Showing papers on "Configuration space published in 2009"

Differentially constrained mobile robot motion planning in state lattices

Abstract: We present an approach to the problem of differentially constrained mobile robot motion planning in arbitrary cost fields The approach is based on deterministic search in a specially discretized state space We compute a set of elementary motions that connects each discrete state value to a set of its reachable neighbors via feasible motions Thus, this set of motions induces a connected search graph The motions are carefully designed to terminate at discrete states, whose dimensions include relevant state variables (eg, position, heading, curvature, and velocity) The discrete states, and thus the motions, repeat at regular intervals, forming a lattice We ensure that all paths in the graph encode feasible motions via the imposition of continuity constraints on state variables at graph vertices and compliance of the graph edges with a differential equation comprising the vehicle model The resulting state lattice permits fast full configuration space cost evaluation and collision detection Experimental results with research prototype rovers demonstrate that the planner allows us to exploit the entire envelope of vehicle maneuverability in rough terrain, while featuring real-time performance © 2009 Wiley Periodicals, Inc

Manipulation planning on constraint manifolds

Abstract: We present the Constrained Bi-directional Rapidly-Exploring Random Tree (CBiRRT) algorithm for planning paths in configuration spaces with multiple constraints. This algorithm provides a general framework for handling a variety of constraints in manipulation planning including torque limits, constraints on the pose of an object held by a robot, and constraints for following workspace surfaces. CBiRRT extends the Bi-directional RRT (BiRRT) algorithm by using projection techniques to explore the configuration space manifolds that correspond to constraints and to find bridges between them. Consequently, CBiRRT can solve many problems that the BiRRT cannot, and only requires one additional parameter: the allowable error for meeting a constraint. We demonstrate the CBiRRT on a 7DOF WAM arm with a 4DOF Barrett hand on a mobile base. The planner allows this robot to perform household tasks, solve puzzles, and lift heavy objects.

Spatiotemporal state lattices for fast trajectory planning in dynamic on-road driving scenarios

Abstract: We present a method for motion planning in the presence of moving obstacles that is aimed at dynamic on-road driving scenarios. Planning is performed within a geometric graph that is established by sampling deterministically from a manifold that is obtained by combining configuration space and time. We show that these graphs are acyclic and shortest path algorithms with linear runtime can be employed. By reparametrising the configuration space to match the course of the road, it can be sampled very economically with few vertices, and this reduces absolute runtime further. The trajectories generated are quintic splines. They are second order continuous, obey nonholonomic constraints and are optimised for minimum square of jerk. Planning time remains below 20 ms on general purpose hardware.

How to quantify energy landscapes of solids.

Abstract: We explore whether the topology of energy landscapes in chemical systems obeys any rules and what these rules are. To answer this and related questions we use several tools: (i) Reduced energy surface and its density of states, (ii) descriptor of structure called fingerprint function, which can be represented as a one-dimensional function or a vector in abstract multidimensional space, (iii) definition of a “distance” between two structures enabling quantification of energy landscapes, (iv) definition of a degree of order of a structure, and (v) definitions of the quasi-entropy quantifying structural diversity. Our approach can be used for rationalizing large databases of crystal structures and for tuning computational algorithms for structure prediction. It enables quantitative and intuitive representations of energy landscapes and reappraisal of some of the traditional chemical notions and rules. Our analysis confirms the expectations that low-energy minima are clustered in compact regions of configuration space (“funnels”) and that chemical systems tend to have very few funnels, sometimes only one. This analysis can be applied to the physical properties of solids, opening new ways of discovering structure-property relations. We quantitatively demonstrate that crystals tend to adopt one of the few simplest structures consistent with their chemistry, providing a thermodynamic justification of Pauling’s fifth rule.

