3,628 research outputs found
Discrete mechanics and variational integrators
This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented
Symplectic-energy-momentum preserving variational integrators
The purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving. To do this, a space–time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy. Criteria for the solvability of the time steps and some numerical examples are given
Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces
A temporally varying discretization often features in discrete gravitational
systems and appears in lattice field theory models subject to a coarse graining
or refining dynamics. To better understand such discretization changing
dynamics in the quantum theory, an according formalism for constrained
variational discrete systems is constructed. While the present manuscript
focuses on global evolution moves and, for simplicity, restricts to Euclidean
configuration spaces, a companion article discusses local evolution moves. In
order to link the covariant and canonical picture, the dynamics of the quantum
states is generated by propagators which satisfy the canonical constraints and
are constructed using the action and group averaging projectors. This projector
formalism offers a systematic method for tracing and regularizing divergences
in the resulting state sums. Non-trivial coarse graining evolution moves lead
to non-unitary, and thus irreversible, projections of physical Hilbert spaces
and Dirac observables such that these concepts become evolution move dependent
on temporally varying discretizations. The formalism is illustrated in a toy
model mimicking a `creation from nothing'. Subtleties arising when applying
such a formalism to quantum gravity models are discussed.Comment: 45 pages, 1 appendix, 6 figures (additional explanations, now matches
published version
The Renormalization-Group peculiarities of Griffiths and Pearce: What have we learned?
We review what we have learned about the "Renormalization-Group
peculiarities" which were discovered about twenty years ago by Griffiths and
Pearce, and which questions they asked are still widely open. We also mention
some related developments.Comment: Proceedings Marseille meeting on mathematical results in statistical
mechanic
Bosons condensed in two modes with flavour-changing interaction
A quantum model is considered for bosons populating two orthogonal
single-particle modes with tunable energy separation in the presence of
flavour-changing contact interaction. The quantum ground state is well
approximated as a coherent superposition (for zero temperature) or a mixture
(at low temperature) of two quasi-classical states. In a mean field
description, the systems realizes one of these states via spontaneous symmetry
breaking. Both mean field states, in a certain parameter range, possess finite
angular momentum and exhibit broken time-reversal symmetry in contrast to the
quantum ground state. The phase diagram is explored at the mean-field level and
by direct diagonalisation. The nature of the quantum ground state at zero and
finite temperature is analyzed by means of the Penrose Onsager criterion. One
of three possible phases shows fragmentation on the single-particle level
together with a finite pair order parameter. Thermal and quantum fluctuations
are characterized with respect to regions of universal scaling behavior. The
non-equilibrium dynamics shows a sharp transition between a self-trapping and a
pair-tunneling regime. A recently realized experimental implementation is
discussed with bosonic atoms condensed in the two inequivalent energy minima
of the second band of a bipartite two-dimensional optical lattice.Comment: 12 pages, 12 figures, extended and revise
Approximate Decoherence of Histories and 't Hooft's Deterministic Quantum Theory
This paper explores the possibility that an exactly decoherent set of
histories may be constructed from an approximately decoherent set by small
distortions of the operators characterizing the histories. In particular, for
the case of histories of positions and momenta, this is achieved by doubling
the set of operators and then finding, amongst this enlarged set, new position
and momentum operators which commute, so decohere exactly, and which are
``close'' to the original operators. The enlarged, exactly decoherent, theory
has the same classical dynamics as the original one, and coincides with the
so-called deterministic quantum theories of the type recently studied by 't
Hooft. These results suggest that the comparison of standard and deterministic
quantum theories may provide an alternative method of characterizing emergent
classicality. A side-product is the surprising result that histories of momenta
in the quantum Brownian motion model (for the free particle in the
high-temperature limit) are exactly decoherent.Comment: 41 pages, plain Te
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