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Random feature neural network approximations of the potential in Hamiltonian systems yield approximations of molecular dynamics correlation observables that have the expected error $\mathcal{O}\big((K^{-1}+J^{-1/2})^{\frac{1}{2}}\big)$, for networks with $K$ nodes using $J$ data points, provided the Hessians of the potential and the observables are bounded. The loss function is based on the least squares error of the potential and regularizations, with the data points sampled from the Gibbs density. The proof uses an elementary new derivation of the generalization error for random feature networks that does not apply the Rademacher or related complexities., Comment: 28 page, 9 figures

We develop a Schwinger-Keldysh field theory (SKFT) for open quantum systems interacting with a dissipative environment and apply it to the spin-boson model as an archetypical example where the environment is composed of a bosonic bath. Prior SKFT developments of this type have been confined to the Markovian regime, as an alternative to a conventional description by the Lindblad quantum master equation (QME) which is a time-local matrix differential equation. Here we combine SKFT with a two-particle irreducible (2PI) action that resums a class of Feynman diagrams to infinite order. We obtain the time-evolution of the spin density matrix in the form of a system of integro-differential equations applicable to both Markovian and non-Markovian regimes. The latter regime--where taking into account memory effects becomes essential--poses a challenge for standard methods when trying to incorporate arbitrary properties of the system, bath, and length of time evolution. The SKFT+2PI-computed time evolution of the spin expectation values in the Markovian regime reproduces the solution of the Lindblad QME, as long as the system-bath coupling in the latter is adjusted by increasing it. In the non-Markovian regime, SKFT+2PI yields a nonperturbative solution that mimics results from both hierarchical equations of motion and tensor networks methods that we employ as benchmarks. Our SKFT+2PI approach can also access challenging cases, such as zero-temperature and sub-Ohmic bath, as well as arbitrary long evolution times. Taking into account favorable numerical cost of solving the integro-differential equations with increasing number of spins, time steps or dimensionality the SKFT+2PI approach offers a promising route for simulation of driven-dissipative systems in quantum computing or quantum magnonics and spintronics in the presence of a variety of (single or multiple) dissipative environments., Comment: 14 pages, 7 figures

All poetic forms come from somewhere. Prosodic templates can be copied for generations, altered by individuals, imported from foreign traditions, or fundamentally changed under the pressures of language evolution. Yet these relationships are notoriously difficult to trace across languages and times. This paper introduces an unsupervised method for detecting structural similarities in poems using local sequence alignment. The method relies on encoding poetic texts as strings of prosodic features using a four-letter alphabet; these sequences are then aligned to derive a distance measure based on weighted symbol (mis)matches. Local alignment allows poems to be clustered according to emergent properties of their underlying prosodic patterns. We evaluate method performance on a meter recognition tasks against strong baselines and show its potential for cross-lingual and historical research using three short case studies: 1) mutations in quantitative meter in classical Latin, 2) European diffusion of the Renaissance hendecasyllable, and 3) comparative alignment of modern meters in 18--19th century Czech, German and Russian. We release an implementation of the algorithm as a Python package with an open license.

We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDE). We propose a novel probabilistic formulation of the operator learning problem based on recently developed generative denoising diffusion probabilistic models (DDPM) in order to learn the input-to-output mapping between problem parameters and solutions of the PDE. To achieve this goal we modify DDPM to supervised learning in which the solution operator for the PDE is represented by a class of conditional distributions. The probabilistic formulation combined with DDPM allows for an automatic quantification of confidence intervals for the learned solutions. Furthermore, the framework is directly applicable for learning from a noisy data set. We compare computational performance of the developed method with the Fourier Network Operators (FNO). Our results show that our method achieves comparable accuracy and recovers the noise magnitude when applied to data sets with outputs corrupted by additive noise.

In addition to being a widely recognised novelist, Milan Kundera has also authored three pieces for theatre: The Owners of the Keys (Majitel\'e kl\'i\v{c}\r{u}, 1961), The Blunder (Pt\'akovina, 1967), and Jacques and his Master (Jakub a jeho p\'an, 1971). In recent years, however, the hypothesis has been raised that Kundera is the true author of a fourth play: Juro J\'ano\v{s}\'ik, first performed in a 1974 production under the name of Karel Steigerwald, who was Kundera's student at the time. In this study, we make use of supervised machine learning to settle the question of authorship attribution in the case of Juro J\'ano\v{s}\'ik, with results strongly supporting the hypothesis of Kundera's authorship.

