Stochastic Processes in Polymeric Fluids

Stochastics is concerned with the mathematical analysis of intuitive notions like “chance,” “randomness,” “fluctuations,” or “noise.” In order to develop a sound mathematical theory of stochastics it is necessary to give rigorous definitions for the following basic concepts, which have an intuitive meaning based on everyday experience, especially from games of chance: event, probability, random variable, expectation, stochastic independence, conditioning, convergence, and stochastic process. The formal definitions of these objects in this chapter are made clear by many examples. The examples and exercises of this chapter play a most important role in this book because they not only illustrate the meaning of abstract concepts, but also contain important information required in the subsequent chapters.

The idea of establishing a relationship between diffusion equations and stochastic differential equations has a long tradition in the kinetic theory of dilute polymer solutions. In 1969, R. Zwanzig [2] described the rigorous equivalence between the two approaches for bead-spring chains with hydrodynamic interaction, and he pointed out the fundamental importance of this equivalence for computer simulations. In particular, Zwanzig rearranged the diffusion tensor in the Fokker-Planck equation so that he obtained the Stratonovich-version of the stochastic differential equation. In 1978, M. Fixman [3] showed in two landmark papers how simulations of polymer dynamics can be constructed and applied (Brownian dynamics simulations).Fixman discussed the relationship between Fokker-Planck equations and stochastic differential equations for bead-spring models with hydrodynamic interaction, even in the presence of constraints. Moreover, he presented numerical integration schemes for stochastic differential equations, in particular, the explicit Euler scheme and the corresponding implicit and predictor-corrector treatments of the diffusion term (which requires an intuitive understanding of the difference between the Itô- and Stratonovich approaches). Fixman also pointed out that the usual integration schemes may not be useful for stochastic differential equations. In the same year, D. L. Ermak and J. A. McCammon carried out a computer simulation for short chains in order to study the effect of hydrodynamic interaction on dynamic equilibrium properties. Their paper [4], which contains a very clear presentation of their explicit Euler scheme for integrating the Itô-version of the stochastic equations of motion (in the mathematical literature, the validity of this most basic integration scheme has been known since 1955) and an explicit procedure for carrying out the Cholesky decomposition of the diffusion tensor, has most frequently been cited in papers on Brownian dynamics. We here show how the rigorous theory of stochastic differential equations can be exploited to obtain more efficient simulation algorithms, even in more general cases where a formulation of the stochastic equations of motion by direct physical arguments is not so straightforward.