The need for a statistical theory

The linear perturbation theory presented in Chapter 7 makes it clear that a fluid configuration can, under certain circumstances, be unstable to perturbations. Once the perturbations grow to sufficiently large amplitudes, the linear theory is no longer applicable. Hence the linear theory is unable to predict what eventually happens to an unstable fluid system.

To understand the effect of an instability on a general dynamical system, let us employ the notion of a phase space introduced in Chapter 1. Figure 8.1 is a schematic representation of the phase space of a dynamical system, within which let P be a point corresponding to an unstable equilibrium. If the state of the system is represented exactly by P, then the state does not change by virtue of equilibrium. If, however, there is some perturbation around the equilibrium, then the state of the system is represented by some point in the neighborhood of P, and as the perturbation grows, the point in the phase space moves away from P. Thus, depending on whether the initial state was exactly at P or slightly away, the final state after some time can lie in very different regions of the phase space. Because of the limited accuracy in any measurement in a realistic situation, one can only assert that the initial state of a system lies in some finite region of the phase region.