Stochastic theory of ignition processes

We consider a closed gaseous system immersed in a heat bath undergoing a thermal explosion. The effects of instantaneous fluctuations in the temperature on the heat removal mechanism and on the reaction rate are considered. The intensity of the fluctuations in situations far from equilibrium is determined by calculating the temperature self-correlation. This quantity scales with the inverse of an effective volume obtained from generalized fluctuation-dissipation theory. This determines a virtual system corresponding to the localized ignition process, possibly leading to a global runaway. The induction period is identified with the Kramers mean passage time for diffusion across a kinetic barrier. The induction period is thus shown to be dependent on the fluctuation volume . The diffusion process is hastened by the critical fluctuations. The explosive decomposition of ethyl azide was selected to test the theory and the results exhibit very good agreement with experimental data. Our treatment resolves the previous discrepancy between the predictions rooted in the classical Frank-Kamenetsky treatment and the premature ignition observed experimentally.     

Stochastic theory of nonlinear nonequilibrium processes

Arguments about the stochastic nature of the changes in thermodynamic parameters are used to construct a nonlinear version of the nonequilibrium thermodynamics of scalar processes, with account taken of crossover effects. The formalism is generalized to the case of vector processes. A cyclic chemical reaction in an ideal gas is treated as an example.Translated from Izvestiya Vyssikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 55–63, September, 1970.     

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 $\text{O}$(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 $\text{O}$(e^N). 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 representation of nearly-Gaussian,nonlinear processes

The use of polynomial functionals of the white noise process is discussed for the treatment of nonlinear random processes. It is noted that such treatments are useful for nearly-Gaussian processes. Applications of such representations to nonlinear systems and to nonlinear fluid mechanics problems (turbulence) are reviewed.     

Topics in data assimilation: Stochastic processes

Stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences. One application is data assimilation, i.e., the combination of model output with observations to form the best picture of the system under study. For any given quantity to be estimated, the relative weights of the model and the data will be adjusted according to estimated model and data error statistics, so implementation of any data assimilation scheme will require some assumption about errors, which are considered to be random. For dynamical models, some assumption about the evolution of errors will be needed. Stochastic models are also applied in studies of predictability.

The formal theory of stochastic processes was well developed in the last half of the twentieth century. One consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification. In some cases the rules of ordinary calculus must be modified.

The formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists. The purpose of this article is to provide an informal introduction to the relevant theory, and to point out those situations in which that theory must be applied in order to model random processes correctly.     

Stochastic theory of nonlinear rate processes with multiple stationary states. II. Relaxation time from a metastable state

We have developed a methodology for obtaining a Fokker-Planck equation for nonlinear systems with multiple stationary states that yields the correct system size dependence, i.e., exponential growth with system size of the relaxation time from a metastable state. We show that this relaxation time depends strongly on the barrier heightU(x) between the metastable and stable states of the system. For a Fokker-Planck (FP) equation to yield the correct result for the relaxation time from a metastable state, it is therefore essential that the free energy functionU(x) of the FP equation not only correctly locate the extrema of U(x), but also have the correct magnitudeU at these extrema. This is accomplished by so choosing the coefficients of the FP equation that its stationary solution is identical to that of the master equation that defines the nonlinear system.This work was supported in part by the National Science Foundation under Grant CHE 75-20624.