17,684 research outputs found
Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling
We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to problems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., submitted]. The characterization of local stability of solutions is complicated by the double saddle-point structure, and here we develop efficiently computable criteria in terms of minimum eigenvalues of certain projected Schur complements. We also propose a modified outer iteration (“Renormalized Newton Method”) in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The Renormalized Newton Method bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two
Low dimensional manifolds for exact representation of open quantum systems
Weakly nonlinear degrees of freedom in dissipative quantum systems tend to
localize near manifolds of quasi-classical states. We present a family of
analytical and computational methods for deriving optimal unitary model
transformations based on representations of finite dimensional Lie groups. The
transformations are optimal in that they minimize the quantum relative entropy
distance between a given state and the quasi-classical manifold. This naturally
splits the description of quantum states into quasi-classical coordinates that
specify the nearest quasi-classical state and a transformed quantum state that
can be represented in fewer basis levels. We derive coupled equations of motion
for the coordinates and the transformed state and demonstrate how this can be
exploited for efficient numerical simulation. Our optimization objective
naturally quantifies the non-classicality of states occurring in some given
open system dynamics. This allows us to compare the intrinsic complexity of
different open quantum systems.Comment: Added section on semi-classical SR-latch, added summary of method,
revised structure of manuscrip
Perturbative quantum gravity with the Immirzi parameter
We study perturbative quantum gravity in the first-order tetrad formalism.
The lowest order action corresponds to Einstein-Cartan plus a parity-odd term,
and is known in the literature as the Holst action. The coupling constant of
the parity-odd term can be identified with the Immirzi parameter of loop
quantum gravity. We compute the quantum effective action in the one-loop
expansion. As in the metric second-order formulation, we find that in the case
of pure gravity the theory is on-shell finite, and the running of Newton's
constant and the Immirzi parameter is inessential. In the presence of fermions,
the situation changes in two fundamental aspects. First, non-renormalizable
logarithmic divergences appear, as usual. Second, the Immirzi parameter becomes
a priori observable, and we find that it is renormalized by a four-fermion
interaction generated by radiative corrections. We compute its beta function
and discuss possible implications. The sign of the beta function depends on
whether the Immirzi parameter is larger or smaller than one in absolute value,
and the values plus or minus one are UV fixed-points (we work in Euclidean
signature). Finally, we find that the Holst action is stable with respect to
radiative corrections in the case of minimal coupling, up to higher order
non-renormalizable interactions.Comment: v2 minor amendment
Fast derivatives of likelihood functionals for ODE based models using adjoint-state method
We consider time series data modeled by ordinary differential equations
(ODEs), widespread models in physics, chemistry, biology and science in
general. The sensitivity analysis of such dynamical systems usually requires
calculation of various derivatives with respect to the model parameters.
We employ the adjoint state method (ASM) for efficient computation of the
first and the second derivatives of likelihood functionals constrained by ODEs
with respect to the parameters of the underlying ODE model. Essentially, the
gradient can be computed with a cost (measured by model evaluations) that is
independent of the number of the ODE model parameters and the Hessian with a
linear cost in the number of the parameters instead of the quadratic one. The
sensitivity analysis becomes feasible even if the parametric space is
high-dimensional.
The main contributions are derivation and rigorous analysis of the ASM in the
statistical context, when the discrete data are coupled with the continuous ODE
model. Further, we present a highly optimized implementation of the results and
its benchmarks on a number of problems.
The results are directly applicable in (e.g.) maximum-likelihood estimation
or Bayesian sampling of ODE based statistical models, allowing for faster, more
stable estimation of parameters of the underlying ODE model.Comment: 5 figure
Semiclassical Phase Reduction Theory for Quantum Synchronization
We develop a general theoretical framework of semiclassical phase reduction
for analyzing synchronization of quantum limit-cycle oscillators. The dynamics
of quantum dissipative systems exhibiting limit-cycle oscillations are reduced
to a simple, one-dimensional classical stochastic differential equation
approximately describing the phase dynamics of the system under the
semiclassical approximation. The density matrix and power spectrum of the
original quantum system can be approximately reconstructed from the reduced
phase equation. The developed framework enables us to analyze synchronization
dynamics of quantum limit-cycle oscillators using the standard methods for
classical limit-cycle oscillators in a quantitative way. As an example, we
analyze synchronization of a quantum van der Pol oscillator under harmonic
driving and squeezing, including the case that the squeezing is strong and the
oscillation is asymmetric. The developed framework provides insights into the
relation between quantum and classical synchronization and will facilitate
systematic analysis and control of quantum nonlinear oscillators.Comment: 20 pages, 5 figure
Delay-induced patterns in a two-dimensional lattice of coupled oscillators
We show how a variety of stable spatio-temporal periodic patterns can be
created in 2D-lattices of coupled oscillators with non-homogeneous coupling
delays. A "hybrid dispersion relation" is introduced, which allows studying the
stability of time-periodic patterns analytically in the limit of large delay.
The results are illustrated using the FitzHugh-Nagumo coupled neurons as well
as coupled limit cycle (Stuart-Landau) oscillators
Fundamental Relativistic Rotator. Hessian singularity and the issue of the minimal interaction with electromagnetic field
There are two relativistic rotators with Casimir invariants of the
Poincar\'{e} group being fixed parameters. The particular models of spinning
particles were studied in the past both at the classical and quantum level.
Recently, a minimal interaction with electromagnetic field has been considered.
We show that the dynamical systems can be uniquely singled out from among other
relativistic rotators by the unphysical requirement that the Hessian referring
to the physical degrees of freedom should be singular. Closely related is the
fact that the equations of free motion are not independent, making the
evolution indeterminate. We show that the Hessian singularity cannot be removed
by the minimal interaction with the electromagnetic field. By making use of a
nontrivial Hessian null space, we show that a single constraint appears in the
external field for consistency of the equations of motion with the Hessian
singularity. The constraint imposes unphysical limitation on the initial
conditions and admissible motions. We discuss the mechanism of appearance of
unique solutions in external fields on an example of motion in the uniform
magnetic field. We give a simple model to illustrate that similarly constrained
evolution cannot be determinate in arbitrary fields.Comment: 16 pages, in v2: shortened, improved presentation, proofs moved to
Appendices, in v3: further text permutations and a comment added concerning
hamiltonization, in v4: language corrections, final for
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