Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

Partial Differential Inclusions Governing Feedback Controls

The authors derive partial differential inclusions of hyperbolic type, the solutions of which are feedbacks governing the viable (controlled invariant) solutions of a control system. They show that the tracking property, another important control problem, leads to such hyperbolic systems of partial differential inclusions. They begin by proving the existence of the largest solution of such a problem, a stability result and provide an explicit solution in the particular case of decomposable systems. They then state a variational principle and an existence theorem of a. (single-valued contingent) solution to such an inclusion, that they apply to assert the existence of a feedback control

Observability of Systems under Uncertainty

The authors observe the evolution of a state of a system under uncertainty governed by a differential inclusion through an observation map. The set-valued character due to uncertainty leads them to introduce the "Sharp Input-Output Map", which is a (usual) product, and the "Hazy Input-Output Map", which is a square product. They provide criteria for both sharp and hazy local observability in terms of (global) sharp and hazy observability of a variational inclusion. They reach their conclusions by implementing the following strategy: (1) Provide a general principle of local injectivity and observability of a set-valued map I, which derives these properties from the fact that the kernel of an adequate derivative of I is equal to 0. (2) Supply chain rule formulas which allow to compute the derivatives of the usual product I_{-} and the square product I_{+} from the derivatives of the observation map H and the solution map S. (3) Characterize the various derivatives of the solution map S in terms of the solution maps of the associated variational inclusions. (4) Piece together these results for deriving local sharp and hazy observability of the original system from sharp and hazy observability of the variational inclusions. (5) Study global sharp and hazy observability of the variational inclusions

Controllability and Observability of Control Systems under Uncertainty

This report surveys the results of nonlinear systems theory (controllability and observability) obtained at IIASA during the last three summers. Classical methods based on differential geometry require some regularity and fail as soon as state-dependent constraints are brought to bear on the controls, or uncertainty and disturbances are involved in the system. Since these important features appear in most realistic control problems, new methods had to be devised, which encompass the classical ones, and allow the presence of a priori feedback into the control systems. This is now possible thanks to new tools, in the development of which IIASA played an important role: differential inclusions and set-valued analysis

Effects of electron-phonon interactions on the electron tunneling spectrum of PbS quantum dots

We present a tunnel spectroscopy study of single PbS Quantum Dots (QDs) as function of temperature and gate voltage. Three distinct signatures of strong electron-phonon coupling are observed in the Electron Tunneling Spectrum (ETS) of these QDs. In the shell-filling regime, the $8\times$ degeneracy of the electronic levels is lifted by the Coulomb interactions and allows the observation of phonon sub-bands that result from the emission of optical phonons. At low bias, a gap is observed in the ETS that cannot be closed with the gate voltage, which is a distinguishing feature of the Franck-Condon (FC) blockade. From the data, a Huang-Rhys factor in the range $S\sim 1.7 - 2.5$ is obtained. Finally, in the shell tunneling regime, the optical phonons appear in the inelastic ETS $d^2I/dV^2$.Comment: 5 pages, 5 figure