Local explicitly correlated coupled-cluster methods: Efficient removal of the basis set incompleteness and domain errors

Abstract: We propose an explicitly correlated local LCCSD-F12 method in which the basis set incompleteness error as well as the error caused by truncating the virtual orbital space to pair-specific local domains are strongly reduced. This is made possible by including explicitly correlated terms that are orthogonalized only to the pair-specific configuration space. Thus, the contributions of excitations outside the domains are implicitly accounted for by the explicitly correlated terms. It is demonstrated for a set of 54 reactions that the reaction energies computed with the new LCCSD-F12 method and triple-zeta basis sets deviate by at most 2.5 kJ/mol from conventional CCSD complete basis set results (RMS: 0.6 kJ/mol). The local approximations should make it possible to achieve linear scaling of the computational cost with molecular size.

Localization Bounds for Multiparticle Systems

Abstract: We consider the spectral and dynamical properties of quantum systems of n particles on the lattice \({\mathbb{Z}^d}\) , of arbitrary dimension, with a Hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite-range interaction. Two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction. It is established here that for all n there are regimes of high disorder, and/or weak enough interactions, for which the system exhibits spectral and dynamical localization. The localization is expressed through bounds on the transition amplitudes, which are uniform in time and decay exponentially in the Hausdorff distance in the configuration space. The results are derived through the analysis of fractional moments of the n-particle Green function, and related bounds on the eigenfunction correlators.

Constrained density functional theory based configuration interaction improves the prediction of reaction barrier heights.

Abstract: In this work, a constrained density functional theory based configuration interaction approach (CDFT-CI) is applied to calculating transition state energies of chemical reactions that involve bond forming and breaking at the same time. At a given point along the reaction path, the configuration space is spanned by two diabaticlike configurations: reactant and product. Each configuration is constructed self-consistently with spin and charge constraints to maximally retain the identities of the reactants or the products. Finally, the total energy is obtained by diagonalizing an effective Hamiltonian constructed in the basis spanned by these two configurations. By design, this prescription does not affect the energies of the reactant or product species but will affect the energy at intermediate points along the reaction coordinate, most notably by modifying the reaction barrier height. When tested with a large set of reactions that include hydrogen transfer, heavy atom transfer, and nucleophilic substitution...

Probabilities in Quantum Cosmological Models: A Decoherent Histories Analysis Using a Complex Potential

Abstract: In the quantization of simple cosmological models (minisuperspace models) described by the Wheeler-DeWitt equation, an important step is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation. This gives sensible semiclassical results but lacks an underlying operator formalism. In this paper, we address the issue of constructing probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in nonrelativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the $S$ matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit, and are closely related to an intersection number operator. The definitions of class operators given here handle the key case in which the underlying classical system has multiple crossings of the boundaries of the regions of interest. We show that oscillatory WKB solutions to the Wheeler-DeWitt equation give approximate decoherence of histories, as do superpositions of WKB solutions, as long as the regions of configuration space are sufficiently large. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures. In brief, we exhibit the well-defined operator formalism underlying the usual heuristic interpretational methods in quantum cosmology.

Symplectic topology of Ma\~n\'e's critical values

Abstract: We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mane critical value c. Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and it is invariant under under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k>c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is non-zero for energy levels k>c but vanishes for k

Localized soft modes and the supercooled liquid's irreversible passage through its configuration space

Abstract: Using computer simulations, we show that the localized low frequency normal modes of a configuration in a supercooled liquid are strongly correlated with the irreversible structural reorganization of the particles within that configuration. Establishing this correlation constitutes the identification of the aspect of a configuration that determines the heterogeneity of the subsequent motion. We demonstrate that the spatial distribution of the summation over the soft local modes can persist in spite of particle reorganization that produces significant changes in individual modes. Along with spatial localization, the persistent influence of soft modes in particle relaxation results in anisotropy in the displacements of mobile particles over the time scale referred to as beta-relaxation.

