Leo P. Kadanoff

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

An experimental study of Rayleigh-Benard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 10 7 Ra 12 ) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra . However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu , proportional to

HYDRODYNAMIC EQUATIONS AND CORRELATION FUNCTIONS

Ra^{\frac{1}{3}}

Notes on Migdal's recursion formulas

while experiment gives an index much closer to

Boolean Dynamics with Random Couplings

\frac{2}{7}

Turbulent heat flow: Structures and scaling

. A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

Quasiperiodicity in dissipative systems: A renormalization group analysis

Abstract The response of a system to an external disturbance can always be expressed in terms of time dependent correlation functions of the undisturbed system. More particularly the linear response of a system disturbed slightly from equilibrium is characterized by the expectation value in the equilibrium ensemble, of a product of two space- and time-dependent operators. When a disturbance leads to a very slow variation in space and time of all physical quantities, the response may alternatively be described by the linearized hydrodynamic equations. The purpose of this paper is to exhibit the complicated structure the correlation functions must have in order that these descriptions coincide. From the hydrodynamic equations the slowly varying part of the expectation values of correlations of densities of conserved quantities is inferred. Two illustrative examples are considered: spin diffusion and transport in an ordinary one-component fluid. Since the descriptions are equivalent, all transport processes which occur in the nonequilibrium system must be exhibited in the equilibrium correlation functions. Thus, when the hydrodynamic equations predict the existence of a diffusion process, the correlation functions will include a part which satisfies a diffusion equation. Similarly when sound waves occur in the nonequilibrium system, they will also be contained in the correlation functions. The description in terms of correlation functions leads naturally to expressions for the transport coefficients like those discussed by Kubo. The analysis also leads to a number of sum rules relating the dissipative linear coefficients to thermodynamic derivatives. It elucidates the peculiarly singular limiting behavior these correlations must have.

Quantum statistical mechanics : Green's function methods in equilibrium and nonequilibrium problems

Abstract A set of renormalization group recursion formulas which were proposed by Migdal are rederived, reinterpreted, and critically analyzed. The new derivation shows the connection between these formulas and previous work on renormalization via decimation and block transformations. The new interpretation which arises from these derivations indicates that Midgals formulas are best understood as referring to systems in which the couplings are anisotropic. A strong indication of the correctness of this reinterpretation comes from the two-dimensional Ising model: The new interpretation gives an exact (!) expression for the critical couplings in this case for all ratios of Jx to Jy. This paper describes the major failings of this approximation which arise from its source as a decimation approximation, in terms of the well-known inadequacy of the fixed points which result from this type of scheme. Some proposals for improvement of the approximation are described. Finally, a new potential-moving scheme is proposed which is used to show that the Migdal approximation is exact when the potentials are strong and ferromagnetic in sign.

Correlation functions on the critical lines of the Baxter and Ashkin-Teller models

This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d,the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.

Singularities and similarities in interface flows

Geometrical structures and scaling behavior provide insights into the nature of convective turbulence and some risky generalizations about “complex systems.”

EXTENDED CHAOS AND DISAPPEARANCE OF KAM TRAJECTORIES

Abstract Dynamical systems with quasiperiodic behavior, i.e., two incommensurate frequencies, may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number. In this paper these curves are followed using renormalization group techniques which are applied to a one-dimensional system (circle) and also to an area-contracting map of an annulus. Two fixed points are found representing different types of universal behavior: a trivial fixed point for smooth motion and a nontrivial fixed point. The latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow. Fixed point functions are determined numerically and via an e-expansion and eigenvalues are calculated.