Brownian Dynamics at Boundaries and Interfaces: In Physics, Chemistry, and Biology

Book Preface

Brownian dynamics serve as mathematical models for the diffusive motion of microscopic particles of various shapes in gaseous, liquid, or solid environments. The renewed interest in Brownian dynamics is due primarily to their key role in molecular and cellular biophysics: diffusion of ions and molecules is the driver of all life. Brownian dynamics simulations are the numerical realizations of stochastic differential equations (SDEs) that model the functions of biological microdevices such as protein ionic channels of biological membranes, cardiac myocytes, neuronal synapses, and many more. SDEs are ubiquitous models in computational physics, chemistry, biophysics, computer science, communications theory, mathematical finance theory, and many other disciplines. Brownian dynamics simulations of the random motion of particles, be it molecules or stock prices, give rise to mathematical problems that neither the kinetic theory of Maxwell and Boltzmann nor Einstein’s and Langevin’s theories of Brownian motion could predict.

Kinetic theory, which assigns probabilities to configurations of ensembles of particles in phase space, assumes that the ensembles are in thermodynamic equilibrium, which means that no net current is flowing through the system. Thus it is not applicable to the description of nonequilibrium situations such as conduction of ions through protein channels, nervous signaling, calcium dynamics in cardiac myocytes, the process of viral infection, and countless other situations in molecular biophysics.

The motion of individual particles in the ensemble is not described in sufficient detail to permit computer simulations of the atomic or molecular individual motions in a way that reproduces all macroscopic phenomena. The Einstein statistical characterization of the motion of a heavy particle undergoing collisions with the much smaller particles of the surrounding medium lays the foundation for computer simulations of the Brownian motion. However, pushing Einstein’s description beyond its range of validity leads to artifacts that baffle the simulators: particles move without velocity, so there is no telling when they enter or leave a given domain. Theoretically, they cross and recross interfaces an infinite number of times in any finite time interval. Thus the simulation of Brownian particles in a small domain surrounded by a continuum becomes problematic. The Langevin description, which includes velocity, partially remedies the problem. There is, however, a price to pay: the dimension, and therefore the computational complexity, is doubled.