Motion Compression using Principal Geodesics Analysis

Abstract: Due to the growing need for large quantities of human animation data in the entertainment industry, it has become a necessity to compress motion capture sequences in order to ease their storage and transmission. We present a novel, lossy compression method for human motion data that exploits both temporal and spatial coherence. Given one motion, we first approximate the poses manifold using Principal Geodesics Analysis (PGA) in the configuration space of the skeleton. We then search this approximate manifold for poses matching end-effectors constraints using an iterative minimization algorithm that allows for real-time, data-driven inverse kinematics. The compression is achieved by only storing the approximate manifold parametrization along with the end-effectors and root joint trajectories, also compressed, in the output data. We recover poses using the IK algorithm given the end-effectors trajectories. Our experimental results show that considerable compression rates can be obtained using our method, with few reconstruction and perceptual errors.

Mathematical aspects of consciousness theory

Abstract: This book discusses general mathematical ideas behind TGD inspired theory of consciousness PART I: New Physics And Mathematics Involved With TGD The Clifford algebra associated with point of configuration space ("world of classical worlds") decomposes to a direct integral of von Neumann algebras known as hyper-finite factors of type II1 This implies strong physical predictions and deep connections with conformal field theories, knot-, braid- and quantum groups, and topological quantum computation In TGD framework dark matter forms a hierarchy with levels characterized partially by the value of Planck constant labeling the pages of the book like structure formed by singular covering spaces of the imbedding space M4× CP2 glued together along four-dimensional back Particles at different pages are dark relative to each other since purely local interactions (vertices of Feynman diagram) involve only particles at the same page The phase transitions changing the value of Planck constant having interpretation as tunneling between different pages of the book would induce phase transitions of gel phases abundant in living matter Part II: TGD Universe as topological quantum computer The braids formed by magnetic flux tubes are ideal for the realization of topological quantum computations (tqc) Bio-systems are basic candidates for topological quantum computers In DNA as tqc vision nucleotides and lipids are connected by flux tubes and the flow of lipids induces tqc programs Part III: Categories, Number Theory And Consciousness Category theory could reflect the basic structures of conscious thought The comparison of the inherent generalized logics associated with categories to the Boolean logic naturally associated with the configuration space spinor fields is also of interest The notion of infinite prime was the first mathematical invention inspired by TGD inspired theory of consciousness The construction of infinite primes is very much analogous to a repeated second quantization of a super-symmetric arithmetic quantum field theory (with analogs of bound states included) Infinite primes form an infinite hierarchy and the physical realization of this hierarchy imply infinite hierarchy of conscious entities and that we represent only a single level in this hierarchy looking infinitesimal from the point of view of higher levels The notion of infinite rational predicts an infinite number of real units with infinitely rich number theoretical anatomy and single space-time point becomes a Platonia able to represent every quantum state of the entire Universe in its structure: kind of algebraic Brahman=Atman identity

The Theory of (Exclusively) Local Beables

Abstract: It is shown how, starting with the de Broglie - Bohm pilot-wave theory, one can construct a new theory of the sort envisioned by several of QM's founders: a Theory of Exclusively Local Beables (TELB). In particular, the usual quantum mechanical wave function (a function on a high-dimensional configuration space) is not among the beables posited by the new theory. Instead, each particle has an associated ``pilot-wave'' field (living in physical space). A number of additional fields (also fields on physical space) maintain what is described, in ordinary quantum theory, as ``entanglement.'' The theory allows some interesting new perspective on the kind of causation involved in pilot-wave theories in general. And it provides also a concrete example of an empirically viable quantum theory in whose formulation the wave function (on configuration space) does not appear -- i.e., it is a theory according to which nothing corresponding to the configuration space wave function need actually exist. That is the theory's \emph{raison d'etre} and perhaps its only virtue. Its vices include the fact that it only reproduces the empirical predictions of the ordinary pilot-wave theory (equivalent, of course, to the predictions of ordinary quantum theory) for spinless non-relativistic particles, and only then for wave functions that are everywhere analytic. The goal is thus not to recommend the TELB proposed here as a replacement for ordinary pilot-wave theory (or ordinary quantum theory), but is rather to illustrate (with a crude first stab) that it might be possible to construct a plausible, empirically viable TELB, and to recommend this as an interesting and perhaps-fruitful program for future research.

