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Physical Review A

Sensitivity analysis in stochastic mechanics

Marvin McClendon, HERSCHEL A. RABITZ

Published May 1, 1988• 2 citations
Publisher

Abstract

The formalism of sensitivity analysis for stochastic differential equations is adapted and extended to the framework of stochastic mechanics. Thus a methodology is provided for analyzing the dependence of expectations of functionals of the stochastic mechanical diffusion process on parameters defining the system. This should prove useful in numerical simulations employing the latter theory. We take into account the fact that in stochastic mechanics typically a probability distribution of initial positions (rather than the initial position itself) is prescribed. The expression for a sensitivity density with respect to a drift variation depending only on time is shown to follow from the Ito-Girsanov formula, which is also employed to develop new formulas for sensitivity densities with respect to position-dependent drift variations. As an example to verify these techniques, the sensitivity coefficients of the harmonic-oscillator correlation function with respect to the angular frequency and the diffusion constant are calculated; the results coincide with those obtained by direct differentiation.

DOI
10.1103/physreva.37.3493

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The formalism of sensitivity analysis for stochastic differential equations is adapted and extended to the framework of stochastic mechanics. Thus a methodology is provided for analyzing the dependence of expectations of functionals of the stochastic mechanical diffusion process on parameters defining the system. This should prove useful in numerical simulations employing the latter theory. We take into account the fact that in stochastic mechanics typically a probability distribution of initial positions (rather than the initial position itself) is prescribed. The expression for a sensitivity density with respect to a drift variation depending only on time is shown to follow from the Ito-Girsanov formula, which is also employed to develop new formulas for sensitivity densities with respect to position-dependent drift variations. As an example to verify these techniques, the sensitivity coefficients of the harmonic-oscillator correlation function with respect to the angular frequency and the diffusion constant are calculated; the results coincide with those obtained by direct differentiation.
An estimation method for unknown parameters in the initial conditions and the potential of a quantal system using the stochastic interpretation of quantum mechanics and some results in system theory are presented. According to this interpretation the possible trajectories of a particle through coordinate space may be represented by the realization of a stochastic process that satisfies a stochastic differential equation. The drift term in this equation is derived from the wave function and consequently contains all unknown parameters in the initial conditions and the potential. The main assumption of the paper is that a continuous sequence of position measurements on the trajectory of the particle can be identified with a realization of this stochastic process over the corresponding period of time. An application of the stochastic filtering theorems subsequently provides a minimum variance estimate of the unknown parameters in the drift conditional on this continuous sequence of measurements. As simple illustrations, this method is used to obtain estimates for the initial momentum of a free particle given measurements on its trajectory and to construct an estimator for the unknown parameters in a harmonic potential. It is shown that an optimal estimator exists if the stochastic processes are associated with a wave function from a potential of the Rellich type. In addition the a posteriori probability density of the parameters in the quantal system is calculated, assuming that all parameters involved prescribe a Rellich potential.
The paper considers the sensitivity of quantum dynamics systems with respect to parameters or features in the underlying Hamiltonians. The analysis allows for the calculation of the gradient of the wave function with respect to any well defined system parameters in the Hamiltonian. Differential sensitivity equations are derived from the Schrödinger equation and their solution is expressed in terms of a Green’s function. For scattering problems the necessary asymptotic Green’s function can be exclusively expressed in terms of the available solution to the original Schrödinger equation. Also considered are problems from time-dependent quantum mechanics. In this latter situation sensitivity theory allows for the calculation of gradients of the time evolution operator with respect to system parameters. The scattering formalism is illustrated with a study of the anisotropy coefficient sensitivity in the collision of an atom and a ridgid rotor. The time-dependent theory was applied to a collision-free molecular beam translating through a region of space containing electric and magnetic fields. This case examined the sensitivity of the molecular transition probabilities to field parameters. These brief examples serve to illustrate the potential usefulness of sensitivity analysis in quantum dynamics.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.
We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic dynamical diffusion coefficients. Explicit computation of the path integral leads to a universal expression for the associated measure regardless of the form of the diffusion coefficient and the drift. This computation also reveals that the drift plays the role of a super potential in the usual super-symmetric quantum mechanics sense. Some simple illustrative examples such as the Ornstein-Uhlenbeck process and the multidimensional Black-Scholes model are also discussed. Basic examples of quantum integrable systems such as the quantum discrete non-linear hierarchy (DNLS) and the XXZ spin chain are presented providing specific connections between quantum (integrable) systems and stochastic differential equations (SDEs). The continuum limits of the SDEs for the first two members of the NLS hierarchy turn out to be the stochastic transport and the stochastic heat equations respectively. The quantum Darboux matrix for the discrete NLS is also computed as a defect matrix and the relevant SDEs are derived.
A diffusion process of which the diffusion property is preserved under time reversal is called a time-reversible diffusion process. A typical example of such a process is found in Nelson’s stochastic mechanics. Within the extended framework of stochastic integrals proposed by Itô, a sort of the interpolation formula, called a stochastic dynamical interpolation formula, to the time-reversible diffusion process, is derived from a pinned process for the diffusion process. It is shown that the stochastic interpolation formula is regarded as an extention of Lévy’s interpolation formula, which is obtained on the basis of a pinned Brownian motion.
It is shown how Adler's trace dynamics can be applied to stochastic mechanics and other complex classical dynamical systems. Emergent non-commutivity due to the fractal nature of sample trajectories is closely related to the fact that the forward and backward time derivatives are different for these diffusions. A new variational approach to stochastic mechanics based on trace dynamics is introduced. It is shown that Yasue's method and Guerra and Morato's method can both be generalized to allow for any diffusion constant in a stochastic model of Schrodinger's equation, and that they can all also describe dissipative diffusion. Then it is shown that the trace dynamical theory seems to only describe dissipative diffusion unless an extra quantum mechanical potential term is added to the Hamiltonian. The differential space theory of Wiener and Siegel is reconsidered as a useful tool in this framework, and is generalized to stochastic processes instead of deterministic ones for the hidden trajectories of observables. It is proposed that the natural measure space for Wiener-Siegel theory is Haar measure for random unitary matrices. A new interpretation of the polychotomic algorithm is given.