This paper concerns the Time-Domain Full Waveform Inversion (FWI) for dispersive and dissipative poroelastic materials. The forward problem is an initial boundary value problem (IBVP) of the poroelastic equations with a memory term; the FWI is formulated as a minimization problem of a least-square misfit function with the (IBVP) as the constraint. In this paper, we derive the adjoint problem of this minimization problem, whose solution can be applied to computed the direction of steepest descent in the iterative process for minimization. The adjoint problem has a similar numerical structure as the forward problem and hence can be solved by the same numerical solver. Because the tracking of the energy evolution plays an important role in the FWI for dissipative and dispersive equations, the energy analysis of the forward system is also carried out in this paper., Comment: 14 pages

We introduce a new class of estimators for the linear response of steady states of stochastic dynamics. We generalize the likelihood ratio approach and formulate the linear response as a product of two martingales, hence the name "martingale product estimators". We present a systematic derivation of the martingale product estimator, and show how to construct such estimator so its bias is consistent with the weak order of the numerical scheme that approximates the underlying stochastic differential equation. Motivated by the estimation of transport properties in molecular systems, we present a rigorous numerical analysis of the bias and variance for these new estimators in the case of Langevin dynamics. We prove that the variance is uniformly bounded in time and derive a specific form of the estimator for second-order splitting schemes for Langevin dynamics. For comparison, we also study the bias and variance of a Green-Kubo estimator, motivated, in part, by its variance growing linearly in time. Presented analysis shows that the new martingale product estimators, having uniformly bounded variance in time, offer a competitive alternative to the traditional Green-Kubo estimator. We compare on illustrative numerical tests the new estimators with results obtained by the Green-Kubo method., Comment: 34 pages, 4 figures

Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be $\mathcal O(M^{-1})$, provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and $M$ is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain $\mathcal O(M^{-1})$ accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace $h:={\rm Tr}(H e^{-\beta H})/{\rm Tr}(e^{-\beta H})$ with respect to the electron degrees of freedom and $H$ is the Weyl symbol corresponding to a quantum many body Hamiltonian $\widehat{H}$. It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy $\mathcal O (M^{-1}+ t\epsilon^2)$, for correlation time $t$ where $\epsilon^2$ is related to the variance of mean value approximation $h$. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue., Comment: 50 pages, 13 figures

Recent advances in cultural analytics and large-scale computational studies of art, literature and film often show that long-term change in the features of artistic works happens gradually. These findings suggest that conservative forces that shape creative domains might be underestimated. To this end, we provide the first large-scale formal evidence of the persistent association between poetic meter and semantics in 18-19th European literatures, using Czech, German and Russian collections with additional data from English poetry and early modern Dutch songs. Our study traces this association through a series of clustering experiments using the abstracted semantic features of 150,000 poems. With the aid of topic modeling we infer semantic features for individual poems. Texts were also lexically simplified across collections to increase generalizability and decrease the sparseness of word frequency distributions. Topics alone enable recognition of the meters in each observed language, as may be seen from highly robust clustering of same-meter samples (median Adjusted Rand Index between 0.48 and 1). In addition, this study shows that the strength of the association between form and meaning tends to decrease over time. This may reflect a shift in aesthetic conventions between the 18th and 19th centuries as individual innovation was increasingly favored in literature. Despite this decline, it remains possible to recognize semantics of the meters from past or future, which suggests the continuity of semantic traditions while also revealing the historical variability of conditions across languages. This paper argues that distinct metrical forms, which are often copied in a language over centuries, also maintain long-term semantic inertia in poetry. Our findings, thus, highlight the role of the formal features of cultural items in influencing the pace and shape of cultural evolution.

This article points out some surprising similarities between a 1944 study by Georgy Udny Yule and modern approaches to authorship attribution., Comment: To appear in: Interf\'erences litteraires/literaire interferenties; forthcoming special issue on "Literature and/as (the) Digital"

Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell+1}=\bar z_\ell + \mathrm{Re}\sum_{k=1}^K\bar b_{\ell k}e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}+ \mathrm{Re}\sum_{k=1}^K\bar c_{\ell k}e^{\mathrm{i}\omega'_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(\omega_{\ell k},\omega'_{\ell k})$ of the random Fourier features $e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}$ and $e^{\mathrm{i}\omega'_{\ell k}\cdot x}$ is derived. This derivation is based on the corresponding generalization error for the approximation of the function values $f(x)$. The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1(\mathbb{R}^d)}}/{(KL)}$ of the generalization error for random Fourier features with one hidden layer and the same total number of nodes $KL$, in the case the $L^\infty$-norm of $f$ is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network. Promising performance of the proposed new algorithm is demonstrated in computational experiments.

The supervised learning problem to determine a neural network approximation $\mathbb{R}^d\ni x\mapsto\sum_{k=1}^K\hat\beta_k e^{{\mathrm{i}}\omega_k\cdot x}$ with one hidden layer is studied as a random Fourier features algorithm. The Fourier features, i.e., the frequencies $\omega_k\in\mathbb{R}^d$, are sampled using an adaptive Metropolis sampler. The Metropolis test accepts proposal frequencies $\omega_k'$, having corresponding amplitudes $\hat\beta_k'$, with the probability $\min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^\gamma\big\}$, for a certain positive parameter $\gamma$, determined by minimizing the approximation error for given computational work. This adaptive, non-parametric stochastic method leads asymptotically, as $K\to\infty$, to equidistributed amplitudes $|\hat\beta_k|$, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods. Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is tested both on synthetic data and a real-world high-dimensional benchmark.

Poetic traditions across languages evolved differently, but we find that certain semantic topics occur in several of them, albeit sometimes with temporal delay, or with diverging trajectories over time. We apply Latent Dirichlet Allocation (LDA) to poetry corpora of four languages, i.e. German (52k poems), English (85k poems), Russian (18k poems), and Czech (80k poems). We align and interpret salient topics, their trend over time (1600--1925 A.D.), showing similarities and disparities across poetic traditions with a few select topics, and use their trajectories over time to pinpoint specific literary epochs.

We introduce time-dependent density matrix renormalization group (tDMRG) as a solution to long standing problem in spintronics -- how to describe spin-transfer torque (STT) between flowing spins of conduction electrons and localized spins within a magnetic material by treating the dynamics of both spin species fully quantum-mechanically. In contrast to conventional Slonczewski-Berger STT, where the localized spins are viewed as classical vectors obeying the Landau-Lifshitz-Gilbert equation and where their STT-driven dynamics is initiated only when the spin-polarization of flowing electrons and localized spins are noncollinear, quantum STT can occur when these vectors are collinear but antiparallel. Using tDMRG, we simulate the time evolution of a many-body quantum state of electrons and localized spins, where the former are injected as a spin-polarized current pulse while the latter comprise a quantum Heisenberg ferromagnetic metallic (FM) spin-$\frac{1}{2}$ XXZ chain initially in the ground state with spin-polarization antiparallel to that of injected electrons. The quantum STT reverses the direction of localized spins, but without rotation from the initial orientation, when the number of injected electrons exceeds the number of localized spins. Such nonclassical reversal, which is absent from LLG dynamics, is strikingly inhomogeneous across the FM chain and it can be accompanied by reduction of the magnetization associated with localized spins, even to zero at specific locations. This is because quantum STT generates a highly entangled nonequilibrium many-body state of all flowing and localized spins, despite starting from the initially unentangled ground state of a mundane FM. Furthermore, the mutual information between localized spins at the FM edges remains nonzero even at infinite separation as the signature of dynamical buildup of long-range entanglement., Comment: 11 pages, 8 figures; four movie available from https://wiki.physics.udel.edu/qttg/Publications

The versified play Henry VIII is nowadays widely recognized to be a collaborative work not written solely by William Shakespeare. We employ combined analysis of vocabulary and versification together with machine learning techniques to determine which authors also took part in the writing of the play and what were their relative contributions. Unlike most previous studies, we go beyond the attribution of particular scenes and use the rolling attribution approach to determine the probabilities of authorship of pieces of texts, without respecting the scene boundaries. Our results highly support the canonical division of the play between William Shakespeare and John Fletcher proposed by James Spedding, but also bring new evidence supporting the modifications proposed later by Thomas Merriam.