Symplectic topology of Mañé's critical values

Abstract: We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mane critical value c . Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and that it is invariant under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k > c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is nonzero for energy levels k > c but vanishes for k < c , so levels above and below c cannot be connected by a stable tame homotopy. Moreover, we show that for strictly 1=4‐pinched negative curvature and nonexact magnetic fields all sufficiently high energy levels are nonstable, provided that the dimension of the base manifold is even and different from two. 53D40; 37D40

de Broglie-Bohm guidance equations for arbitrary Hamiltonians

Abstract: In a pilot-wave theory, an individual closed system is described by a wavefunction ψ(q) and configuration q. The evolution of the wavefunction and configuration are respectively determined by the Schrodinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density |ψ(q)|2. We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schrodinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space.

Molecular dissociation of hydrogen peroxide (HOOH) on a neural network ab initio potential surface with a new configuration sampling method involving gradient fitting

Abstract: The O–O bond dissociation of HOOH is investigated on an analytic ab initio potential-energy surface obtained by fitting the energies of 25 608 configurations using neural network (NN) methods. The electronic structure calculations are executed using MP2 calculations with the 6-31G∗ basis set. A new data-sampling technique is introduced to collect HOOH configurations in the six-dimensional hyperspace. This method is based on a comparison of the NN-computed gradients at configuration points currently in the database with the target gradients. By requiring that the NN gradients closely fit the MP2 target gradients, both the potential and the gradients are more accurately fitted. The selection criteria also ensure a more uniform distribution of configuration points throughout the important regions of configuration space. Molecular dynamics (MD) trajectories are not involved in the sampling. The final NN fitting yields average absolute and root-mean-squared testing set errors of 0.0060 eV (0.58 kJ mol−1) and 0...

Control of Probabilistic Diffusion in Motion Planning

Abstract: The paper presents a method to control probabilistic diffusion in motion planning algorithms. The principle of the method is to use on line the results of a diffusion algorithm to describe the free space in which the planning takes place. Given that description, it makes the diffusion go faster in favoured directions. That way, if the free space appears as a small volume around a submanifold of a highly dimensioned configuration space, the method overcomes the usual limitations of diffusion algorithms and finds a solution quickly. The presented method is theoretically analyzed and experimentally compared to known motion planning algorithms.

Interpolating moving least-squares methods for fitting potential energy surfaces: using classical trajectories to explore configuration space.

Abstract: We develop two approaches for growing a fitted potential energy surface (PES) by the interpolating moving least-squares (IMLS) technique using classical trajectories. We illustrate both approaches by calculating nitrous acid (HONO) cis-->trans isomerization trajectories under the control of ab initio forces from low-level HF/cc-pVDZ electronic structure calculations. In this illustrative example, as few as 300 ab initio energy/gradient calculations are required to converge the isomerization rate constant at a fixed energy to approximately 10%. Neither approach requires any preliminary electronic structure calculations or initial approximate representation of the PES (beyond information required for trajectory initial conditions). Hessians are not required. Both approaches rely on the fitting error estimation properties of IMLS fits. The first approach, called IMLS-accelerated direct dynamics, propagates individual trajectories directly with no preliminary exploratory trajectories. The PES is grown "on the fly" with the computation of new ab initio data only when a fitting error estimate exceeds a prescribed tight tolerance. The second approach, called dynamics-driven IMLS fitting, uses relatively inexpensive exploratory trajectories to both determine and fit the dynamically accessible configuration space. Once exploratory trajectories no longer find configurations with fitting error estimates higher than the designated accuracy, the IMLS fit is considered to be complete and usable in classical trajectory calculations or other applications.

A Gauge Model for Quantum Mechanics on a Stratified Space

Abstract: In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified Kahler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure yields a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is SU(2), we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.

