arXiv subjects

R. S. MacKay

Publications and source records attributed to R. S. MacKay.

15 recordsLinked to original sources

arXiv2025

Thermal Macroeconomics: An axiomatic theory of aggregate economic phenomena

An axiomatic approach to macroeconomics based on the mathematical structure of thermodynamics is presented. It deduces relations between aggregate properties of an economy, concerning quantities and flows of goods and money, prices and the value of money, without any recourse to microeconomic foundations about the preferences and actions of individual economic agents. The approach has three important payoffs. 1) it provides a new and solid foundation for aspects of standard macroeconomic theory such as the existence of market prices, the value of money, the meaning of inflation, the symmetry and negative-definiteness of the macro-Slutsky matrix, and the Le Chatelier-Samuelson principle, without relying on implausibly strong rationality assumptions over individual microeconomic agents. 2) the approach generates new results, including implications for money flow and trade when two or more economies are put in contact, in terms of new concepts such as economic entropy, economic temperature, goods' values and money capacity. Some of these are related to standard economic concepts (eg marginal utility of money, market prices). Yet our approach derives them at a purely macroeconomic level and gives them a meaning independent of usual restrictions. Others of the concepts, such as economic entropy and temperature, have no direct counterparts in standard economics, but they have important economic interpretations and implications, as aggregate utility and the inverse marginal aggregate utility of money, respectively. 3) this analysis promises to open up new frontiers in macroeconomics by building a bridge to ideas from non-equilibrium thermodynamics. More broadly, we hope that the economic analogue of entropy (governing the possible transitions between states of economic systems) may prove to be as fruitful for the social sciences as entropy has been in the natural sciences.

econ.GN↗

arXiv2025

Convergence to equilibrium for a class of exchange economies

For a class of stochastic dynamical models of exchange economies that we call ``fully connected Cobb-Douglas'', the paper proves convergence of the probability distribution to an equilibrium, in total variation metric as time goes to infinity. The convergence is exponential and the equilibrium is determined uniquely by the number of agents, their ``exponents'', and the initial amounts of money and goods in the economy.

math.PR↗

arXiv2024

Tests of thermal macroeconomic theory on simulated micro-economies

In this paper, we test predictions of a new theory of macroeconomics, called "thermal macroeconomics." The theory aims to apply the mathematical structure of classical thermodynamics, including analogues of temperature and entropy, to predict aspects of the aggregate behaviour of populations of economic agents without analyzing their detailed interactions. We test the theory by comparing its predictions with the behaviour of a variety of simulated micro-economies in which goods and money can be exchanged between agents, confirming the predictions of the theory. The paper serves also to illustrate and make more tangible the predictions of thermal macroeconomics.

econ.GN↗

arXiv2024

Escape from a potential well under forcing and damping with application to ship capsize

Escape from a potential well occurs in a wide variety of physical systems from chemical reactions to ship capsize. In these situations escape often occurs by passage over a normally hyperbolic submanifold (NHS). This paper describes the computational implementation of an algorithm to identify the NHS of a two degree of freedom model for ship motion under the influence of damping and aperiodic forcing. Additionally we demonstrate how the stable manifolds of these submanifolds can be used to classify initial ship states as safe or unsafe.

math.DS↗

arXiv2024

Characterization of admissible quasisymmetries

We solve "half" the problem of finding three-dimensional quasisymmetric magnetic fields that do not necessarily satisfy force balance. This involves determining which hidden symmetries are admissible as quasisymmetries, and then showing explicitly how to construct quasisymmetric magnetic fields given an admissible symmetry. The admissibility conditions take the form of a system of overdetermined nonlinear partial differential equations involving second derivatives of the symmetry's infinitesimal generator.

physics.plasm-ph↗

arXiv2023

Isodrastic Magnetic fields for suppressing transitions in guiding-centre motion

In a magnetic field, transitions between classes of guiding-centre motion can lead to cross-field diffusion and escape. We say a magnetic field is isodrastic if guiding centres make no transitions between classes of motion. Therefore, this is an important ideal for enhancing confinement. First, we present a weak formulation, based on the longitudinal adiabatic invariant, generalising omnigenity. To demonstrate that isodrasticity is strictly more general than omnigenity, we construct weakly isodrastic mirror fields that are not omnigenous. Then we present a strong formulation that is exact for guiding-centre motion. We develop a first-order treatment of the strong version via a Melnikov function and show that it recovers the weak version. The theory provides quantification of deviations from isodrasticity that can be used as objective functions in optimal design. The theory is illustrated on some simple examples.

