Kurt E. Shuler

Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I Theory

A method has been developed to investigate the sensitivity of the solutions of large sets of coupled nonlinear rate equations to uncertainties in the rate coefficients. This method is based on varying all the rate coefficients simultaneously through the introduction of a parameter in such a way that the output concentrations become periodic functions of this parameter at any given time t. The concentrations of the chemical species are then Fourier analyzed at time t. We show via an application of Weyls ergodic theorem that a subset of the Fourier coefficients is related to 〈∂ci/∂kl〉, the rate of change of the concentration of species i with respect to the rate constant for reaction l averaged over the uncertainties of all the other rate coefficients. Thus a large Fourier coefficient corresponds to a large sensitivity, and a small Fourier coefficient corresponds to a small sensitivity. The amount of numerical integration required to calculate these Fourier coefficients is considerably less than that requi...

Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. III. Analysis of the approximations

In Parts I and II of this series [J. Chem. Phys. 59, 3873, 3879 (1973)] we developed a new method of sensitivity analysis for large sets of coupled nonlinear equations with many parameters. In developing this theory and in carrying out the computer calculations involved in this analysis we made a number of approximations. We present here a quantitative analysis of these approximations and, where applicable, develop rigorous error bounds. Our analysis shows that we can specify the approximations which enter into our theory so as to obtain sensitivity measures of known accuracy. On this basis we feel that the techniques developed in this series of papers provide a useful and efficient method of sensitivity analysis of large systems with many parameters.

Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. II Applications

The Fourier amplitude method developed in Paper I as a diagnostic tool for determining the sensitivity of the results of complex calculations to the parameters which enter these calculations has been applied to two chemical reaction systems involving sets of coupled, nonlinear rate equations. These were: (a) a five reaction set describing the high temperature (6000 °K) dissociation of air and (b) a nine reaction set describing the constant temperature (2000 °K) combustion of H2 and O2. We have evaluated the Fourier amplitudes for all the species at a number of different times for both reaction systems. The analysis of these results verifies the claims made in Paper I. The relative magnitudes of the Fourier amplitudes showed a several order of magnitude distribution which permitted a clear distinction of the relative sensitivity of the species concentration to uncertainties in the rate coefficients. The conclusions based on the Fourier amplitude method for these two reaction systems are in excellent agreem...

On the Relation Between Master Equations and Random Walks and Their Solutions.

It is shown that there is a simple relation between master equation and random walk solutions. We assume that the random walker takes steps at random times, with the time between steps governed by a probability density ψ(Δt). Then, if the random walk transition probability matrix M and the master equation transition rate matrix A are related by A = (M − 1)/τ1, where τ1 is the first moment of Ψ(t) and thus the average time between steps, the solutions of the random walk and the master equation approach each other at long times and are essentially equal for times much larger than the maximum of (τn/n!)1/n, where τn is the nth moment of ψ(t). For a Poisson probability density ψ(t), the solutions are shown to be identical at all times. For the case where A ≠ (M − 1)/τ1, the solutions of the master equation and the random walk approach each other at long times and are approximately equal for times much larger than the maximum of (τn/n!)1/n if the eigenvalues and eigenfunctions of A and (M − 1)/τ1 are approxima...

Stochastic theory of nonlinear rate processes with multiple stationary states

A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of O(N0) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is O(eN). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.

Stochastic and Deterministic Formulation of Chemical Rate Equations

It is shown, on the basis of some examples, that the commonly used stochastic theory of gas‐phase chemical rate equations reduces to the deterministic formulation in the thermodynamic limit, N → ∞, V → ∞, N / V fixed. Since the commonly used deterministic collision theory of chemical kinetics is derivable from the Boltzmann equation with reactive scattering terms, the stochastic formulation of chemical kinetics is thus shown to be equivalent to the results of the Boltzmann equation in the thermodynamic limit. However, since the stochastic theory has not been derived from the Liouville equation for finite systems, the validity of the calculations of the deviation of the stochastic mean from the deterministric results and the fluctuations about that mean for finite N has not been established.

Stochastic processes with non-additive fluctuations

We present a comparison of the Fokker-Planck equations obtained by the Ito prescription and by the Stratonovich prescription for physical systems described by a Langevin equation with non-additive fluctuations. Our main conclusion is that the Stratonovich prescription is the one that should always be used to describe physical systems. This conclusion is shown to be consistent with results obtained from path integral and Master equation approaches.

First Passage Time and Extremum Properties of Markov and Independent Processes

It was shown by Newell in 1962 that the extreme value and first passage time distributions of various types of common Markov processes asymptotically approach those for independent random variables. In view of the great simplification this occasions in the calculation of a number of important properties of Markov processes, it is clearly of interest to determine in some detail the conditions on both the time and space variables under which this equivalence holds. In this paper we investigate and establish these conditions for Markov processes described by the Fokker-Planck equation and express them in simple analytic forms which are directly related to the coefficients of the Fokker-Planck equation. To demonstrate the usefulness of these conditions, we apply them to two representative examples of Fokker-Planck equations, the Ornstein-Uhlenbeck process and the Montroll-Shuler model for harmonic oscillator dissociation. It is shown very clearly in these examples that the extreme value and first passage time

Studies in nonlinear stochastic processes. II. The duffing oscillator revisited

We have applied the approximation method of statistical linearization and various higher order corrections thereto to the study of a nonlinear oscillator perturbed by Gaussian, delta-correlated noise. We compute the second-order statistics of the response, i.e., the variances, autocorrelation functions, and spectral densities for various forms of the nonlinearity and compare our results with the few more exact calculations which are available in the literature. We show that a very simple modification of statistical linearization, based upon the use of the variance as obtained from the appropriate Fokker-Planck equation, yields results which are in better agreement with the “exact” literature results than either statistical linearization or first-order corrections thereto. This modified method of statistical linearization has the significant advantage of great computational simplicity as compared to other attempts of accurate calculations of second-order statistics of nonlinear stochastic equations now in the literature.

On the Microscopic Conditions for Linear Macroscopic Laws

We have investigated the conditions which must be imposed on the microscopic equations of motion to obtain exact linear laws for macroscopic (phase averaged) variables. The starting point in this study has been the lowest order master equation (Pauli equation) which is a linear microscopic equation in the state probabilities with a time‐independent transition matrix. Discrete and continuous variable master equations as well as their multivariate generalizations have been considered. In the case of continuum state variables, we have used various Fokker‐Planck equations and their corresponding Langevin equations as our starting microscopic equation of motion. In each case the conditions which must be imposed to obtain linear macroscopic transport equations have been derived and discussed. The problem of the derivation of linear macroscopic laws from microscopic laws which are nonlinear in the dynamical variables has been discussed in the context of our results. We find that exact linear macroscopic laws can...