11 - Open systems and escape rates

Summary

The escape-rate formalism

One of the most interesting developments in the theory of irreversible processes is a connection, first explored by Gaspard and Nicolis, between the dynamical properties of open systems and the hydrodynamic or transport properties of such systems. We will explore this connection in Chapter 13, but first we must consider the dynamical properties of open systems. We consider a system to be open if the phase space of the system has physical boundaries and there is a mapping or transformation which can take phase points inside the boundaries to phase points outside the boundaries. Further, we will assume that the boundaries are such that once a phase point passes a boundary, it can never return to the bounded system. Thus the boundaries on the phase space region may be considered to be absorbing. To get some idea of the motivation for considering open systems, we might imagine a Brownian particle diffusing in a fluid inside a container with absorbing boundaries. The motion of the particle is really deterministic and can be described – microscopically – by some transformation in the phase space of the entire system. Now, when we describe the motion of the phase point, we lose the Brownian particle whenever it encounters the boundary of the container. If we were to describe the motion macroscopically, we would solve the diffusion equation for the probability density of the Brownian particle, in the fluid, with absorbing boundaries. The probability of finding the particle inside the container is an exponentially decreasing function of time with decay coefficient depending on the diffusion coefficient of the Brownian particle in the fluid, and on the geometry of the container.