In Nature Human Behaviour 3/2019, an article was published entitled "Large-scale quantitative profiling of the Old English verse tradition" dealing with (besides other things) the question of the authorship of the Old English poem Beowulf. The authors provide various textual measurements that they claim present "serious obstacles to those who would advocate for composite authorship or scribal recomposition" (p. 565). In what follows we raise doubts about their methods and address serious errors in both their data and their code. We show that reliable stylometric methods actually identify significant stylistic heterogeneity in Beowulf. In what follows we discuss each method separately following the order of the original article.

Graphene sandwiched between semiconducting monolayers of ferromagnet Cr$_2$Ge$_2$Te$_6$ and transition-metal dichalcogenide WS$_2$ acquires both spin-orbit (SO), of valley-Zeeman and Rashba types, and exchange couplings. Using first-principles combined with quantum transport calculations, we predict that such doubly proximitized graphene within van der Waals heterostructure will exhibit SO torque driven by unpolarized charge current. This system lacking spin Hall current, putatively considered to be necessary for efficient damping-like (DL) SO torque that plays a key role in magnetization switching, demonstrates how DL torque component can be generated solely by skew-scattering off spin-independent potential barrier or impurities in purely two-dimensional electronic transport due to the presence of proximity SO coupling and its spin texture tilted out-of-plane. This leads to current-driven nonequilibrium spin density emerging in all spatial directions, whose cross product with proximity magnetization yields DL SO torque, unlike the ballistic regime with no scatterers in which only field-like (FL) SO torque appears. In contrast to SO torque on conventional metallic ferromagnets in contact with three dimensional SO-coupled materials, the ratio of FL and DL torque can be tuned by more than an order of magnitude via combined top and back gates., Comment: 12 pages, 9 figures

We consider numerical schemes for computing the linear response of steady-state averages of stochastic dynamics with respect to a perturbation of the drift part of the stochastic differential equation. The schemes are based on Girsanov's change-of-measure theory to reweight trajectories with factors derived from a linearization of the Girsanov weights. We investigate both the discretization error and the finite time approximation error. The designed numerical schemes are shown to be of bounded variance with respect to the integration time, which is a desirable feature for long time simulation. We also show how the discretization error can be improved to second order accuracy in the time step by modifying the weight process in an appropriate way.

The recently discovered two-dimensional (2D) magnetic insulator CrI$_3$ is an intriguing case for basic research and spintronic applications since it is a ferromagnet in the bulk, but an antiferromagnet in bilayer form, with its magnetic ordering amenable to external manipulations. Using first-principles quantum transport approach, we predict that injecting unpolarized charge current parallel to the interface of bilayer-CrI$_3$/monolayer-TaSe$_2$ van der Waals heterostructure will induce spin-orbit torque (SOT) and thereby driven dynamics of magnetization on the first monolayer of CrI$_3$ in direct contact with TaSe$_2$. By combining calculated complex angular dependence of SOT with the Landau-Lifshitz-Gilbert equation for classical dynamics of magnetization, we demonstrate that current pulses can switch the direction of magnetization on the first monolayer to become parallel to that of the second monolayer, thereby converting CrI$_3$ from antiferromagnet to ferromagnet while not requiring any external magnetic field. We explain the mechanism of this reversible current-driven nonequilibrium phase transition by showing that first monolayer of CrI$_3$ carries current due to evanescent wavefunctions injected by metallic transition metal dichalcogenide TaSe$_2$, while concurrently acquiring strong spin-orbit coupling (SOC) via such proximity effect, whereas the second monolayer of CrI$_3$ remains insulating. The transition can be detected by passing vertical read current through the vdW heterostructure, encapsulated by bilayer of hexagonal boron nitride and sandwiched between graphite electrodes, where we find tunneling magnetoresistance of $\simeq 240$%., Comment: 8 pages, 6 figures, one movie (available from https://wiki.physics.udel.edu/qttg/Publications), PDFLaTeX

