The stochastic description of the dynamics of a many-body system consisting of a
single, large particle, suspended in a fluid made up of an enormous number of much
smaller, molecular-sized particles was a major advance in Physics. This approach to
describe Brownian motion was pioneered by Einstein [1] and Langevin [2], and their
work marks the beginning of a new way to treat the dynamics of many-body systems
where a dynamical variable of interest (in this case the velocity of the large particle) is
singled out and some other aspects of the problem are treated by the theory of random
processes. This treatment is in contrast with the more traditional, kinetic-theory
approach of considering the explicit dynamics of all the particles in the system.
Since this early work these stochastic methods have been generalised and refined into a
field of major significance. Many areas of research have been influenced by this Brownian
approach and stochastic methods have been applied to problems in such diverse areas of
physics, chemistry, biology and engineering [3–5]. One of the main uses of these methods
in the theory of matter is to treat many-body systems in which different degrees of freedom
operate on very different time scales. This is because this approach allows the separation of
these very different degrees of freedom and the treatment of each of them by different methods. This separation is achieved by averaging over some variables, which results in equations of motion for only the chosen variables. This approach to singling out particular
variables is, however, quite general and not restricted to situations where there is a clear distinction between “fast” and “slow” variables, and can be applied to any system whose
dynamics is described by equations of motion such as Newton’s equations. The equations
resulting from this alternative approach are exact and equivalent to the starting equations
from which they are derived and, thus, may be used to provide an exact, but alternative
description of the time evolution of any dynamical variable. However, the form of these new
equations of motion also enables certain variables to be treated approximately and simply,
while allowing the remaining variables to be treated in greater detail. Many methods of this
type have been developed under the heading of generalised Langevin methods (GLE) and
have provided insight into the properties of many-body systems ranging from atomic liquids, atoms on surfaces and polymeric solutions to colloidal suspensions.