physics.plasm-ph↗

arXiv2021

Surfaces of locally minimal flux

For exact area-preserving twist maps, curves were constructed through the gaps of cantori in \cite{MMP84}, which were conjectured to have minimal flux subject to passing through the points of the cantorus. It was pointed out by \cite{Pol} that these curves do {\em not} have minimal flux if there coexists a rotational invariant circle of a different rotation number, but if hyperbolic they do have {\em locally} minimal flux even without the constraint of passing through the points of the cantorus. Following the criterion of \cite{M94} for surfaces of locally minimal flux for 3D volume-preserving flows, I revisit this result and show that in general the analogous curves through the points of rotationally-ordered periodic orbits or their heteroclinic orbits do {\em not} have locally minimal flux. Along the way, various questions are posed. Some results for more degrees of freedom are summarised.

nlin.CD↗

arXiv2021

Dobrushin and Steif metrics are equal

It is proved that two useful and apparently different metrics on the set of Borel probabilities on countable products of Polish spaces of bounded diameters are equal. This paves the way for advances in their computation.

math.PR↗

arXiv2020

Generalized Grad-Shafranov equation for non-axisymmetric MHD equilibria

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad-Shafranov equation, which is a single elliptic PDE in two space dimensions. By revealing a hidden symmetry, we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad-Shafranov equation. In contrast to solutions of the classical Grad-Shafranov equation, solutions of he generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad-Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

physics.plasm-ph↗

arXiv2020

How directed is a directed network?

The trophic levels of nodes in directed networks can reveal their functional properties. Moreover, the trophic coherence of a network, defined in terms of trophic levels, is related to properties such as cycle structure, stability and percolation. The standard definition of trophic levels, however, borrowed from ecology, suffers from drawbacks such as requiring source nodes, which limit its applicability. Here we propose a simple new definition of trophic level that can be computed on any directed network. We demonstrate how the method can identify node function in examples including ecosystems, supply chain networks, gene expression, and global language networks. We also explore how trophic levels and coherence relate to other topological properties, such as non-normality and cycle structure, and show that our method reveals the extent to which the edges in a directed network are aligned in a global direction.

physics.soc-ph↗

arXiv2010

Quantifying Emergence in terms of Persistent Mutual Information

We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long term prediction, which we propose as the primary signature of (Strong) Emergent Behaviour. The key feature of our definition of PMI is the omission of an interval of 'present' time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behaviour or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map we compare predicted with measured long-time PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalisation of the notion of ordered phases.

nlin.AO↗

arXiv2003

Symplectic twist maps without conjugate points

We consider a sequences of symplectic twist maps corresponding to Frenkel Kontorova variational functional. For such a sequence without conjugate points we construct an invariant Lagrangian subbundle. Using the ideas by E.Hopf from Riemannian geometry we show that absence of conjugate points is a rare property for this class of maps.

math.SG↗

arXiv2000

Approximate renormalization for the break-up of invariant tori with three frequencies

We construct an approximate renormalization transformation for Hamiltonian systems with three degrees of freedom in order to study the break-up of invariant tori with three incommensurate frequencies which belong to the cubic field $Q(τ)$, where $τ^3+τ^2-2τ-1=0$. This renormalization has two fixed points~: a stable one and a hyperbolic one with a codimension one stable manifold. We compute the associated critical exponents that characterize the universality class for the break-up of the invariant tori we consider.

nlin.CD↗

arXiv1999

Variational properties of a nonlinear elliptic equation and rigidity

We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally minimising. However, in dimension two the situation is different. We show that for radially symmetric potentials there always exist solutions which are not locally minimal unless the potential vanishes identically. We discuss the relations of this with the so-called Hopf rigidity phenomenon.

math.DS↗

arXiv1997

Energy thresholds for discrete breathers in one-, two- and three-dimensional lattices

Discrete breathers are time-periodic, spatially localized solutions of equations of motion for classical degrees of freedom interacting on a lattice. They come in one-parameter families. We report on studies of energy properties of breather families in one-, two- and three-dimensional lattices. We show that breather energies have a positive lower bound if the lattice dimension of a given nonlinear lattice is greater than or equal to a certain critical value. These findings could be important for the experimental detection of discrete breathers.

patt-sol↗