We not only reproduce burst of short-wavelength spin waves (SWs) observed in recent experiment [S. Woo et al., Nat. Phys. 13, 448 (2017)] on magnetic-field-driven annihilation of two magnetic domain walls (DWs) but, furthermore, we predict that this setup additionally generates highly unusual} pumping of electronic spin currents in the absence of any bias voltage. Prior to the instant of annihilation, their power spectrum is ultrabroadband, so they can be converted into rapidly changing in time charge currents, via the inverse spin Hall effect, as a source of THz radiation of bandwidth $\simeq 27$ THz where the lowest frequency is controlled by the applied magnetic field. The spin pumping stems from time-dependent fields introduced into the quantum Hamiltonian of electrons by the classical dynamics of localized magnetic moments (LMMs) comprising the domains. The pumped currents carry spin-polarized electrons which, in turn, exert backaction on LMMs in the form of nonlocal damping which is more than twice as large as conventional local Gilbert damping. The nonlocal damping can substantially modify the spectrum of emitted SWs when compared to widely-used micromagnetic simulations where conduction electrons are completely absent. Since we use fully microscopic (i.e., Hamiltonian-based) framework, self-consistently combining time-dependent electronic nonequilibrium Green functions with the Landau-Lifshitz-Gilbert equation, we also demonstrate that previously derived phenomenological formulas miss ultrabroadband spin pumping while underestimating the magnitude of nonlocal damping due to nonequilibrium electrons., Comment: 7 pages, 4 figures, Supplemental Material (one movie + one note available from https://wiki.physics.udel.edu/qttg/Publications)

We examine challenges to sampling from Boltzmann distributions associated with multiscale energy landscapes. The multiscale features, or "roughness," corresponds to highly oscillatory, but bounded, perturbations of a smooth landscape. Through a combination of numerical experiments and analysis we demonstrate that the performance of Metropolis Adjusted Langevin Algorithm can be severely attenuated as the roughness increases. In contrast, we prove that Random Walk Metropolis is insensitive to such roughness. We also formulate two alternative sampling strategies that incorporate large scale features of the energy landscape, while resisting the impact of fine scale roughness; these also outperform Random Walk Metropolis. Numerical experiments on these landscapes are presented that confirm our predictions. Open questions and numerical challenges are also highlighted., Comment: 34 pages, first revision

Multiscale modeling is a systematic approach to describe the behavior of complex systems by coupling models from different scales. The approach has been demonstrated to be very effective in areas of science as diverse as materials science, climate modeling and chemistry. However, routine use of multiscale simulations is often hindered by the very high cost of individual at-scale models. Approaches aiming to alleviate that cost by means of Gaussian process regression based surrogate models have been proposed. Yet, many of these surrogate models are expensive to construct, especially when the number of data needed is large. In this article, we employ a hierarchical sparse Cholesky decomposition to develop a sparse Gaussian process regression method and apply the method to approximate the equation of state of an energetic material in a multiscale model of dynamic deformation. We demonstrate that the method provides a substantial reduction both in computational cost and solution error as compared with previous methods.

The standard spin-transfer torque (STT)---where spin-polarized current drives dynamics of magnetization viewed as a classical vector---requires noncollinearity between electron spins carried by the current and magnetization of a ferromagnetic layer. However, recent experiments [A. Zholud et al., Phys. Rev. Lett. 119, 257201 (2017)] observing magnetization dynamics in spin valves at cryogenic temperatures, even when electron spin is collinear to magnetization, point at overlooked quantum effects in STT which can lead to highly nonclassical magnetization states. Using fully quantum many-body treatment, where an electron injected as spin-polarized wave packet interacts with local spins comprising the anisotropic quantum Heisenberg ferromagnetic chain, we define quantum STT as any time evolution of local spins due to initial many-body state not being an eigenstate of electron+local-spins system. For time evolution caused by injected spin-down electron scattering off local up-spins, entanglement between electron subsystem and local spins subsystem takes place leading to decoherence and, therefore, shrinking of the total magnetization but without rotation from its initial orientation which explains the experiments. Furthermore, the same processes---entanglement and thereby induced decoherence---are present also in standard noncollinear geometry, together with the usual magnetization rotation. This is because STT in quantum many-body picture is caused only by electron spin-down factor state, and the only difference between collinear and noncollinear geometries is in relative size of the contribution of initial many-body state containing such factor state to superpositions of separable many-body quantum states generated during time evolution., Comment: 6 pages, 4 figures; PDFLaTeX

