Modeling Fluctuations in Macroscopic Systems,
Abstract
Macroscopic systems are often modeled by deterministic differential equations (DDE), such as x-dot=f(x,t). Here, x(t) 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 X(t) satisfies a stochastic differential equation (SDE), X-dot=f(X,t)+a(X,t)xi(t). In some sense, x(t) should be the average of X(t). If f is linear and a(X,t) is a constant then E(X(t))=x(t). If f(X,t) is non-linear, then E(f(X,t)) not = f(E(X),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 X(t) is x(t). Thus, there is an underlying consistency between the deterministic and stochastic formulations. The procedure also provides a prescription for the calculation of a(X,t), which is usually not constant if f(x,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)
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1979
- Accession Number
- ADA070821
Entities
People
- Marc Mangel
Organizations
- Center for Naval Analyses