Potential field guide for humanoid multicontacts acyclic motion planning

Abstract: We present a motion planning algorithm that computes rough trajectories used by a contact-points planner as a guide to grow its search graph. We adapt collision-free motion planning algorithms to plan a path within the guide space, a submanifold of the configuration space included in the free space in which the configurations are subject to static stability constraint. We first discuss the definition of the guide space. Then we detail the different techniques and ideas involved: relevant C-space sampling for humanoid robot, task-driven projection process, static stability test based on polyhedral convex cones theory's double description method. We finally present results from our implementation of the algorithm.

Harmonic oscillator in a background magnetic field in noncommutative quantum phase-space

Abstract: We solve explicitly the two-dimensional harmonic oscillator and the harmonic oscillator in a background magnetic field in noncommutative phase-space without making use of any type of representation. A key observation that we make is that for a specific choice of the noncommutative parameters, the time-reversal symmetry of the systems get restored since the energy spectrum becomes degenerate. This is in contrast to the noncommutative configuration space where the time-reversal symmetry of the harmonic oscillator is always broken.

Understanding interference experiments with polarized light through photon trajectories

Abstract: Bohmian mechanics allows to visualize and understand the quantum-mechanical behavior of massive particles in terms of trajectories. As shown by Bialynicki-Birula, Electromagnetism also admits a hydrodynamical formulation when the existence of a wave function for photons (properly defined) is assumed. This formulation thus provides an alternative interpretation of optical phenomena in terms of photon trajectories, whose flow yields a pictorial view of the evolution of the electromagnetic energy density in configuration space. This trajectory-based theoretical framework is considered here to study and analyze the outcome from Young-type diffraction experiments within the context of the Arago-Fresnel laws. More specifically, photon trajectories in the region behind the two slits are obtained in the case where the slits are illuminated by a polarized monochromatic plane wave. Expressions to determine electromagnetic energy flow lines and photon trajectories within this scenario are provided, as well as a procedure to compute them in the particular case of gratings totally transparent inside the slits and completely absorbing outside them. As is shown, the electromagnetic energy flow lines obtained allow to monitor at each point of space the behavior of the electromagnetic energy flow and, therefore, to evaluate the effects caused on it by the presence (right behind each slit) of polarizers with the same or different polarization axes. This leads to a trajectory-based picture of the Arago-Fresnel laws for the interference of polarized light.

Single surface beyond Born–Oppenheimer equation for a three-state model Hamiltonian of Na3 cluster

Abstract: When a set of three states is coupled with each other but shows negligibly weak interaction with other states of the Hilbert space, these states form a sub-Hilbert space. In case of such subspace [J. Chem. Phys. 124, 074101 (2006)], (a) the adiabatic-diabatic transformation (ADT) condition, ∇A+τA=0 [Chem. Phys. Lett. 35, 112 (1975)], provides the explicit forms of the nonadiabatic coupling (NAC) elements in terms of electronic basis function angles, namely, the ADT angles, and (b) those NAC terms satisfy the so-called curl conditions [Chem. Phys. Lett. 35, 112 (1975)], which ensure the removal of the NAC elements [could be singular also at specific point(s) or along a seam in the configuration space] during the ADT to bring the diabatic representation of the nuclear Schrodinger equation with a smooth functional form of coupling elements among the electronic states. Since the diabatic to adiabatic representation of the Hamiltonian is related through the same unitary transformation (∇A+τA=0), it could be ...