In this paper we study Monte Carlo estimators based on the likelihood ratio approach for steady-state sensitivity. We first extend the result of Glynn and Olvera-Cravioto [doi:doi: 10.1287/stsy.2018.002] to the setting of continuous time Markov chains with a countable state space which include models such as stochastic reaction kinetics and kinetic Monte Carlo lattice system. We show that the variance of the centered likelihood ratio estimators does not grow in time. This result suggests that the centered likelihood ratio estimators should be favored for sensitivity analysis when the mixing time of the underlying continuous time Markov chain is large, which is typically the case when systems exhibit multi-scale behavior. We demonstrate a practical implication of this analysis on a numerical benchmark of two examples for the biochemical reaction networks.

We introduce a multiscale framework which combines time-dependent nonequilibrium Green function (TD-NEGF) algorithms, scaling linearly in the number of time steps and describing quantum-mechanically conduction electrons in the presence of time-dependent fields of arbitrary strength or frequency, with classical time evolution of localized magnetic moments described by the Landau-Lifshitz-Gilbert (LLG) equation. The TD-NEGF+LLG framework can be applied to a variety of problems where current-driven spin torque induces dynamics of magnetic moments as the key resource for next generation spintronics. Using magnetic domain wall (DW) as an example, we predict that its motion will pump time-dependent spin and charge currents (on the top of unpolarized DC charge current injected through normal metal leads to drive the DW motion). The conversion of AC components of spin current, whose amplitude increases (decreases) as the DW approaches (distances from) the normal metal lead, into AC voltage via the inverse spin Hall effect offers a tool to precisely track the DW position along magnetic nanowire. We also quantify the DW transient inertial displacement due to its acceleration and deceleration by pulse current and the entailed spin and charge pumping. Finally, TD-NEGF+LLG as a nonperturbative (i.e., numerically exact) framework allows us to establish the limits of validity of the so-called spin-motive force (SMF) theory for pumped charge current by time-dependent magnetic textures---the perturbative analytical formula of SMF theory becomes inapplicable for large frequencies (but unrealistic in magnetic system) and, more importantly, for increasing noncollinearity when the angles between neighboring magnetic moments exceed $\simeq 10^\circ$., Comment: 11 pages, 8 figures, PDFLaTeX; three additional movies of domain wall motion driven by injecting DC or pulse charge current are available from https://wiki.physics.udel.edu/qttg/Publications

We review a unified approach for computing: (i) spin-transfer torque in magnetic trilayers like spin-valves and magnetic tunnel junction, where injected charge current flows perpendicularly to interfaces; and (ii) spin-orbit torque in magnetic bilayers of the type ferromagnet/spin-orbit-coupled-material, where injected charge current flows parallel to the interface. Our approach requires to construct the torque operator for a given Hamiltonian of the device and the steady-state nonequilibrium density matrix, where the latter is expressed in terms of the nonequilibrium Green's functions and split into three contributions. Tracing these contributions with the torque operator automatically yields field-like and damping-like components of spin-transfer torque or spin-orbit torque vector, which is particularly advantageous for spin-orbit torque where the direction of these components depends on the unknown-in-advance orientation of the current-driven nonequilibrium spin density in the presence of spin-orbit coupling. We provide illustrative examples by computing spin-transfer torque in a one-dimensional toy model of a magnetic tunnel junction and realistic Co/Cu/Co spin-valve, both of which are described by first-principles Hamiltonians obtained from noncollinear density functional theory calculations; as well as spin-orbit torque in a ferromagnetic layer described by a tight-binding Hamiltonian which includes spin-orbit proximity effect within ferromagnetic monolayers assumed to be generated by the adjacent monolayer transition metal dichalcogenide., Comment: 22 pages, 9 figures, PDFLaTeX; prepared for Springer Handbook of Materials Modeling, Volume 2 Applications: Current and Emerging Materials

