Non-Markovian stochastic processes: colored noise

We survey classical non-Markovian processes driven by thermal equilibrium or nonequilibrium (nonthermal) colored noise. Examples of colored noise are presented. For processes driven by thermal equilibrium noise, the fluctuation-dissipation relation holds. In consequence, the system has to be described by a generalized (integro-differential) Langevin equation with a restriction on the damping integral kernel: Its form depends on the correlation function of noise. For processes driven by nonequilibrium noise, there is no such a restriction: They are considered to be described by stochastic differential (Ito- or Langevin-type) equations with an independent noise term. For the latter, we review methods of analysis of one-dimensional systems driven by Ornstein-Uhlenbeck noise.     

Quantum stochastic processes

In order to describe rigorously certain measurement procedures, where observations of the arrival of quanta at a counter are made throughout an interval of time, it is necessary to introduce the concept of a quantum stochastic process. While fully quantum mechanical in nature, these have a great deal of similarity with classical stochastic processes and can be characterized by and constructed from their infinitesimal generators. The infinitestimal generators are naturally obtained from certain fields which we prove must be of the boson or fermion type.This work was supported by a National Science Foundation grant GP-7952X.     

Quantum stochastic processes

We first define a class of processes which we call regular quantum Markov processes. We next prove some basic results concerning such processes. A method is given for constructing quantum Markov processes using transition amplitude kernels. Finally we show that the Feynman path integral formalism can be clarified by approximating it with a quantum stochastic process.     

On an inverse method in one dimension

Given a sequence of bound-state energies, the inverse Gel'fand-Levitan theory for the one-dimensional Schrödinger equation enables us to uniquely construct a symmetric reflectionless potential which supports those bound states. We test the reliability of this technique by applying it to some known symmetric potentials (confining and non-confining) in one dimension. We also study the systematics under which the method may be reliable.     

Quantum stochastic processes II

We investigate properties of a class of quantum stochastic processes subject to a condition of irreducibility. These processes must be recurrent or transient and an equilibrium state can only exist in the former case. Every finite dimensional process is recurrent and it is possible to establish convergence in time to a unique equilibrium state. We study particularly the class of transition processes, which describe photon emissions of simple quantum mechanical systems in excited states.Work supported by U.S.A.F. contract number F 44620-67-C-0029.     

Quantum stochastic processes III

We construct an example of a quantum stochastic process with a non-zero, linear, time-independent source, for a massive scalar Boson field in four space-time dimensions. Also we study in detail a similar process with only a single degree of freedom.Work supported by A.F.O.S.R. contract no. F44620-67-C-0029     

Grating profile reconstruction by an inverse scattering method

An inverse scattering method is used to reconstruct grating profiles from efficiency measurements. The computed profiles are compared to electron micrographs. The accuracy of the method is shown to be very good.