Invariant manifolds and the response of spiral arms in barred galaxies

Abstract: The unstable invariant manifolds of the short-period family of periodic orbits around the unstable Lagrangian points L1 and L2 of a barred galaxy define loci in the configuration space which tak e the form of a trailing spiral pattern. In previous works we have explored the association of such a pattern to the observed spiral patt ern in N-body models of barred-spiral galaxies and found it to be quite relevant. Our aims in the present paper are: a) to investigat e this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. b) to examine the dynamical role played by each of the non-axisymmetric components of the potential, i.e. the bar and the spiral perturbation, and their consequences on the form of the invariant manifolds, and c) to examine the relation of ‘r esponse’ models of barred-spiral galaxies with the theory o f the invariant manifolds. Our method relies on calculating the invariant manifolds for values of the Jacobi constant close to its value for L1 and L2. Our main results are the following: a) The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term tur ned on. We provide a theoretical model explaining the form of the invariant manifolds that supports the spiral structure. The azimuthal displacement of the Lagrangian points with respect to the bar’s major axis is a crucial parameter in this modeling. When this is taken into account, the manifolds necessarily develop in a spiral-l ike domain of the configuration space, delimited from below by the bound ary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. On the contrary, if the whole non-axisymmetric perturbation is artificially ‘aligned’ with the bar (i.e. there is no azimuthal shift of th e Lagrangian manifolds), the manifolds support a ring rather than a spiral structure. b) We construct ‘spiral response’ models on the b asis of the theory of the invariant manifolds and examine the connection of the latter to the ‘response’ models (Patsis 2006) used to fi t real barred-spiral galaxies, explaining how are the manif olds related to a number of morphological features seen in such models.

Triangleland: I. Classical dynamics with exchange of relative angular momentum

Abstract: In Euclidean relational particle mechanics, only relative times, relative angles and relative separations are meaningful. Barbour?Bertotti (1982 Proc. R. Soc. Lond. A 382 295) theory is of this form and can be viewed as a recovery of (a portion of) Newtonian mechanics from relational premises. This is of interest in the absolute versus relative motion debate and also shares a number of features with the geometrodynamical formulation of general relativity, making it suitable for some modelling of the problem of time in quantum gravity. I also study similarity relational particle mechanics ('dynamics of pure shape'), in which only relative times, relative angles and ratios of relative separations are meaningful. This I consider first as it is simpler, particularly in 1 and 2D, for which the configuration space geometry turns out to be well known, e.g. for the 'triangleland' (3-particle) case that I consider in detail. Second, the similarity model occurs as a sub-model within the Euclidean model: that admits a shape-scale split. For harmonic oscillator like potentials, similarity triangleland model turns out to have the same mathematics as a family of rigid rotor problems, while the Euclidean case turns out to have parallels with the Kepler?Coulomb problem in spherical and parabolic coordinates. Previous work on relational mechanics covered cases where the constituent subsystems do not exchange relative angular momentum, which is a simplifying (but in some ways undesirable) feature paralleling centrality in ordinary mechanics. In this paper I lift this restriction. In each case I reduce the relational problem to a standard one, thus obtain various exact, asymptotic and numerical solutions, and then recast these into the original mechanical variables for physical interpretation.

Comparison of three enveloping distribution sampling Hamiltonians for the estimation of multiple free energy differences from a single simulation

Abstract: We test the performance of three different reference state Hamiltonians for enveloping distribution sampling (EDS). EDS is an implementation of umbrella sampling which allows estimation of various free energy differences “on the fly” from a single simulation. This is achieved by construction of a reference state, which envelopes the regions of configuration space important to the various end states of interest. The proposed Hamiltonians differ in the way energy barriers separating these regions of configuration space are reduced. The test system consisted of 17 disubstituted benzenes in water and in complex with α-cyclodextrin. The calculated free energy differences correlate with thermodynamic integration results (R2 > 0.99 for the ligands in water and R2 > 0.98 for the ligands in complex with α-cyclodextrin). One of the reference state Hamiltonians outperformed the others in sampling the configuration space important to the various end states. In this reference state not all barriers between all pairs of states are reduced. Instead a minimum spanning tree of states is calculated, which connects states that are “closest” in configuration space. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009

Harmonic oscillator in a background magnetic field in noncommutative quantum phase-space

Abstract: We solve explicitly the two-dimensional harmonic oscillator and the harmonic oscillator in a background magnetic field in noncommutative phase-space without making use of any type of representation. A key observation that we make is that for a specific choice of the noncommutative parameters, the time reversal symmetry of the systems get restored since the energy spectrum becomes degenerate. This is in contrast to the noncommutative configuration space where the time reversal symmetry of the harmonic oscillator is always broken.