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.

Spectrum of a Feinberg-Zee random hopping matrix

This paper provides a new proof of a theorem of Chandler-Wilde, Chonchaiya, and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra contain a hole around the origin. The presence of the hole forces substantial changes to the analysis.

Random-walk-based characterisation of multiple access interference in a DS/SSMA system

The problem of calculating the probability of error in a DS/SSMA system has been extensively studied for more than two decades. When random sequences are employed some conditioning must be done before the application of the central limit theorem is attempted, leading to a Gaussian distribution. The authors seek to characterise the multiple access interference as a random-walk with a random number of steps, for random and deterministic sequences. Using results from random-walk theory, they model the interference as a K-distributed random variable and use it to calculate the probability of error in the form of a series, for a DS/SSMA system with a coherent correlation receiver and BPSK modulation under Gaussian noise. The asymptotic properties of the proposed distribution agree with other analyses. This is, to the best of the authors' knowledge, the first attempt to propose a non-Gaussian distribution for the interference. The modelling can be extended to consider multipath fading and general modulation

A theoretical framework for energy and momentum consistency in subgrid-scale parameterization for climate models

A theoretical framework for the joint conservation of energy and momentum in the parameterization of subgrid-scale processes in climate models is presented. The framework couples a hydrostatic resolved (planetary scale) flow to a nonhydrostatic subgrid-scale (mesoscale) flow. The temporal and horizontal spatial scale separation between the planetary scale and mesoscale is imposed using multiple-scale asymptotics. Energy and momentum are exchanged through subgrid-scale flux convergences of heat, pressure, and momentum. The generation and dissipation of subgrid-scale energy and momentum is understood using wave-activity conservation laws that are derived by exploiting the (mesoscale) temporal and horizontal spatial homogeneities in the planetary-scale flow. The relations between these conservation laws and the planetary-scale dynamics represent generalized nonacceleration theorems. A derived relationship between the wave-activity fluxes-which represents a generalization of the second Eliassen-Palm theorem-is key to ensuring consistency between energy and momentum conservation. The framework includes a consistent formulation of heating and entropy production due to kinetic energy dissipation.

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.

Correlations between two sets of angular relation equations

It is shown here that the angular relation equations between direct and reciprocal vectors are very similar to the angular relation equations in Euler's theorem. These two sets of equations are usually treated separately as unrelated equations in different fields. In this careful study, the connection between the two sets of angular equations is revealed by considering the cosine rule for the spherical triangle. It is found that understanding of the correlation is hindered by the facts that the same variables are defined differently and different symbols are used to represent them in the two fields. Understanding the connection between different concepts is not only stimulating and beneficial, but also a fundamental tool in innovation and research, and has historical significance. The background of the work presented here contains elements of many scientific disciplines. This work illustrates the common ground of two theories usually considered separately and is therefore of benefit not only for its own sake but also to illustrate a general principle that a theory relevant to one discipline can often be used in another. The paper works with chemistry related concepts using mathematical methodologies unfamiliar to the usual audience of mainstream experimental and theoretical chemists.

When do export subsidies have a redistributional role? Reply

Our differences are three. The first arises from the belief that "... a nonzero value for the optimally chosen policy instrument implies that the instrument is efficient for redistribution" (Alston, Smith, and Vercammen, p. 543, paragraph 3). Consider the two equations: (1) o* = f(P3) and (2) = -f(3) ++r h* (a, P3) representing the solution to the problem of maximizing weighted, Marshallian surplus using, simultaneously, a per-unit border intervention, 9, and a per-unit domestic intervention, wr. In the solution, parameter ot denotes the weight applied to producer surplus; parameter p denotes the weight applied to government revenues; consumer surplus is implicitly weighted one; and the country in question is small in the sense that it is unable to affect world price by any of its domestic adjustments (see the Appendix). Details of the forms of the functions f((P) and h(ot, p) are easily derived, but what matters in the context of Alston, Smith, and Vercammen's Comment is: Redistributivep referencest hatf avorp roducers are consistent with higher values "alpha," and whereas the optimal domestic intervention, 7r*, has both "alpha and beta effects," the optimal border intervention, r*, has only a "beta effect,"-it does not have a redistributional role. Garth Holloway is reader in agricultural economics and statistics, Department of Agricultural and Food Economics, School of Agriculture, Policy, and Development, University of Reading. The author is very grateful to Xavier Irz, Bhavani Shankar, Chittur Srinivasan, Colin Thirtle, and Richard Tiffin for their comments and their wisdom; and to Mario Mazzochi, Marinos Tsigas, and Cal Turvey for their scholarship, including help in tracking down a fairly complete collection of the papers that cite Alston and Hurd. They are not responsible for any errors or omissions. Note, in equation (1), that the border intervention is positive whenever a distortion exists because 8 > 0 implies 3 - 1 + 8 > 1 and, thus, f((P) > 0 (see Appendix). Using Alston, Smith, and Vercammen's definition, the instrument is now "efficient," and therefore has a redistributive role. But now, suppose that the distortion is removed so that 3 - 1 + 8 = 1, 8 = 0, and consequently the border intervention is zero. According to Alston, Smith, and Vercammen, the instrument is now "inefficient" and has no redistributive role. The reader will note that this thought experiment has said nothing about supporting farm incomes, and so has nothing whatsoever to do with efficient redistribution. Of course, the definition is false. It follows that a domestic distortion arising from the "excess-burden argument" 3 = 1 + 8, 8 > 0 does not make an export subsidy "efficient." The export subsidy, having only a "beta effect," does not have a redistributional role. The second disagreement emerges from the comment that Holloway "... uses an idiosyncratic definition of the relevant objective function of the government (Alston, Smith, and Vercammen, p. 543, paragraph 2)." The objective function that generates equations (1) and (2) (see the Appendix) is the same as the objective function used by Gardner (1995) when he first questioned Alston, Carter, and Smith's claim that a "domestic distortion can make a border intervention efficient in transferring surplus from consumers and taxpayers to farmers." The objective function used by Gardner (1995) is the same objective function used in the contributions that precede it and thus defines the literature on the debate about borderversus- domestic intervention (Streeten; Yeh; Paarlberg 1984, 1985; Orden; Gardner 1985). The objective function in the latter literature is the same as the one implied in another literature that originates from Wallace and includes most notably Gardner (1983), but also Alston and Hurd. Amer. J. Agr. Econ. 86(2) (May 2004): 549-552 Copyright 2004 American Agricultural Economics Association This content downloaded on Tue, 15 Jan 2013 07:58:41 AM All use subject to JSTOR Terms and Conditions 550 May 2004 Amer. J. Agr. Econ. The objective function in Holloway is this same objective function-it is, of course, Marshallian surplus.1 The third disagreement concerns scholarship. The Comment does not seem to be cognizant of several important papers, especially Bhagwati and Ramaswami, and Bhagwati, both of which precede Corden (1974, 1997); but also Lipsey and Lancaster, and Moschini and Sckokai; one important aspect of Alston and Hurd; and one extremely important result in Holloway. This oversight has some unfortunate repercussions. First, it misdirects to the wrong origins of intellectual property. Second, it misleads about the appropriateness of some welfare calculations. Third, it prevents Alston, Smith, and Vercammen from linking a finding in Holloway (pp. 242-43) with an old theorem (Lipsey and Lancaster) that settles the controversy (Alston, Carter, and Smith 1993, 1995; Gardner 1995; and, presently, Alston, Smith, and Vercammen) about the efficiency of border intervention in the presence of domestic distortions.

Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.

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.

Strong structural controllability and the multilink inverted pendulum

This technical note investigates the controllability of the linearized dynamics of the multilink inverted pendulum as the number of links and the number and location of actuators changes. It is demonstrated that, in some instances, there exist sets of parameter values that render the system uncontrollable and so usual methods for assessing controllability are difficult to employ. To assess the controllability, a theorem on strong structural controllability for single-input systems is extended to the multiinput case.

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship A E =cA P between pseudoenergy A E and pseudomomentum A P, where c is the horizontal phase speed in the direction of symmetry associated with A P, has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

Balitskiǐ-Fadin-Kuraev-Lipatov equation versus data from DESY HERA

The BFKL equation and the kT-factorization theorem are used to obtain predictions for F2 in the small Bjo/rken-x region over a wide range of Q2. The dependence on the parameters, especially on those concerning the infrared region, is discussed. After a background fit to recent experimental data obtained at DESY HERA and at Fermilab (E665 experiment) we find that the predicted, almost Q2 independent BFKL slope λ≳0.5 appears to be too steep at lower Q2 values. Thus there seems to be a chance that future HERA data can distinguish between pure BFKL and conventional field theoretic renormalization group approaches. © 1995 The American Physical Society.

Noise propagation from a cutting of arbitrary cross-section and impedance

A boundary integral equation is described for the prediction of acoustic propagation from a monofrequency coherent line source in a cutting with impedance boundary conditions onto surrounding flat impedance ground. The problem is stated as a boundary value problem for the Helmholtz equation and is subsequently reformulated as a system of boundary integral equations via Green's theorem. It is shown that the integral equation formulation has a unique solution at all wavenumbers. The numerical solution of the coupled boundary integral equations by a simple boundary element method is then described. The convergence of the numerical scheme is demonstrated experimentally. Predictions of A-weighted excess attenuation for a traffic noise spectrum are made illustrating the effects of varying the depth of the cutting and the absorbency of the surrounding ground surface.

On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method. © 2000 American Institute of Physics.

Averaging, slaving and balance dynamics in a simple atmospheric model

We report numerical results from a study of balance dynamics using a simple model of atmospheric motion that is designed to help address the question of why balance dynamics is so stable. The non-autonomous Hamiltonian model has a chaotic slow degree of freedom (representing vortical modes) coupled to one or two linear fast oscillators (representing inertia-gravity waves). The system is said to be balanced when the fast and slow degrees of freedom are separated. We find adiabatic invariants that drift slowly in time. This drift is consistent with a random-walk behaviour at a speed which qualitatively scales, even for modest time scale separations, as the upper bound given by Neishtadt’s and Nekhoroshev’s theorems. Moreover, a similar type of scaling is observed for solutions obtained using a singular perturbation (‘slaving’) technique in resonant cases where Nekhoroshev’s theorem does not apply. We present evidence that the smaller Lyapunov exponents of the system scale exponentially as well. The results suggest that the observed stability of nearly-slow motion is a consequence of the approximate adiabatic invariance of the fast motion.