The theory of fluctuations is extended to nonlinear systems far from equilibrium. Systems whose evolution involves two separate time scales, e.g., chemically reacting mixtures near a local equilibrium regime, are studied in detail. It is shown that the usual stochastic description of chemical kinetics based on a "birth and death" model is inadequate and has to be replaced by a more detailed phase-space description. This enables one to develop for such systems a plausible mechanism for the emergence of instabilities, in which the departure from the steady state is governed by large fluctuations of macroscopic size, while small thermal fluctuations are still described by a generalization of Einstein's equilibrium theory. On the other hand, far from a local equilibrium regime, infinitesimal fluctuations may increase and attain macroscopic values. In this case the system evolves to a state of "generalized turbulence", in which the distinction between macroscopic averages and fluctuations becomes meaningless.
The theory of hydrodynamic instability has always been an important part of fluid dynamics [see, e.g., Chandrasekhar, in Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, England, 1961) and Non-Equilibrium Thermodynamics, Variation Techniques, and Stability, R. J. Donnelly, R. Herman, and I. Prigogine, Eds. (University of Chicago Press, Chicago, Ill., 1966)]. Such instabilities involve both convective processes (such as mechanical flow) and dissipative processes (such as viscous dissipation). We investigate the possibility of an instability in purely dissipative systems involving chemical reactions and transport processes such as diffusion, but no hydrodynamic motion. We demonstrate that for well-defined values of the constraints such as the chemical affinities of the over-all reactions and the constants involved, such systems can indeed become unstable. Such an instability is investigated following an example of autocatalytic reactions first proposed by Turing. The major feature of this instability is its symmetry breaking character. Indeed, beyond the transition point the stable steady state is inhomogenous, the diffusion compensating the differences in reaction rates. The existence of such instabilities has far-reaching consequences which are briefly discussed.
For dynamical systems of large spatial extension giving rise to transport phenomena, like the Lorentz gas, we establish a relationship between the transport coefficient and the difference between the positive Lyapunov exponent and the Kolmogorov-Sinai entropy per unit time, characterizing the fractal and chaotic repeller of trapped trajectories. Consequences for nonequilibrium statistical mechanics are discussed.
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