15,736 research outputs found
The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
In this Letter we show that an arbitrarily good approximation to the
propagator e^{itH} for a 1D lattice of n quantum spins with hamiltonian H may
be obtained with polynomial computational resources in n and the error
\epsilon, and exponential resources in |t|. Our proof makes use of the finitely
correlated state/matrix product state formalism exploited by numerical
renormalisation group algorithms like the density matrix renormalisation group.
There are two immediate consequences of this result. The first is that the
Vidal's time-dependent density matrix renormalisation group will require only
polynomial resources to simulate 1D quantum spin systems for logarithmic |t|.
The second consequence is that continuous-time 1D quantum circuits with
logarithmic |t| can be simulated efficiently on a classical computer, despite
the fact that, after discretisation, such circuits are of polynomial depth.Comment: 4 pages, 2 figures. Simplified argumen
Exact relaxation in a class of non-equilibrium quantum lattice systems
A reasonable physical intuition in the study of interacting quantum systems
says that, independent of the initial state, the system will tend to
equilibrate. In this work we study a setting where relaxation to a steady state
is exact, namely for the Bose-Hubbard model where the system is quenched from a
Mott quantum phase to the strong superfluid regime. We find that the evolving
state locally relaxes to a steady state with maximum entropy constrained by
second moments, maximizing the entanglement, to a state which is different from
the thermal state of the new Hamiltonian. Remarkably, in the infinite system
limit this relaxation is true for all large times, and no time average is
necessary. For large but finite system size we give a time interval for which
the system locally "looks relaxed" up to a prescribed error. Our argument
includes a central limit theorem for harmonic systems and exploits the finite
speed of sound. Additionally, we show that for all periodic initial
configurations, reminiscent of charge density waves, the system relaxes
locally. We sketch experimentally accessible signatures in optical lattices as
well as implications for the foundations of quantum statistical mechanics.Comment: 8 pages, 3 figures, replaced with final versio
Review of the Supply of and Demand for Further Education in Scotland
These documents provide are an Executive Summary and Full Report of the background to, methodology for, and overall conclusions and recommendations of a review of the supply of and demand for Further Education (FE) provision in Scottish Further Education colleges in 2000. The review was commissioned by the Scottish Further Education Funding Council (SFEFC), and carried out between November 1999 and June 2000 by a team of researchers drawn from the Scottish Further Education Unit (SFEU), the Centre for Research in Lifelong Learning, Glasgow Caledonian University/University of Stirling, and the Applied Statistics Group, Napier University
Solving frustration-free spin systems
We identify a large class of quantum many-body systems that can be solved
exactly: natural frustration-free spin-1/2 nearest-neighbor Hamiltonians on
arbitrary lattices. We show that the entire ground state manifold of such
models can be found exactly by a tensor network of isometries acting on a space
locally isomorphic to the symmetric subspace. Thus, for this wide class of
models real-space renormalization can be made exact. Our findings also imply
that every such frustration-free spin model satisfies an area law for the
entanglement entropy of the ground state, establishing a novel large class of
models for which an area law is known. Finally, we show that our approach gives
rise to an ansatz class useful for the simulation of almost frustration-free
models in a simple fashion, outperforming mean field theory.Comment: 5 pages, 1 figur
Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues
The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to PDE-based continuum methods, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for non-crystalline materials, it may still be used as a first order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to 2-D cellular-scale models by assessing the mechanical behaviour of model biological tissues, including crystalline (honeycomb) and non-crystalline reference states. The numerical procedure consists in precribing an affine deformation on the boundary cells and computing the position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For centre-based models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete/continuous modelling, adaptation of atom-to-continuum (AtC) techniques to biological tissues and model classification
Colorectal Cancer Through Simulation and Experiment
Colorectal cancer has continued to generate a huge amount of research interest over several decades, forming a canonical example of tumourigenesis since its use in Fearon and Vogelstein’s linear model of genetic mutation. Over time, the field has witnessed a transition from solely experimental work to the inclusion of mathematical biology and computer-based modelling. The fusion of these disciplines has the potential to provide valuable insights into oncologic processes, but also presents the challenge of uniting many diverse perspectives. Furthermore, the cancer cell phenotype defined by the ‘Hallmarks of Cancer’ has been extended in recent times and provides an excellent basis for future research. We present a timely summary of the literature relating to colorectal cancer, addressing the traditional experimental findings, summarising the key mathematical and computational approaches, and emphasising the role of the Hallmarks in current and future developments. We conclude with a discussion of interdisciplinary work, outlining areas of experimental interest which would benefit from the insight that mathematical and computational modelling can provide
Bounds on Information Propagation in Disordered Quantum Spin Chains
We investigate the propagation of information through the disordered XY
model. We find, with a probability that increases with the size of the system,
that all correlations, both classical and quantum, are suppressed outside of an
effective lightcone whose radius grows at most polylogarithmically with |t|.Comment: 4 pages, pdflatex, 1 pdf figure. Corrected the bound for the
localised propagator and quantified the probability it bound occur
Solute transport within porous biofilms: diffusion or dispersion?
Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behaviour by controllling nutrient supply, evacuation of waste products and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilmscale. We show that solute transport may be described via two coupled partial differential equations for the averaged concentrations, or telegrapher’s equations. These models are particularly relevant for chemical species, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterised by a second-order tensor whose components depend on: (1) the topology of the channels’ network; (2) the solute’s diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion-dominated, this analysis shows that dispersion effects may significantly contribute to transport
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