Instabilities and Non-Reversibility of Molecular Dynamics Trajectories

The theoretical justification of the Hybrid Monte Carlo algorithm depends upon the molecular dynamics trajectories within it being exactly reversible. If computations were carried out with exact arithmetic then it would be easy to ensure such reversibility, but the use of approximate floating point arithmetic inevitably introduces violations of reversibility. In the absence of evidence to the contrary, we are usually prepared to accept that such rounding errors can be made small enough to be innocuous, but in certain circumstances they are exponentially amplified and lead to blatantly erroneous results. We show that there are two types of instability of the molecular dynamics trajectories which lead to this behavior, instabilities due to insufficiently accurate numerical integration of Hamilton's equations, and intrinsic chaos in the underlying continuous fictitious time equations of motion themselves. We analyze the former for free field theory, and show that it is essentially a finite volume effect. For the latter we propose a hypothesis as to how the Liapunov exponent describing the chaotic behavior of the fictitious time equations of motion for an asymptotically free quantum field theory behaves as the system is taken to its continuum limit, and explain why this means that instabilities in molecular dynamics trajectories are not a significant problem for Hybrid Monte Carlo computations. We present data for pure $SU(3)$ gauge theory and for QCD with dynamical fermions on small lattices to illustrate and confirm some of our results.Comment: 28 pages latex with 19 color postscript figures included by eps

Cost of the Generalised Hybrid Monte Carlo Algorithm for Free Field Theory

We study analytically the computational cost of the Generalised Hybrid Monte Carlo (GHMC) algorithm for free field theory. We calculate the Metropolis acceptance probability for leapfrog and higher-order discretisations of the Molecular Dynamics (MD) equations of motion. We show how to calculate autocorrelation functions of arbitrary polynomial operators, and use these to optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize for the special cases of linear and quadratic operators. We show that long trajectories are optimal for GHMC, and that standard HMC is more efficient than algorithms based on Second Order Langevin Monte Carlo (L2MC), sometimes known as Kramers Equation. We show that contrary to naive expectations HMC and L2MC have the same volume dependence, but their dynamical critical exponents are z = 1 and z = 3/2 respectively.Comment: 54 pages, 3 figure

Cost of Generalised HMC Algorithms for Free Field Theory

We study analytically the computational cost of the Generalised Hybrid Monte Carlo (GHMC) algorithm for free field theory. We calculate the autocorrelation functions of operators quadratic in the fields, and optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize. We show that long trajectories are optimal for GHMC, and that standard HMC is much more efficient than algorithms based on the Second Order Langevin (L2MC) or Kramers Equation. We show that contrary to naive expectations HMC and L2MC have the same volume dependence, but their dynamical critical exponents are z=1 and z=3/2 respectively.Comment: LATTICE99(Algorithms and Machines) - 3 pages, 1 PostScript figur

Preserving and Extending the Commodification of Football Supporter Relations: a Cultural Economy of Supporters Direct

This paper examines the role of Supporters Direct, a sports policy initiative launched by the British Labour government in 2000. The objective of Supporters Direct is to democratise football clubs by intervening in what it views as the unequal relationship that exists between the relatively powerless supporters of football clubs and private shareholders who have organisational control of clubs. They hope to achieve this by facilitating mutual forms of ownership and control of clubs via supporters\' trusts. With respect to this objective, established research concerning Supporters Direct emphasises this initiative as an inherently progressive development for the football industry. The aim of this paper is to situate the development of Supporters Direct in the wider context of the British Labour Government\'s policy of social inclusion. On the basis of a textual analysis that draws on current literature in the area of culture and economy - with specific reference to processes of commodification - we reveal an alternative view of Supporters Direct. The Supporters Direct initiative, we conclude, is an integral part of a social policy aimed at the preservation and extension of commodified social relations.Supporters Direct; Supporters\' Trusts, Social Inclusion, Mutualism, Commodification

Investigations into the assembly behaviour of a 'rigidified': P-carboxylatocalix[4]arene

The p-carboxylatocalix[4]arenes have been shown to be versatile supramolecular building blocks capable of forming a range of bi-layers, capsules and nanoscale tubules in the solid state. Here we report the synthesis of a new 'rigidified' analogue, as well as investigations into its self-assembly and related coordination chemistry. These behaviours are reminiscent of other p-carboxylatocalix[4]arenes despite the presence of rigidifying groups at the lower-rim, suggesting that this building block may be further exploited in the assembly of a range of new metal-organic cages and coordination polymers

Viewing the world through a wider lens: Māori and council planning documents

My effort to convey an iwi perspective on environmental resource management - what we call kaitiakitanga – should highlight for new planners about to enter the profession that the environmental perspectives of hapū and iwi (which are provided for in the RMA), are currently not well covered in either mainstream local government planning or education

Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo Tests

Simulations of the self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's Stochastic Loewner Evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance would imply that if we map the walks in the cut-plane into the half plane using the conformal map z -> sqrt(z), then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.Comment: Second version added more simulations and a proof of irreducibility. 25 pages, 16 figure

ORGANIC FARMING AND SOCIAL CAPITAL APPROACH IN THE RESTORATION OF SUSTAINABLE AGRICULTURAL LIVELIHOODS IN A POST-CONFLICT SETTING: A CASE OF NORTHERN UGANDA

This report presents a discussion of how organic farming and social capital development can contribute towards the restoration of sustainable agricultural livelihoods in a post-conflict setting; with a case study of Northern Uganda. Strictly speaking, the paper goes beyond a simple exposition of the value of organic farming, but it attempts to explain the complex ways in which social capital relates with organic farming to revitalize sustainable agricultural systems, and how this can impact on the livelihoods of communities in a post-conflict situation, with respect to household food security and income