Nonequilibrium static growing length scales in supercooled liquids on approaching the glass transition

The small wavenumber $k$ behavior of the structure factor $S(k)$ of overcompressed amorphous hard-sphere configurations was previously studied for a wide range of densities up to the maximally random jammed state, which can be viewed as a prototypical glassy state [A. Hopkins, F. H. Stillinger and S. Torquato, Phys. Rev. E, 86, 021505 (2012)]. It was found that a precursor to the glassy jammed state was evident long before the jamming density was reached as measured by a growing nonequilibrium length scale extracted from the volume integral of the direct correlation function $c(r)$, which becomes long-ranged as the critical jammed state is reached. The present study extends that work by investigating via computer simulations two different atomic models: the single-component Z2 Dzugutov potential in three dimensions and the binary-mixture Kob-Andersen potential in two dimensions. Consistent with the aforementioned hard-sphere study, we demonstrate that for both models a signature of the glass transition is apparent well before the transition temperature is reached as measured by the length scale determined from from the volume integral of the direct correlation function in the single-component case and a generalized direct correlation function in the binary-mixture case. The latter quantity is obtained from a generalized Orstein-Zernike integral equation for a certain decoration of the atomic point configuration. We also show that these growing length scales, which are a consequence of the long-range nature of the direct correlation functions, are intrinsically nonequilibrium in nature as determined by an index $X$ that is a measure of deviation from thermal equilibrium. It is also demonstrated that this nonequilibrium index, which increases upon supercooling, is correlated with a characteristic relaxation time scale.Comment: 26 pages, 14 figure

A new perturbative approach to the adiabatic approximation

A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of "adiabatic" evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well-known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to be not widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.Comment: 9 pages. Added references, discussion of previous results, expanded upon discussion of main result and application of i

Second best toll and capacity optimisation in network: solution algorithm and policy implications

This paper looks at the first and second-best jointly optimal toll and road capacity investment problems from both policy and technical oriented perspectives. On the technical side, the paper investigates the applicability of the constraint cutting algorithm for solving the second-best problem under elastic demand which is formulated as a bilevel programming problem. The approach is shown to perform well despite several problems encountered by our previous work in Shepherd and Sumalee (2004). The paper then applies the algorithm to a small sized network to investigate the policy implications of the first and second-best cases. This policy analysis demonstrates that the joint first best structure is to invest in the most direct routes while reducing capacities elsewhere. Whilst unrealistic this acts as a useful benchmark. The results also show that certain second best policies can achieve a high proportion of the first best benefits while in general generating a revenue surplus. We also show that unless costs of capacity are known to be low then second best tolls will be affected and so should be analysed in conjunction with investments in the network

Dynamical mechanism of atrial fibrillation: a topological approach

While spiral wave breakup has been implicated in the emergence of atrial fibrillation, its role in maintaining this complex type of cardiac arrhythmia is less clear. We used the Karma model of cardiac excitation to investigate the dynamical mechanisms that sustain atrial fibrillation once it has been established. The results of our numerical study show that spatiotemporally chaotic dynamics in this regime can be described as a dynamical equilibrium between topologically distinct types of transitions that increase or decrease the number of wavelets, in general agreement with the multiple wavelets hypothesis. Surprisingly, we found that the process of continuous excitation waves breaking up into discontinuous pieces plays no role whatsoever in maintaining spatiotemporal complexity. Instead this complexity is maintained as a dynamical balance between wave coalescence -- a unique, previously unidentified, topological process that increases the number of wavelets -- and wave collapse -- a different topological process that decreases their number.Comment: 15 pages, 14 figure

Reconstructing Cardiac Electrical Excitations from Optical Mapping Recordings

The reconstruction of electrical excitation patterns through the unobserved depth of the tissue is essential to realizing the potential of computational models in cardiac medicine. We have utilized experimental optical-mapping recordings of cardiac electrical excitation on the epicardial and endocardial surfaces of a canine ventricle as observations directing a local ensemble transform Kalman Filter (LETKF) data assimilation scheme. We demonstrate that the inclusion of explicit information about the stimulation protocol can marginally improve the confidence of the ensemble reconstruction and the reliability of the assimilation over time. Likewise, we consider the efficacy of stochastic modeling additions to the assimilation scheme in the context of experimentally derived observation sets. Approximation error is addressed at both the observation and modeling stages, through the uncertainty of observations and the specification of the model used in the assimilation ensemble. We find that perturbative modifications to the observations have marginal to deleterious effects on the accuracy and robustness of the state reconstruction. Further, we find that incorporating additional information from the observations into the model itself (in the case of stimulus and stochastic currents) has a marginal improvement on the reconstruction accuracy over a fully autonomous model, while complicating the model itself and thus introducing potential for new types of model error. That the inclusion of explicit modeling information has negligible to negative effects on the reconstruction implies the need for new avenues for optimization of data assimilation schemes applied to cardiac electrical excitation.Comment: main text: 18 pages, 10 figures; supplement: 5 pages, 9 figures, 2 movie

Novel Ground-State Crystals with Controlled Vacancy Concentrations: From Kagom\'{e} to Honeycomb to Stripes

We introduce a one-parameter family, $0 \leq H \leq 1$, of pair potential functions with a single relative energy minimum that stabilize a range of vacancy-riddled crystals as ground states. The "quintic potential" is a short-ranged, nonnegative pair potential with a single local minimum of height $H$ at unit distance and vanishes cubically at a distance of \rt. We have developed this potential to produce ground states with the symmetry of the triangular lattice while favoring the presence of vacancies. After an exhaustive search using various optimization and simulation methods, we believe that we have determined the ground states for all pressures, densities, and $0 \leq H \leq 1$. For specific areas below 3\rt/2, the ground states of the "quintic potential" include high-density and low-density triangular lattices, kagom\'{e} and honeycomb crystals, and stripes. We find that these ground states are mechanically stable but are difficult to self-assemble in computer simulations without defects. For specific areas above 3\rt/2, these systems have a ground-state phase diagram that corresponds to hard disks with radius \rt. For the special case of H=0, a broad range of ground states is available. Analysis of this case suggests that among many ground states, a high-density triangular lattice, low-density triangular lattice, and striped phases have the highest entropy for certain densities. The simplicity of this potential makes it an attractive candidate for experimental realization with application to the development of novel colloidal crystals or photonic materials.Comment: 25 pages, 11 figure

Application of regulatory sequence analysis and metabolic network analysis to the interpretation of gene expression data

We present two complementary approaches for the interpretation of clusters of co-regulated genes, such as those obtained from DNA chips and related methods. Starting from a cluster of genes with similar expression profiles, two basic questions can be asked: 1. Which mechanism is responsible for the coordinated transcriptional response of the genes? This question is approached by extracting motifs that are shared between the upstream sequences of these genes. The motifs extracted are putative cis-acting regulatory elements. 2. What is the physiological meaning for the cell to express together these genes? One way to answer the question is to search for potential metabolic pathways that could be catalyzed by the products of the genes. This can be done by selecting the genes from the cluster that code for enzymes, and trying to assemble the catalyzed reactions to form metabolic pathways. We present tools to answer these two questions, and we illustrate their use with selected examples in the yeast Saccharomyces cerevisiae. The tools are available on the web (http://ucmb.ulb.ac.be/bioinformatics/rsa-tools/; http://www.ebi.ac.uk/research/pfbp/; http://www.soi.city.ac.uk/~msch/)