11 research outputs found
Pseudo generators of spatial transfer operators
Metastable behavior in dynamical systems may be a significant challenge for a
simulation based analysis. In recent years, transfer operator based approaches
to problems exhibiting metastability have matured. In order to make these
approaches computationally feasible for larger systems, various reduction
techniques have been proposed: For example, Sch\"utte introduced a spatial
transfer operator which acts on densities on configuration space, while Weber
proposed to avoid trajectory simulation (like Froyland et al.) by considering a
discrete generator.
In this manuscript, we show that even though the family of spatial transfer
operators is not a semigroup, it possesses a well defined generating structure.
What is more, the pseudo generators up to order 4 in the Taylor expansion of
this family have particularly simple, explicit expressions involving no
momentum averaging. This makes collocation methods particularly easy to
implement and computationally efficient, which in turn may open the door for
further efficiency improvements in, e.g., the computational treatment of
conformation dynamics. We experimentally verify the predicted properties of
these pseudo generators by means of two academic examples
Preparations for Quantum Simulations of Quantum Chromodynamics in 1+1 Dimensions: (II) Single-Baryon -Decay in Real Time
A framework for quantum simulations of real-time weak decays of hadrons and
nuclei in a 2-flavor lattice theory in one spatial dimension is presented. A
single generation of the Standard Model is found to require 16 qubits per
spatial lattice site after mapping to spin operators via the Jordan-Wigner
transformation. Both quantum chromodynamics and flavor-changing weak
interactions are included in the dynamics, the latter through four-Fermi
effective operators. Quantum circuits which implement time evolution in this
lattice theory are developed and run on Quantinuum's H1-1 20-qubit trapped ion
system to simulate the -decay of a single baryon on one lattice site.
These simulations include the initial state preparation and are performed for
both one and two Trotter time steps. The potential intrinsic error-correction
properties of this type of lattice theory are discussed and the leading lattice
Hamiltonian required to simulate -decay of nuclei induced by a
neutrino Majorana mass term is provided.Comment: 26 pages, 11 figure
Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics
We consider complex dynamical systems showing metastable behavior but no
local separation of fast and slow time scales. The article raises the question
of whether such systems exhibit a low-dimensional manifold supporting its
effective dynamics. For answering this question, we aim at finding nonlinear
coordinates, called reaction coordinates, such that the projection of the
dynamics onto these coordinates preserves the dominant time scales of the
dynamics. We show that, based on a specific reducibility property, the
existence of good low-dimensional reaction coordinates preserving the dominant
time scales is guaranteed. Based on this theoretical framework, we develop and
test a novel numerical approach for computing good reaction coordinates. The
proposed algorithmic approach is fully local and thus not prone to the curse of
dimension with respect to the state space of the dynamics. Hence, it is a
promising method for data-based model reduction of complex dynamical systems
such as molecular dynamics
Spectral Properties of Effective Dynamics from Conditional Expectations
The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting
Diffusion maps tailored to arbitrary non-degenerate Ito processes
We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling