Constrained Hybrid Monte Carlo algorithms for gauge-Higgs models

We develop Hybrid Monte Carlo (HMC) algorithms for constrained Hamiltonian systems of gauge- Higgs models and introduce a new observable for the constraint effective Higgs potential. We use an extension of the so-called Rattle algorithm to general Hamiltonians for constrained systems, which we adapt to the 4D Abelian-Higgs model and the 5D SU(2) gauge theory on the torus and on the orbifold. The derivative of the potential is measured via the expectation value of the Lagrange multiplier for the constraint condition and allows a much more precise determination of the effective potential than conventional histogram methods. With the new method, we can access the potential over the full domain of the Higgs variable, while the histogram method is restricted to a short region around the expectation value of the Higgs field in unconstrained simulations, and the statistical precision does not deteriorate when the volume is increased. We further verify our results by comparing to the one-loop Higgs potential of the 4D Abelian-Higgs model in unitary gauge and find good agreement. To our knowledge, this is the first time this problem has been addressed for theories with gauge fields. The algorithm can also be used in four dimensions to study finite temperature and density transitions via effective Polyakov loop actions.Comment: added comparison to one-loop potential in section 3.3, improved text; version accepted for publication in Computer Physics Communication

The Leray- Gårding method for finite difference schemes

International audienceIn [Ler53] and [ Går56], Leray and Gårding have developed a multiplier technique for deriving a priori estimates for solutions to scalar hyperbolic equations in either the whole space or the torus. In particular, the arguments in [Ler53, Går56 ] provide with at least one local multiplier and one local energy functional that is controlled along the evolution. The existence of such a local multiplier is the starting point of the argument by Rauch in [Rau72] for the derivation of semigroup estimates for hyperbolic initial boundary value problems. In this article, we explain how this multiplier technique can be adapted to the framework of finite difference approximations of transport equations. The technique applies to numerical schemes with arbitrarily many time levels, and encompasses a somehow magical trick that has been known for a long time for the leapfrog scheme. More importantly, the existence and properties of the local multiplier enable us to derive optimal semigroup estimates for fully discrete hyperbolic initial boundary value problems, which answers a problem raised by Trefethen, Kreiss and Wu [Tre84, KW93]

NVU dynamics. I. Geodesic motion on the constant-potential-energy hypersurface

An algorithm is derived for computer simulation of geodesics on the constant potential-energy hypersurface of a system of N classical particles. First, a basic time-reversible geodesic algorithm is derived by discretizing the geodesic stationarity condition and implementing the constant potential energy constraint via standard Lagrangian multipliers. The basic NVU algorithm is tested by single-precision computer simulations of the Lennard-Jones liquid. Excellent numerical stability is obtained if the force cutoff is smoothed and the two initial configurations have identical potential energy within machine precision. Nevertheless, just as for NVE algorithms, stabilizers are needed for very long runs in order to compensate for the accumulation of numerical errors that eventually lead to "entropic drift" of the potential energy towards higher values. A modification of the basic NVU algorithm is introduced that ensures potential-energy and step-length conservation; center-of-mass drift is also eliminated. Analytical arguments confirmed by simulations demonstrate that the modified NVU algorithm is absolutely stable. Finally, simulations show that the NVU algorithm and the standard leap-frog NVE algorithm have identical radial distribution functions for the Lennard-Jones liquid

What makes nonholonomic integrators work?

A nonholonomic system is a mechanical system with velocity constraints not originating from position constraints; rolling without slipping is the typical example. A nonholonomic integrator is a numerical method specifically designed for nonholonomic systems. It has been observed numerically that many nonholonomic integrators exhibit excellent long-time behaviour when applied to various test problems. The excellent performance is often attributed to some underlying discrete version of the Lagrange--d'Alembert principle. Instead, in this paper, we give evidence that reversibility is behind the observed behaviour. Indeed, we show that many standard nonholonomic test problems have the structure of being foliated over reversible integrable systems. As most nonholonomic integrators preserve the foliation and the reversible structure, near conservation of the first integrals is a consequence of reversible KAM theory. Therefore, to fully evaluate nonholonomic integrators one has to consider also non-reversible nonholonomic systems. To this end we construct perturbed test problems that are integrable but no longer reversible (with respect to the standard reversibility map). Applying various nonholonomic integrators from the literature to these problems we observe that no method performs well on all problems. This further indicates that reversibility is the main mechanism behind near conservation of first integrals for nonholonomic integrators. A list of relevant open problems is given.Comment: 27 pages, 9 figure

A nonholonomic Newmark method

Using the nonholonomic exponential map, we obtain a new version of Newmark-type methods for nonholonomic systems (see also Jay and Negrut(2009) for a different extension). We give numerical examples including a test problem where the structure of reversible integrability responsible for good energy behaviour as described in Modin and Verdier (2020) is lost. We observe that the composition of two Newmark methods is able to produce good energy behaviour on this test problem.Fil: Anahory Simoes, Alexandre. IE Universidad; EspañaFil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Marrero González, Juan Carlos. Universidad de La Laguna; EspañaFil: Martín de Diego, David. Universidad Nebrija; España. Instituto de Ciencias Matemáticas; Españ

Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves

AbstractWe consider systems of equations which arise in modelling strong interactions of weakly nonlinear long waves in dispersive media. For a certain class of such systems, we prove the existence and stability of localized solutions representing coupled solitary waves travelling at a common speed. Our results apply in particular to the systems derived by Gear and Grimshaw and by Liu, Kubota and Ko as models for interacting gravity waves in a density-stratified fluid. For the latter system, we also prove that any coupled solitary-wave solution must have components which are all symmetric about a common vertical axis

Discrete Hamiltonian evolution and quantum gravity

We study constrained Hamiltonian systems by utilizing general forms of time discretization. We show that for explicit discretizations, the requirement of preserving the canonical Poisson bracket under discrete evolution imposes strong conditions on both allowable discretizations and Hamiltonians. These conditions permit time discretizations for a limited class of Hamiltonians, which does not include homogeneous cosmological models. We also present two general classes of implicit discretizations which preserve Poisson brackets for any Hamiltonian. Both types of discretizations generically do not preserve first class constraint algebras. Using this observation, we show that time discretization provides a complicated time gauge fixing for quantum gravity models, which may be compared with the alternative procedure of gauge fixing before discretization.Comment: 8 pages, minor changes, to appear in CQ

The fictitious domain method and applications in wave propagation

International audienceThis paper deals with the convergence analysis of the fictitious domain method used for taking into account the Neumann boundary condition on the surface of a crack (or more generally an object) in the context of acoustic and elastic wave propagation. For both types of waves we consider the first order in time formulation of the problem known as mixed velocity-pressure formulation for acoustics and velocity-stress formulation for elastodynamics. The convergence analysis for the discrete problem depends on the mixed finite elements used. We consider here two families of mixed finite elements that are compatible with mass lumping. When using the first one which is less expensive and corresponds to the choice made in a previous paper, it is shown that the fictitious domain method does not always converge. For the second one a theoretical convergence analysis was carried out in [7] for the acoustic case. Here we present numerical results that illustrate the convergence of the method both for acoustic and elastic waves

The Kramers equation simulation algorithm II. An application to the Gross-Neveu model

We continue the investigation on the applications of the Kramers equation to the numerical simulation of field theoretic models. In a previous paper we have described the theory and proposed various algorithms. Here, we compare the simplest of them with the Hybrid Monte Carlo algorithm studying the two-dimensional lattice Gross-Neveu model. We used a Symanzik improved action with dynamical Wilson fermions. Both the algorithms allow for the determination of the critical mass. Their performances in the definite phase simulations are comparable with the Hybrid Monte Carlo. For the two methods, the numerical values of the measured quantities agree within the errors and are compatible with the theoretical predictions; moreover, the Kramers algorithm is safer from the point of view of the numerical precision.Comment: 20 pages + 1 PostScript figure not included, REVTeX 3.0, IFUP-TH-2

Direct numerical simulation of the dynamics of sliding rough surfaces

The noise generated by the friction of two rough surfaces under weak contact pressure is usually called roughness noise. The underlying vibration which produces the noise stems from numerous instantaneous shocks (in the microsecond range) between surface micro-asperities. The numerical simulation of this problem using classical mechanics requires a fine discretization in both space and time. This is why the finite element method takes much CPU time. In this study, we propose an alternative numerical approach which is based on a truncated modal decomposition of the vibration, a central difference integration scheme and two algorithms for contact: The penalty algorithm and the Lagrange multiplier algorithm. Not only does it reproduce the empirical laws of vibration level versus roughness and sliding speed found experimentally but it also provides the statistical properties of local events which are not accessible by experiment. The CPU time reduction is typically a factor of 10.Comment: 16 pages, 16 figures, accepted versio