2,628 research outputs found
WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction
WavePacket is an open-source program package for numeric simulations in
quantum dynamics. It can solve time-independent or time-dependent linear
Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions.
Also coupled equations can be treated, which allows, e.g., to simulate
molecular quantum dynamics beyond the Born-Oppenheimer approximation.
Optionally accounting for the interaction with external electric fields within
the semi-classical dipole approximation, WavePacket can be used to simulate
experiments involving tailored light pulses in photo-induced physics or
chemistry. Being highly versatile and offering visualization of quantum
dynamics 'on the fly', WavePacket is well suited for teaching or research
projects in atomic, molecular and optical physics as well as in physical or
theoretical chemistry. Building on the previous Part I which dealt with closed
quantum systems and discrete variable representations, the present Part II
focuses on the dynamics of open quantum systems, with Lindblad operators
modeling dissipation and dephasing. This part also describes the WavePacket
function for optimal control of quantum dynamics, building on rapid
monotonically convergent iteration methods. Furthermore, two different
approaches to dimension reduction implemented in WavePacket are documented
here. In the first one, a balancing transformation based on the concepts of
controllability and observability Gramians is used to identify states that are
neither well controllable nor well observable. Those states are either
truncated or averaged out. In the other approach, the H2-error for a given
reduced dimensionality is minimized by H2 optimal model reduction techniques,
utilizing a bilinear iterative rational Krylov algorithm
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Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms
In this paper, we study Jarzynski's equality and fluctuation theorems for
diffusion processes. While some of the results considered in the current work
are known in the (mainly physics) literature, we review and generalize these
nonequilibrium theorems using mathematical arguments, therefore enabling
further investigations in the mathematical community. On the numerical side,
variance reduction approaches such as importance sampling method are studied in
order to compute free energy differences based on Jarzynski's equality.Comment: journal versio
Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems
We study linear-quadratic stochastic optimal control problems with bilinear
state dependence for which the underlying stochastic differential equation
(SDE) consists of slow and fast degrees of freedom. We show that, in the same
way in which the underlying dynamics can be well approximated by a reduced
order effective dynamics in the time scale limit (using classical
homogenziation results), the associated optimal expected cost converges in the
time scale limit to an effective optimal cost. This entails that we can well
approximate the stochastic optimal control for the whole system by the reduced
order stochastic optimal control, which is clearly easier to solve because of
lower dimensionality. The approach uses an equivalent formulation of the
Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs
(FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares
Monte Carlo algorithm and show its applicability by a suitable numerical
example
Partitioning a macroscopic system into independent subsystems
We discuss the problem of partitioning a macroscopic system into a collection
of independent subsystems. The partitioning of a system into replica-like
subsystems is nowadays a subject of major interest in several field of
theoretical and applied physics, and the thermodynamic approach currently
favoured by practitioners is based on a phenomenological definition of an
interface energy associated with the partition, due to a lack of easily
computable expressions for a microscopic (i.e.~particle-based) interface
energy. In this article, we outline a general approach to derive sharp and
computable bounds for the interface free energy in terms of microscopic
statistical quantities. We discuss potential applications in nanothermodynamics
and outline possible future directions.Comment: This is an author-created, un-copyedited version of an article
accepted for publication in JSTA
Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations
Model reduction methods for bilinear control systems are compared by means of
practical examples of Liouville-von Neumann and Fokker--Planck type. Methods
based on balancing generalized system Gramians and on minimizing an H2-type
cost functional are considered. The focus is on the numerical implementation
and a thorough comparison of the methods. Structure and stability preservation
are investigated, and the competitiveness of the approaches is shown for
practically relevant, large-scale examples
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