Showing papers on "Configuration space published in 2013"

Identification of slow molecular order parameters for Markov model construction

Abstract: A goal in the kinetic characterization of a macromolecular system is the description of its slow relaxation processes via (i) identification of the structural changes involved in these processes and (ii) estimation of the rates or timescales at which these slow processes occur. Most of the approaches to this task, including Markov models, master-equation models, and kinetic network models, start by discretizing the high-dimensional state space and then characterize relaxation processes in terms of the eigenvectors and eigenvalues of a discrete transition matrix. The practical success of such an approach depends very much on the ability to finely discretize the slow order parameters. How can this task be achieved in a high-dimensional configuration space without relying on subjective guesses of the slow order parameters? In this paper, we use the variational principle of conformation dynamics to derive an optimal way of identifying the "slow subspace" of a large set of prior order parameters - either generic internal coordinates or a user-defined set of parameters. Using a variational formulation of conformational dynamics, it is shown that an existing method-the time-lagged independent component analysis-provides the optional solution to this problem. In addition, optimal indicators-order parameters indicating the progress of the slow transitions and thus may serve as reaction coordinates-are readily identified. We demonstrate that the slow subspace is well suited to construct accurate kinetic models of two sets of molecular dynamics simulations, the 6-residue fluorescent peptide MR121-GSGSW and the 30-residue intrinsically disordered peptide kinase inducible domain (KID). The identified optimal indicators reveal the structural changes associated with the slow processes of the molecular system under analysis.

Double field theory: a pedagogical review

Abstract: Double field theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk–Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions and present a brief parcours on worldsheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review.

Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review

Abstract: We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetric ideally hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal knife-edge rolling point contact with the ground level. The mass distribution and geometry are otherwise arbitrary. This conservative non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrized by rates of frame lean, steer angle and rear wheel rotation. We construct the terms in the governing equations methodically for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equations by hand in two ways and also checked them against two nonlinear dynamics simulations. In the century-old literature, several sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternative formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as a check for general purpose dynamic programs. For the benchmark bicycles, we accurately calculate the eigenvalues (the roots of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-old result that this conservative system can have asymptotic stability.

Fast Marching Tree: a Fast Marching Sampling-Based Method for Optimal Motion Planning in Many Dimensions

Abstract: In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm (FMT*). The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM* and RRT*. The FMT* algorithm performs a "lazy" dynamic programming recursion on a predetermined number of probabilistically-drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds--the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order $O(n^{-1/d+\rho})$, where $n$ is the number of sampled points, $d$ is the dimension of the configuration space, and $\rho$ is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT*, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our theoretical and heuristic arguments by showing that FMT*, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive.

Hard-disk equation of state: first-order liquid-hexatic transition in two dimensions with three simulation methods.

Abstract: We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.

Motivic Amplitudes and Cluster Coordinates

Abstract: In this paper we study motivic amplitudes--objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf_n(P^3) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude A_{7,2} and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster X-coordinates on Conf_n(P^3). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to A_{7,2} being expressible in terms of classical polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster X-coordinates of a single algebra. In this respect it is similar to Abel's 5-term dilogarithm identity.

Identification of slow molecular order parameters for Markov model construction

Abstract: A goal in the kinetic characterization of a macromolecular system is the description of its slow relaxation processes, involving (i) identification of the structural changes involved in these processes, and (ii) estimation of the rates or timescales at which these slow processes occur. Most of the approaches to this task, including Markov models, Master-equation models, and kinetic network models, start by discretizing the high-dimensional state space and then characterize relaxation processes in terms of the eigenvectors and eigenvalues of a discrete transition matrix. The practical success of such an approach depends very much on the ability to finely discretize the slow order parameters. How can this task be achieved in a high-dimensional configuration space without relying on subjective guesses of the slow order parameters? In this paper, we use the variational principle of conformation dynamics to derive an optimal way of identifying the "slow subspace" of a large set of prior order parameters - either generic internal coordinates (distances and dihedral angles), or a user-defined set of parameters. It is shown that a method to identify this slow subspace exists in statistics: the time-lagged independent component analysis (TICA). Furthermore, optimal indicators-order parameters indicating the progress of the slow transitions and thus may serve as reaction coordinates-are readily identified. We demonstrate that the slow subspace is well suited to construct accurate kinetic models of two sets of molecular dynamics simulations, the 6-residue fluorescent peptide MR121-GSGSW and the 30-residue natively disordered peptide KID. The identified optimal indicators reveal the structural changes associated with the slow processes of the molecular system under analysis.

Metrics for measuring distances in configuration spaces.

Abstract: In order to characterize molecular structures we introduce configurational fingerprint vectors which are counterparts of quantities used experimentally to identify structures. The Euclidean distance between the configurational fingerprint vectors satisfies the properties of a metric and can therefore safely be used to measure dissimilarities between configurations in the high dimensional configuration space. In particular we show that these metrics are a perfect and computationally cheap replacement for the root-mean-square distance (RMSD) when one has to decide whether two noise contaminated configurations are identical or not. We introduce a Monte Carlo approach to obtain the global minimum of the RMSD between configurations, which is obtained from a global minimization over all translations, rotations, and permutations of atomic indices.

Path Planning Under Kinematic Constraints by Rapidly Exploring Manifolds

Abstract: The situation arising in path planning under kinematic constraints, where the valid configurations define a manifold embedded in the joint ambient space, can be seen as a limit case of the well-known narrow corridor problem. With kinematic constraints, the probability of obtaining a valid configuration by sampling in the joint ambient space is not low but null, which complicates the direct application of sampling-based path planners. This paper presents the AtlasRRT algorithm, which is a planner especially tailored for such constrained systems that builds on recently developed tools for higher-dimensional continuation. These tools provide procedures to define charts that locally parametrize a manifold and to coordinate the charts, forming an atlas that fully covers it. AtlasRRT simultaneously builds an atlas and a bidirectional rapidly exploring random tree (RRT), using the atlas to sample configurations and to grow the branches of the RRTs, and the RRTs to devise directions of expansion for the atlas. The efficiency of AtlasRRT is evaluated in several benchmarks involving high-dimensional manifolds embedded in large ambient spaces. The results show that the combined use of the atlas and the RRTs produces a more rapid exploration of the configuration space manifolds than existing approaches.

A fully variational spin-orbit coupled complete active space self-consistent field approach: Application to electron paramagnetic resonance g-tensors

Abstract: In this work, a relativistic version of the state-averaged complete active space self-consistent field method is developed (spin-orbit coupled state-averaged complete active space self-consistent field; CAS-SOC). The program follows a “one-step strategy” and treats the spin-orbit interaction (SOC) on the same footing as the electron-electron interaction. As opposed to other existing approaches, the program employs an intermediate coupling scheme in which spin and space symmetry adapted configuration space functions are allowed to interact via SOC. This adds to the transparency and computational efficiency of the procedure. The approach requires the utilization of complex-valued configuration interaction coefficients, but the molecular orbital coefficients can be kept real-valued without loss of generality. Hence, expensive arithmetic associated with evaluation of complex-valued transformed molecular integrals is completely avoided. In order to investigate the quality of the calculated wave function, we extended the method to the calculation of electronic g-tensors. As the SOC is already treated to all orders in the SA-CASSCF process, first order perturbation theory with the Zeeman operator is sufficient to accomplish this task. As a test-set, we calculated g-tensors of a set of diatomics, a set of d1 transition metal complexes MOX4n−, and a set of 5f1 actinide complexes AnX6n−. These calculations reveal that the effect of the wavefunction relaxation due to variation inclusion of SOC is of the same order of magnitude as the effect of inclusion of dynamic correlation and hence cannot be neglected for the accurate prediction of electronic g-tensors.

A new approach to simulating collisionless dark matter fluids

Abstract: Recently, we have shown how current cosmological N-body codes already follow the fine grained phase-space information of the dark matter fluid. Using a tetrahedral tesselation of the three-dimensional manifold that describes perfectly cold fluids in six-dimensional phase space, the phase-space distribution function can be followed throughout the simulation. This allows one to project the distribution function into configuration space to obtain highly accurate densities, velocities, and velocity dispersions. Here, we exploit this technique to show first steps on how to devise an improved particle-mesh technique. At its heart, the new method thus relies on a piecewise linear approximation of the phase space distribution function rather than the usual particle discretisation. We use pseudo-particles that approximate the masses of the tetrahedral cells up to quadrupolar order as the locations for cloud-in-cell (CIC) deposit instead of the particle locations themselves as in standard CIC deposit. We demonstrate that this modification already gives much improved stability and more accurate dynamics of the collisionless dark matter fluid at high force and low mass resolution. We demonstrate the validity and advantages of this method with various test problems as well as hot/warm-dark matter simulations which have been known to exhibit artificial fragmentation. This completely unphysical behaviour is much reduced in the new approach. The current limitations of our approach are discussed in detail and future improvements are outlined.

Double Field Theory: A Pedagogical Review

Abstract: Double Field Theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk-Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions, and present a brief parcours on world-sheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review.

"Sur une forme nouvelle des ´ equations de la M´ ecanique"

Abstract: We present in modern language the contents of the famous note published by Henri Poincare in 1901 "Sur une forme nouvelle des ´ equations de la Mecanique", in which he proves that, when a Lie algebra acts locally transitively on the config- uration space of a Lagrangian mechanical system, the well known Euler-Lagrange equations are equivalent to a new system of differential equations defined on the product of the configuration space with the Lie algebra. We wr ite these equations, called the Euler-Poincarequations, under an intrinsic form, without any reference to a particular system of local coordinates, and prove that t hey can be conveniently expressed in terms of the Legendre and momentum maps. We discuss the use of the Euler-Poincare equation for reduction (a procedure sometimes called Lagrangian re- duction by modern authors), and compare this procedure with the well known Hamil- tonian reduction procedure (formulated in modern terms in 1974 by J.E. Marsden and A. Weinstein). We explain how a break of symmetry in the phase space produces the appearance of a semi-direct product of groups.

Stochastic determination of effective Hamiltonian for the full configuration interaction solution of quasi-degenerate electronic states.

Abstract: We propose a novel quantum Monte Carlo method in configuration space, which stochastically samples the contribution from a large secondary space to the effective Hamiltonian in the energy dependent partitioning of Lowdin. The method treats quasi-degenerate electronic states on a target energy with bond dissociations and electronic excitations avoiding significant amount of the negative sign problem. The performance is tested with small model systems of H4 and N2 at various configurations with quasi-degeneracy.

Rapid exploration of configuration space with diffusion-map-directed molecular dynamics.

Abstract: The gap between the time scale of interesting behavior in macromolecular systems and that which our computational resources can afford often limits molecular dynamics (MD) from understanding experimental results and predicting what is inaccessible in experiments. In this paper, we introduce a new sampling scheme, named diffusion-map-directed MD (DM-d-MD), to rapidly explore molecular configuration space. The method uses a diffusion map to guide MD on the fly. DM-d-MD can be combined with other methods to reconstruct the equilibrium free energy, and here, we used umbrella sampling as an example. We present results from two systems: alanine dipeptide and alanine-12. In both systems, we gain tremendous speedup with respect to standard MD both in exploring the configuration space and reconstructing the equilibrium distribution. In particular, we obtain 3 orders of magnitude of speedup over standard MD in the exploration of the configurational space of alanine-12 at 300 K with DM-d-MD. The method is reaction c...

Lefschetz thimbles and stochastic quantization: Complex actions in the complex plane

Abstract: Lattice field theories with a complex action can be studied numerically by allowing a complexified configuration space to be explored. Here we compare the recently introduced formulation on a Lefschetz thimble with the result from stochastic quantisation (or complex Langevin dynamics) in the case of a simple model and contrast the distributions being sampled. We also study the role of the residual phase on the Lefschetz thimble.