Stochastic reaction networks that exhibit bistability are common in many fields such as systems biology and materials science. Sampling of the stationary distribution is crucial for understanding and characterizing the long term dynamics of bistable stochastic dynamical systems. However, this is normally hindered by the insufficient sampling of the rare transitions between the two metastable regions. In this paper, we apply the parallel replica (ParRep) method for continuous time Markov chain to accelerate the stationary distribution sampling of bistable stochastic reaction networks. The proposed method uses parallel computing to accelerate the sampling of rare transitions and it is very easy to implement. We combine ParRep with the path space information bounds for parametric sensitivity analysis. We demonstrate the efficiency and accuracy of the method by studying the Schl\"{o}gl model and the genetic switches network., Comment: 7 figures

In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables., Comment: 31 pages. The solution of the nonlinear eigenvalue problem, previously based on the Cauchy-Kovalevskaya theorem, is replaced by an approximate solution in (5.3)

Metastable condensed matter typically fluctuates about local energy minima at the femtosecond time scale before transitioning between local minima after nanoseconds or microseconds. This vast scale separation limits the applicability of classical molecular dynamics methods and has spurned the development of a host of approximate algorithms. One recently proposed method is diffusive molecular dynamics which aims to integrate a system of ordinary differential equations describing the likelihood of occupancy by one of two species, in the case of a binary alloy, while quasistatically evolving the locations of the atoms. While diffusive molecular dynamics has shown to be efficient and provide agreement with observations, it is fundamentally a model, with unclear connections to classical molecular dynamics. In this work, we formulate a spin-diffusion stochastic process and show how it can be connected to diffusive molecular dynamics. The spin-diffusion model couples a classical overdamped Langevin equation to a kinetic Monte Carlo model for exchange amongst the species of a binary alloy. Under suitable assumptions and approximations, spin-diffusion can be shown to lead to diffusive molecular molecular dynamics type models. The key assumptions and approximations include a well defined time scale separation, a choice of spin exchange rates, a low temperature approximation, and a mean field type approximation. We derive several models from different assumptions and show their relationship to diffusive molecular dynamics. Differences and similarities amongst the models are explored in a simple test problem., Comment: 26 Pages, corrected a typo from the previous version

We propose a Multi-level Monte Carlo technique to accelerate Monte Carlo sampling for approximation of properties of materials with random defects. The computational efficiency is investigated on test problems given by tight-binding models of a single layer of graphene or of $MoS_2$ where the integrated electron density of states per unit area is taken as a representative quantity of interest. For the chosen test problems the multi-level Monte Carlo estimators significantly reduce the computational time of standard Monte Carlo estimators to obtain a given accuracy.

It is known that ab initio molecular dynamics based on the electron ground state eigenvalue can be used to approximate quantum observables in the canonical ensemble when the temperature is low compared to the first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature. The proof uses the semiclassical Weyl law to show that canonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The result covers observables that depend on time-correlations. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's law shows that the error estimate holds for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei position and the observables are in the diagonal form with respect to the electron eigenstates., Comment: 55 pages, 5 figures, Theorem 3.2 previously used a solution of the nonlinear eigenvalue problem (3.8) based on the Cauchy-Kovalevskaya theorem. It is replaced by an approximate solution in (3.9), derived in Section 3.1, which is also used in Theorems 3.6 and 3.7

We propose two algorithms for simulating continuous time Markov chains in the presence of metastability. We show that the algorithms correctly estimate, under the ergodicity assumption, stationary averages of the process. Both algorithms, based on the idea of the parallel replica method, use parallel computing in order to explore metastable sets more efficiently. The algorithms require no assumptions on the Markov chains beyond ergodicity and the presence of identifiable metastability. In particular, there is no assumption on reversibility. For simpler illustration of the algorithms, we assume that a synchronous architecture is used throughout of the paper. We present error analyses, as well as numerical simulations on multi-scale stochastic reaction network models in order to demonstrate consistency of the method and its efficiency.

