Thermalisation of Local Observables in Small Hubbard Lattices

We present a study of thermalisation of a small isolated Hubbard lattice cluster prepared in a pure state with a well-defined energy. We examine how a two-site subsystem of the lattice thermalises with the rest of the system as its environment. We explore numerically the existence of thermalisation over a range of system parameters, such as the interaction strength, system size and the strength of the coupling between the subsystem and the rest of the lattice. We find thermalisation over a wide range of parameters and that interactions are crucial for efficient thermalisation of small systems. We relate this thermalisation behaviour to the eigenstate thermalisation hypothesis and quantify numerically the extent to which eigenstate thermalisation holds. We also verify our numerical results theoretically with the help of previously established results from random matrix theory for the local density of states, particularly the finite-size scaling for the onset of thermalisation.Comment: 22 pages, 23 figure

Ground state cooling of atoms in optical lattices

We propose two schemes for cooling bosonic and fermionic atoms that are trapped in a deep optical lattice. The first scheme is a quantum algorithm based on particle number filtering and state dependent lattice shifts. The second protocol alternates filtering with a redistribution of particles by means of quantum tunnelling. We provide a complete theoretical analysis of both schemes and characterize the cooling efficiency in terms of the entropy. Our schemes do not require addressing of single lattice sites and use a novel method, which is based on coherent laser control, to perform very fast filtering.Comment: 12 pages, 7 figure

Natural noise and external wake field seeding in a proton-driven plasma accelerator

We discuss the level of natural shot noise in a proton bunch-driven plasma accelerator. The required seeding for the plasma wake field must be larger than the cumulative shot noise. This is the necessary condition for the axial symmetry of the generated wake and the acceleration quality. We develop an analytical theory of the noise field and compare it with multi-dimensional simulations. It appears that the natural noise wake field generated in plasma by the available at CERN super-protons-synchrotron (SPS) bunches is very low, at the level of a few 10 kV/m. This fortunate fact eases the requirements on the seed. Our three dimensional simulations show that even a few tens MeV electron bunch precursor of a very moderate intensity is sufficient to seed the proton bunch self-modulation in plasma.Comment: 5 pages, 5 figure

A framework for teaching epistemic insight in schools

This paper gives the rationale and a draft outline for a framework for education to teach epistemic insight into schools in England. The motivation to research and propose a strategy to teach and assess epistemic insight followed research that investigated how students and teachers in primary and secondary schools respond to big questions about the nature of reality and human personhood. The research revealed that there are pressures in schools that dampen students’ expressed curiosity in these types of questions and limit their developing epistemic insight into how science, religion and the wider humanities relate. These findings prompted the construction of a framework for education for students aged 5–16 designed to encourage students’ expressed interest in big questions and develop their understanding of the ways that science interacts with other ways of knowing. The centrepiece of the framework is a sequence of learning objectives for epistemic insight, organised into three categories. The categories are, firstly, the nature of science in real world contexts and multidisciplinary arenas; secondly, ways of knowing and how they interact; and thirdly, the relationships between science and religion. Our current version of the Framework is constructed to respond to the way that teaching is organised in England. The key principles and many of the activities could be adopted and tailored to work in many other countries

Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

Asymptotic behaviour of estimators of the parameters of nearly unstable INAR(1) models

A sequence of first-order integer-valued autoregressive type (INAR(1)) processes is investigated, where the autoregressive type coefficients converge to 1. It is shown that the limiting distribution of the joint conditional least squares estimators for this coefficient and for the mean of the innovation is normal. Consequences for sequences of Galton{Watson branching processes with unobservable immigration, where the mean of the offspring distribution converges to 1 (which is the critical value), are discussed

Measuring Information Transfer

An information theoretic measure is derived that quantifies the statistical coherence between systems evolving in time. The standard time delayed mutual information fails to distinguish information that is actually exchanged from shared information due to common history and input signals. In our new approach, these influences are excluded by appropriate conditioning of transition probabilities. The resulting transfer entropy is able to distinguish driving and responding elements and to detect asymmetry in the coupling of subsystems.Comment: 4 pages, 4 Figures, Revte