Stochastic determination of effective Hamiltonian for the full configuration interaction solution of quasi-degenerate electronic states

Abstract: We propose a novel quantum Monte Carlo method in configuration space, which stochastically samples the contribution from a large secondary space to the effective Hamiltonian in the energy dependent partitioning of Lowdin. The method treats quasi-degenerate electronic states on a target energy with bond dissociations and electronic excitations avoiding significant amount of the negative sign problem. The performance is tested with small model systems of H$_4$ and N$_2$ at various configurations with quasi-degeneracy.

Spatial Continuum Models of Rods Undergoing Large Deformation and Inflation

Abstract: Quaternions without unity constraint are used as configuration variables for rotational degrees of freedom of Cosserat rod models, thereby naturally incorporating inflation as well as bending, twisting, extension, and shear deformations of elongate robotic manipulators. The configuration space becomes isomorphic to a subspace of 7-D real-valued functions; thus, an unconstrained local minimizer of total potential energy is a static equilibrium. The ensuing calculus of variations is automated by computer algebra to derive weak-form integral equations that are easily translated to a finite-element package for efficient computation using internal forces. Discontinuities in strain variables are handled in a numerically reliable way. Inextensible, unshearable rod models are derived simply by taking limits of corresponding stiffness parameters. The same procedure facilitates unified software code for both flexible and rigid segments. Simulation experiments with an inflating tube, a helical coil, and a magnetic catheter produce good-quality results and indicate that the computational effort of the proposed method is about two orders of magnitude less than common 3-D finite-element models of large deformation nonlinear elasticity.

Diffusion along the splitting/commitment probability reaction coordinate.

Abstract: The splitting or commitment probabilities of states in the region of configuration space that separates reactants and products play an important role in the theory of chemical reactions. Assuming that the splitting probability changes more slowly than any other coordinate, we project multidimensional diffusive dynamics onto it. The resulting one-dimensional diffusion equation is not exact because the assumed separation of time scales does not hold in general. Nevertheless, this equation has the remarkable property that it always predicts the exact value of the number of transitions between reactants and products per unit time at equilibrium and hence the exact reaction rate. In the special case of two deep basins separated by a harmonic saddle, this equation is equivalent to the one that describes diffusion along a coordinate perpendicular to the transition state, defined as the surface starting from which reactants and products are reached with equal probability.

Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods

Abstract: We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.

Going beyond “no-pair relativistic quantum chemistry”

Abstract: The current field of relativistic quantum chemistry (RQC) has been built upon the no-pair and no-retardation approximations. While retardation effects must be treated in a time-dependent manner through quantum electrodynamics (QED) and are hence outside RQC, the no-pair approximation (NPA) has to be removed from RQC for it has some fundamental defects. Both configuration space and Fock space formulations have been proposed in the literature to do this. However, the former is simply wrong, whereas the latter is still incomplete. To resolve the old problems pertinent to the NPA itself and new problems beyond the NPA, we propose here an effective many-body (EMB) QED approach that is in full accordance with standard methodologies of electronic structure. As a first application, the full second order energy E2 of a closed-shell many-electron system subject to the instantaneous Coulomb-Breit interaction is derived, both algebraically and diagrammatically. It is shown that the same E2 can be obtained by means of...

Configuration-Based Optimization for Six Degree-of-Freedom Haptic Rendering for Fine Manipulation

Abstract: Six-degree-of-freedom (6-DOF) haptic rendering for fine manipulation in narrow space is a challenging topic because of frequent constraint changes caused by small tool movement and the requirement to preserve the feel of fine-features of objects. In this paper, we introduce a configuration-based constrained optimization method for solving this rendering problem. We represent an object using a hierarchy of spheres, i.e., a sphere tree, which allows faster detection of multiple contacts/collisions among objects than polygonal mesh and facilitates contact constraint formulation. Given a moving graphic tool as the avatar of the haptic tool in the virtual environment, we compute its quasi-static motion by solving a configuration-based optimization. The constraints in the 6D configuration space of the graphic tool is obtained and updated through online mapping of the nonpenetration constraint between the spheres of the graphic tool and those of the other objects in the three-dimensional physical space, based on the result of collision detection. This problem is further modeled as a quadratic programming optimization and solved by the classic active-set methods. Our algorithm has been implemented and interfaced with a 6-DOF Phantom Premium 3.0. We demonstrate its performance in several benchmarks involving complex, multiregion contacts. The experimental results show both the high efficiency and stability of haptic rendering by our method for complex scenarios. Nonpenetration between the graphic tool and the object is maintained under frequent contact switches. Update rate of the simulation loop including optimization and constraint identification is maintained at about 1 kHz.

