"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

Detecting strange attractors in turbulence

Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.

Synchronization in chaotic systems

/pdf/determining-lyapunov-exponents-from-a-time-series-3bxxq6d0wt.pdf

Determining Lyapunov exponents from a time series

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

Statistical mechanics away from equilibrium is in a formative stage, where general concepts slowly emerge. We restrict ourselves here to the study of transport phenomena, especially heat transport, and consider several well-studied classes of systems: finite systems with isokinetic thermostats, infinite classical Hamiltonian systems, infinite quantum spin systems. For those classes we discuss how various physical quantities can be defined. Our review however leaves several basic questions quite open. * Math. Dept., Rutgers University, and IHES, 91440 Bures sur Yvette, France. email: ruelle@ihes.fr

What physical quantities make sense in nonequilibrium statistical mechanics

The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems describing units of fluid of smaller and smaller spatial extent. These units are macroscopic but have a few degrees of freedom, and they can be studied by the methods of (microscopic) nonequilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specifically, we obtain the formula for the exponents of the structure functions (). The meaning of the adjustable parameter κ is that when an eddy of size r has decayed to eddies of size , their energies have a thermal distribution. The above formula, with is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture that can thus also be used in related problems.

https://www.pnas.org/content/pnas/109/50/20344.full.pdf

Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics

The free energy of a system of particies interacting by a two-body potential was investigated for the canonical ensemble The existence of a limit for the free energy per particle when the system becomes infinite and the stability conditions were proved rigorously for a large class of potentials. It was also shown that the pressure is a continuous function of the volume for bounded potentials. The grand canonical ensemble was investigated in a similar manner and a generaiization of the resuits of Yang and Lee is given. (auth)

Classical statistical mechanics of a system of particles

Techniques due toR. L. Dobrushin andR. Griffiths are combined to prove the existence of a first order phase transition at low temperature for a class of lattice systems with non nearest-neighbour interaction.

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Condensation of lattice gases

We study nonequilibrium statistical mechanics in the presence of a thermostat acting by random forces, and propose a formula for the rate of entropy productione(μ) in a state μ. When μ is a natural nonequilibrium steady state we show thate(μ)≥0, and sometimes we can provee(μ)>0.

/pdf/positivity-of-entropy-production-in-the-presence-of-a-random-hnsyqjmnqq.pdf

David Ruelle

Papers

What physical quantities make sense in nonequilibrium statistical mechanics

Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics

Classical statistical mechanics of a system of particles

Condensation of lattice gases

Positivity of entropy production in the presence of a random thermostat