In the presence of multiscale dynamics in a reaction network, direct simulation methods become inefficient as they can only advance the system on the smallest scale. This work presents stochastic averaging techniques to accelerate computations for obtaining estimates of expected values and sensitivities with respect to the steady state distribution. A two-time-scale formulation is used to establish bounds on the bias induced by the averaging method. Further, this formulation provides a framework to create an accelerated `averaged' version of most single-scale sensitivity estimation method. In particular, we propose a new lower-variance ergodic likelihood ratio type estimator for steady-state estimation and show how one can adapt it to accelerated simulations of multiscale systems.Lastly, we develop an adaptive "batch-means" stopping rule for determining when to terminate the micro-equilibration process., Comment: 20 pages, 6 figures, 2 tables, in this version: corrigendum of Proposition III.1 - only convergence is established, at present no rate available for general case

In this paper, we discuss information-theoretic tools for obtaining optimized coarse-grained molecular models for both equilibrium and non-equilibrium molecular dynamics. The latter are ubiquitous in physicochemical and biological applications, where they are typically associated with coupling mechanisms, multi-physics and/or boundary conditions. In general the non-equilibrium steady states are not known explicitly as they do not necessarily have a Gibbs structure. The presented approach can compare microscopic behavior of molecular systems to parametric and non-parametric coarse-grained one using the relative entropy between distributions on the path space and setting up a corresponding path space variational inference problem. The methods can become entirely data-driven when the microscopic dynamics are replaced with corresponding correlated data in the form of time series. Furthermore, we present connections and generalizations of force matching methods in coarse-graining with path-space information methods, as well as demonstrate the enhanced transferability of information-based parameterizations to general observables due to information inequalities. We further discuss methodological connections between information-based coarse-graining of molecular systems and variational inference methods primarily developed in the machine learning community. However, we note that the work presented here addresses variational inference for correlated time series due to the focus on dynamics. The applicability of the proposed methods is demonstrated on high-dimensional stochastic processes given by Langevin, overdamped and driven Langevin dynamics of interacting particles.

Using the probabilistic language of conditional expectations we reformulate the force matching method for coarse-graining of molecular systems as a projection on spaces of coarse observables. A practical outcome of this probabilistic description is the link of the force matching method with thermodynamic integration. This connection provides a way to systematically construct a local mean force in order to optimally approximate the potential of mean force through force matching. We introduce a generalized force matching condition for the local mean force in the sense that allows the approximation of the potential of mean force under both linear and non-linear coarse graining mappings (e.g., reaction coordinates, end-to-end length of chains). Furthermore, we study the equivalence of force matching with relative entropy minimization which we derive for general non-linear coarse graining maps. We present in detail the generalized force matching condition through applications to specific examples in molecular systems.

Uncertainty quantification is a primary challenge for reliable modeling and simulation of complex stochastic dynamics. Such problems are typically plagued with incomplete information that may enter as uncertainty in the model parameters, or even in the model itself. Furthermore, due to their dynamic nature, we need to assess the impact of these uncertainties on the transient and long-time behavior of the stochastic models and derive corresponding uncertainty bounds for observables of interest. A special class of such challenges is parametric uncertainties in the model and in particular sensitivity analysis along with the corresponding sensitivity bounds for stochastic dynamics. Moreover, sensitivity analysis can be further complicated in models with a high number of parameters that render straightforward approaches, such as gradient methods, impractical. In this paper, we derive uncertainty and sensitivity bounds for path-space observables of stochastic dynamics in terms of new goal-oriented divergences; the latter incorporate both observables and information theory objects such as the relative entropy rate. These bounds are tight, depend on the variance of the particular observable and are computable through Monte Carlo simulation. In the case of sensitivity analysis, the derived sensitivity bounds rely on the path Fisher Information Matrix, hence they depend only on local dynamics and are gradient-free. These features allow for computationally efficient implementation in systems with a high number of parameters, e.g., complex reaction networks and molecular simulations., Comment: 29 pages, 2 figures