Topology-based representations for motion planning and generalization in dynamic environments with interactions

Abstract: Motion can be described in several alternative representations, including joint configuration or end-effector spaces, but also more complex topology-based representations that imply a change of Voronoi bias, metric or topology of the motion space Certain types of robot interaction problems, eg wrapping around an object, can suitably be described by so-called writhe and interaction mesh representations However, considering motion synthesis solely in a topology-based space is insufficient since it does not account for additional tasks and constraints in other representations In this paper, we propose methods to combine and exploit different representations for synthesis and generalization of motion in dynamic environments Our motion synthesis approach is formulated in the framework of optimal control as an approximate inference problem This allows for consistent combination of multiple representations (eg across task, end-effector and joint space) Motion generalization to novel situations and kinematics is similarly performed by projecting motion from topology-based to joint configuration space We demonstrate the benefit of our methods on problems where direct path finding in joint configuration space is extremely hard whereas local optimal control exploiting a representation with different topology can efficiently find optimal trajectories In real-world demonstrations, we highlight the benefits of using topology-based representations for online motion generalization in dynamic environments

Localised distributions and criteria for correctness in complex Langevin dynamics

Abstract: Complex Langevin dynamics can solve the sign problem appearing in numerical simulations of theories with a complex action. In order to justify the procedure, it is important to understand the properties of the real and positive distribution, which is effectively sampled during the stochastic process. In the context of a simple model, we study this distribution by solving the Fokker-Planck equation as well as by brute force and relate the results to the recently derived criteria for correctness. We demonstrate analytically that it is possible that the distribution has support in a strip in the complexified configuration space only, in which case correct results are expected.

Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space

Abstract: The main point of this paper is to examine a “hidden” dynamical symmetry connected with the conservation of Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of non-commutative quantum mechanics (NCQM). The basic features of NCQM will be introduced to the reader, the key one being the fact that the notion of a point, or a zero distance in the considered configuration space, is abandoned and replaced with a “fuzzy” structure in such a way that the rotational invariance is preserved. The main facts about the conservation of LRL vector in both classical and quantum theory will be reviewed. Finally, we will search for an analogy in the NCQM, provide our results and their comparison with the QM predictions. The key notions we are going to deal with are non-commutative space, Coulomb-Kepler problem, and symmetry.

Renormalization of Massless Feynman Amplitudes in Configuration Space

Abstract: A systematic study of recursive renormalization of Feynman amplitudes is carried out both in Euclidean and in Minkowski configuration space. For a massless quantum field theory (QFT) we use the technique of extending associate homogeneous distributions to complete the renormalization recursion. A homogeneous (Poincare covariant) amplitude is said to be convergent if it admits a (unique covariant) extension as a homogeneous distribution. For any amplitude without subdivergences - i.e. for a Feynman distribution that is homogeneous off the full (small) diagonal - we define a renormalization invariant residue. Its vanishing is a necessary and sufficient condition for the convergence of such an amplitude. It extends to arbitrary - not necessarily primitively divergent - Feynman amplitudes. This notion of convergence is finer than the usual power counting criterion and includes cancellation of divergences.

Noether symmetries in extended gravity quantum cosmology

Abstract: We summarize the use of Noether symmetries in Minisuperspace Quantum Cosmology. In particular, we consider minisuperspace models, showing that the existence of conserved quantities gives selection rules that allow to recover classical behaviors in cosmic evolution according to the so called Hartle criterion. Such a criterion selects correlated regions in the configuration space of dynamical variables whose meaning is related to the emergence of classical observable universes. Some minisuperspace models are worked out starting from Extended Gravity, in particular coming from scalar tensor, f(R) and f(T) theories. Exact cosmological solutions are derived.