Improving the efficiency of variational tensor network algorithms

We present several results relating to the contraction of generic tensor networks and discuss their application to the simulation of quantum many-body systems using variational approaches based upon tensor network states. Given a closed tensor network $\mathcal{T}$, we prove that if the environment of a single tensor from the network can be evaluated with computational cost $\kappa$, then the environment of any other tensor from $\mathcal{T}$ can be evaluated with identical cost $\kappa$. Moreover, we describe how the set of all single tensor environments from $\mathcal{T}$ can be simultaneously evaluated with fixed cost $3\kappa$. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for the ground state of a 1D quantum system, where they are shown to substantially reduce the computation time.Comment: 12 pages, 8 figures, RevTex 4.1, includes reference implementation. Software updated to v1.02: Resolved two scenarios in which multienv would generate errors for valid input

Thermomechanical behavior of plasma-sprayed ZrO2-Y2O3 coatings influenced by plasticity, creep and oxidation

Thermocycling of ceramic-coated turbomachine components produces high thermomechanical stresses that are mitigated by plasticity and creep but aggravated by oxidation, with residual stresses exacerbated by all three. These residual stresses, coupled with the thermocyclic loading, lead to high compressive stresses that cause the coating to spall. A ceramic-coated gas path seal is modeled with consideration given to creep, plasticity, and oxidation. The resulting stresses and possible failure modes are discussed

FMRI Clustering and False Positive Rates

Recently, Eklund et al. (2016) analyzed clustering methods in standard FMRI packages: AFNI (which we maintain), FSL, and SPM [1]. They claimed: 1) false positive rates (FPRs) in traditional approaches are greatly inflated, questioning the validity of "countless published fMRI studies"; 2) nonparametric methods produce valid, but slightly conservative, FPRs; 3) a common flawed assumption is that the spatial autocorrelation function (ACF) of FMRI noise is Gaussian-shaped; and 4) a 15-year-old bug in AFNI's 3dClustSim significantly contributed to producing "particularly high" FPRs compared to other software. We repeated simulations from [1] (Beijing-Zang data [2], see [3]), and comment on each point briefly.Comment: 3 pages, 1 figure. A Letter accepted in PNA

Complexity of classical dynamics of molecular systems. I. Methodology

Methods for the calculation of complexity have been investigated as a possible alternative for the analysis of the dynamics of molecular systems. “Computational mechanics” is the approach chosen to describe emergent behavior in molecular systems that evolve in time. A novel algorithm has been developed for symbolization of a continuous physical trajectory of a dynamic system. A method for calculating statistical complexity has been implemented and tested on representative systems. It is shown that the computational mechanics approach is suitable for analyzing the dynamic complexity of molecular systems and offers new insight into the process

Water network dynamics at the critical moment of a peptide's ß-turn formation:a molecular dynamics study

All-atom molecular dynamics simulations for a single molecule of Leu-Enkephalin in aqueous solution have been used to study the role of the water network during the formation of ß-turns. We give a detailed account of the intramolecular hydrogen bonding, the water-peptide hydrogen bonding, and the orientation and residence times of water molecules focusing on the short critical periods of transition to the stable ß-turns. These studies suggest that, when intramolecular hydrogen bonding between the first and fourth residue of the ß-turn is not present, the disruption of the water network and the establishment of water bridges constitute decisive factors in the formation and stability of the ß-turn. Finally, we provide possible explanations and mechanisms for the formations of different kinds of ß-turns

Complexity of classical dynamics of molecular systems. II. Finite statistical complexity of a water-Na+ system

The computational mechanics approach has been applied to the orientational behavior of water molecules in a molecular dynamics simulated water–Na + system. The distinctively different statistical complexity of water molecules in the bulk and in the first solvation shell of the ion is demonstrated. It is shown that the molecules undergo more complex orientational motion when surrounded by other water molecules compared to those constrained by the electric field of the ion. However the spatial coordinates of the oxygen atom shows the opposite complexity behavior in that complexity is higher for the solvation shell molecules. New information about the dynamics of water molecules in the solvation shell is provided that is additional to that given by traditional methods of analysis

