconstruct the North Atlantic Deep Water (NADW) OFF state behavior under a varying external salt-flux forcing. This part (Part I) of the paper examines the steady-state solution, which gives insight into the mechanisms that sustain the NADW OFF state in this coupled model; Part II deals with the transient behavior predicted by the evolution equation. The nonlinear behavior of the Antarctic Intermediate Water (AAIW) reverse cell is critical to the OFF state. Higher Atlantic salinity leads both to a reduced AAIW reverse cell and to a greater vertical salinity gradient in the South Atlantic. The former tends to reduce Atlantic salt export to the Southern Ocean, while the latter tends to increases it. These competing effects produce a nonlinear response of Atlantic salinity and salt export to salt forcing, and the existence of maxima in these quantities. Thus the authors obtain a natural and accurate analytical saddle-node condition for the maximal surface salt flux for which a NADW OFF state exists. By contrast, the bistability indicator proposed by De Vries and Weber does not generally work in this model. It is applicable only when the effect of the AAIW reverse cell on the Atlantic salt budget is weak.

Market power and its effects on equilibrium in the food system

When food processors have conjectures about rival firms' responses, their profit functions can be used to estimate the degree of market power in the food system. The effects of this power are investigated analytically and through applying the results to the U.S. pork sector

Evolution of discrimination in populations at equilibrium between selfishness and altruism

Where there is genetically based variation in selfishness and altruism, as in man, altruists with an innate ability to recognise and thereby only help their altruistic relatives may evolve. Here we use diploid population genetic models to chart the evolution of genetically-based discrimination in populations initially in stable equilibrium between altruism and selfishness. The initial stable equilibria occur because help is assumed subject to diminishing returns. Similar results were obtained whether we used a model with two independently inherited loci, one controlling altruism the other discrimination, or a one locus model with three alleles. The latter is the opposite extreme to the first model, and can be thought of as involving complete linkage between two loci on the same chromosome. The introduction of discrimination reduced the benefits obtained by selfish individuals, more so as the number of discriminators increased, and selfishness was eventually eliminated in some cases. In others selfishness persisted and the evolutionary outcome was a stable equilibrium involving selfish individuals and both discriminating and non-discriminating altruists. Heritable variation in selfishness, altruism and discrimination is predicted to be particularly evident among full sibs. The suggested coexistence of these three genetic dispositions could explain widespread interest within human social groups as to who will and who will not help others. These predictions merit experimental and observational investigation by primatologists, anthropologists and psychologists. Keywords: Population genetics, Diploid, Heritability, Prosocial, Behaviour genetics

Departures from convective equilibrium with a rapidly-varying surface forcing

Convective equilibrium is a long-standing and useful concept for understanding many aspects of the behaviour of deep moist convection. For example, it is often invoked in developing parameterizations for large-scale models. However, the equilibrium assumption may begin to break down as models are increasingly used with shorter timesteps and finer resolutions. Here we perform idealized cloud-system resolving model simulations of deep convection with imposed time variations in the surface forcing. A range of rapid forcing timescales from 1 − 36hr are used, in order to induce systematic departures from equilibrium. For the longer forcing timescales, the equilibrium assumption remains valid, in at least the limited sense that cycle-integrated measures of convective activity are very similar from cycle to cycle. For shorter forcing timescales, cycle-integrated convection becomes more variable, with enhanced activity on one cycle being correlated with reduced activity on the next, suggesting a role for convective memory. Further investigation shows that the memory does not appear to be carried by the domain-mean thermodynamic fields but rather by structures on horizontal scales of 5 − 20km. Such structures are produced by the convective clouds and can persist beyond the lifetime of the cloud, even through to the next forcing cycle.

A mathematical approach to chemical equilibrium theory for gaseous systems—I: theory

Equilibrium theory occupies an important position in chemistry and it is traditionally based on thermodynamics. A novel mathematical approach to chemical equilibrium theory for gaseous systems at constant temperature and pressure is developed. Six theorems are presented logically which illustrate the power of mathematics to explain chemical observations and these are combined logically to create a coherent system. This mathematical treatment provides more insight into chemical equilibrium and creates more tools that can be used to investigate complex situations. Although some of the issues covered have previously been given in the literature, new mathematical representations are provided. Compared to traditional treatments, the new approach relies on straightforward mathematics and less on thermodynamics, thus, giving a new and complementary perspective on equilibrium theory. It provides a new theoretical basis for a thorough and deep presentation of traditional chemical equilibrium. This work demonstrates that new research in a traditional field such as equilibrium theory, generally thought to have been completed many years ago, can still offer new insights and that more efficient ways to present the contents can be established. The work presented here can be considered appropriate as part of a mathematical chemistry course at University level.

A mathematical approach to chemical equilibrium theory for gaseous systems—II: extensions and applications

Straightforward mathematical techniques are used innovatively to form a coherent theoretical system to deal with chemical equilibrium problems. For a systematic theory it is necessary to establish a system to connect different concepts. This paper shows the usefulness and consistence of the system by applications of the theorems introduced previously. Some theorems are shown somewhat unexpectedly to be mathematically correlated and relationships are obtained in a coherent manner. It has been shown that theorem 1 plays an important part in interconnecting most of the theorems. The usefulness of theorem 2 is illustrated by proving it to be consistent with theorem 3. A set of uniform mathematical expressions are associated with theorem 3. A variety of mathematical techniques based on theorems 1–3 are shown to establish the direction of equilibrium shift. The equilibrium properties expressed in initial and equilibrium conditions are shown to be connected via theorem 5. Theorem 6 is connected with theorem 4 through the mathematical representation of theorem 1.