# Accession Number:

## ADA070821

# Title:

## Modeling Fluctuations in Macroscopic Systems,

# Descriptive Note:

## Professional paper,

# Corporate Author:

## CENTER FOR NAVAL ANALYSES ARLINGTON VA OPERATIONS EVALUATION GROUP

# Personal Author(s):

# Report Date:

## 1979-04-01

# Pagination or Media Count:

## 31.0

# Abstract:

Macroscopic systems are often modeled by deterministic differential equations DDE, such as x-dotfx,t. Here, xt is a macrovariable and represents an average over some set of ensembles. A possible extention of such a model to include fluctuations is to assume that a random variable Xt satisfies a stochastic differential equation SDE, X-dotfX,taX,txit. In some sense, xt should be the average of Xt. If f is linear and aX,t is a constant then EXtxt. If fX,t is non-linear, then EfX,t not fEX,t generally and some authors feel that the SDE is not a correct extension of the DDE. A procedure will be introduced here so that an appropriate conditional average of Xt is xt. Thus, there is an underlying consistency between the deterministic and stochastic formulations. The procedure also provides a prescription for the calculation of aX,t, which is usually not constant if fx,t is nonlinear. Two examples are studied to illustrate the application of the procedure. First, the logistic equation of population dynamics is studied in deterministic and stochastic versions. Second, stochastic effects on a chemical oscillator are analyzed. Author

# Descriptors:

# Subject Categories:

- Statistics and Probability