Entanglement renormalization, scale invariance, and quantum criticality

The use of entanglement renormalization in the presence of scale invariance is investigated. We explain how to compute an accurate approximation of the critical ground state of a lattice model, and how to evaluate local observables, correlators and critical exponents. Our results unveil a precise connection between the multi-scale entanglement renormalization ansatz (MERA) and conformal field theory (CFT). Given a critical Hamiltonian on the lattice, this connection can be exploited to extract most of the conformal data of the CFT that describes the model in the continuum limit.Comment: 4 pages, 3 figures, RevTeX 4. Revised for greater clarit

Studies on the structure and reactivity of bicyclic systems

Factors controlling the regio- and stereo selectivity in small polycyclic molecules have been investigated by X-ray diffraction, reactions of model compounds and molecular mechanics calculations. The bicyclo[ 3.2.0]heptene system was investigated in detail. The products on addition of HOBr were isolated and characterised. Similarly, the analogous exo- and endo-epoxides were synthesised and reacted with hydrobromic acid. Product ratios were determined by g.l.c. and confirmed by isolated yields. The compounds displaying high selectivity have the possibility of a transannular 0...C=0 interaction which could stabilise the transition state leading to the major product. Investigations of these interactions showed them to be a consequence of the molecular conformation. The energies involved are not sufficient to affect the product ratios. One compound which displayed an enhanced 0...C=0 inter action due to the presence of two chlorines vicinal to the carbonyl crystallised as a 1:1 mixture of ring and chain forms in a single crystal. This is the first example of ring-chain tautomers (or reactant and product of a chemical reaction) being isolated in a crystal lattice. The ground state structures of seven compounds were determined by X-ray diffraction and the transition states were modelled by molecular-mechanics calculations. The stereoselectivity was explained by steric congestion around the double bond. X I . Regioselectivity depends on the preferred conforma tion. From the available data (n.m.r., X-ray structures, molecular-mechanics) the endo-envelope conformer of the bicyclo[ 3.2.0]heptane system is preferred to the exo-twist conformer. Assuming an antiperiplanar arrangement of the reacting species, this preferred conformation explains the observed regioselectivity. The exo-bromonium ions and epoxides display higher selectivity than their endo analogues. This has been explained by conformational and steric factors

Sensitivity of peptide conformational dynamics on clustering of a classical molecular dynamics trajectory

We investigate the sensitivity of a Markov model with states and transition probabilities obtained from clustering a molecular dynamics trajectory. We have examined a 500 ns molecular dynamics trajectory of the peptide valine-proline-alanine-leucine in explicit water. The sensitivity is quantified by varying the boundaries of the clusters and investigating the resulting variation in transition probabilities and the average transition time between states. In this way, we represent the effect of clustering using different clustering algorithms. It is found that in terms of the investigated quantities, the peptide dynamics described by the Markov model is sensitive to the clustering; in particular, the average transition times are found to vary up to 46%. Moreover, inclusion of nonphysical sparsely populated clusters can lead to serious errors of up to 814%. In the investigation, the time step used in the transition matrix is determined by the minimum time scale on which the system behaves approximately Markovian. This time step is found to be about 100 ps. It is concluded that the description of peptide dynamics with transition matrices should be performed with care, and that using standard clustering algorithms to obtain states and transition probabilities may not always produce reliable results

Calculating mean first passage times from Markov models of proteins

Assuming that the dynamics of peptides and proteins can be described by Markov transitions between different configurational states, we present a method which can calculate the mean first passage time (MFPT) between sets of initial and final states. The method is described in detail and differences between this and a method commonly employed [1] are explained. It is shown that the proposed method is (i) more general in allowing sets of final states, (ii) significantly faster, (iii) more accurate since it does not involve the calculation of infinite summations. Particular attention is given to the biologically important case of multiple final states