Long-time, large-scale properties of the random-force-driven Burgers equation

Field theory of the inverse cascade in two-dimensional turbulence

A two-dimensional fluid, stirred at high wave numbers and damped by both viscosity and linear friction, is modeled by a statistical field theory. The fluid's long-distance behavior is studied using renormalization-group (RG) methods, as begun by Forster, Nelson, and Stephen [Phys. Rev. A 16, 732 (1977)]. With friction, which dissipates energy at low wave numbers, one expects a stationary inverse energy cascade for strong enough stirring. While such developed turbulence is beyond the quantitative reach of perturbation theory, a combination of exact and perturbative results suggests a coherent picture of the inverse cascade. The zero-friction fluctuation-dissipation theorem (FDT) is derived from a generalized time-reversal symmetry and implies zero anomalous dimension for the velocity even when friction is present. Thus the Kolmogorov scaling of the inverse cascade cannot be explained by any RG fixed point. The beta function for the dimensionless coupling ĝ is computed through two loops; the ĝ3 term is positive, as already known, but the ĝ5 term is negative. An ideal cascade requires a linear beta function for large ĝ, consistent with a Padé approximant to the Borel transform. The conjecture that the Kolmogorov spectrum arises from an RG flow through large ĝ is compatible with other results, but the accurate k(-5/3) scaling is not explained and the Kolmogorov constant is not estimated. The lack of scale invariance should produce intermittency in high-order structure functions, as observed in some but not all numerical simulations of the inverse cascade. When analogous RG methods are applied to the one-dimensional Burgers equation using an FDT-preserving dimensional continuation, equipartition is obtained instead of a cascade--in agreement with simulations.

Renormalization-group and numerical analysis of a noisy Kuramoto-Sivashinsky equation in 1 + 1 dimensions

The long-wavelength properties of a noisy Kuramoto-Sivashinsky (KS) equation in 1 + 1 dimensions are investigated by use of the dynamic renormalization group (RG) and direct numerical simulations. It is shown that the noisy KS equation is in the same universality class as the Kardar-Parisi-Zhang (KPZ) equation in the sense that they have scale invariant solutions with the same scaling exponents in the long-wavelength limit. The RG analysis reveals that the RG flow for the parameters of the noisy KS equation rapidly approach the KPZ fixed point with increasing strength of the noise. This is supplemented by numerical simulations of the KS equation with a stochastic noise, in which scaling behavior close to the KPZ scaling can be observed even in a moderate system size and time.

Stochastic equation for a jumping process with long-time correlations

A jumping process, defined in terms of the jump size and waiting time distributions, is presented. The jumping rate depends on the process value. The process, which is Markovian and stationary, relaxes to an equilibrium and is characterized by a power-law autocorrelation function. Therefore, it can serve as a model of 1/f noise as well as of the stochastic force in the generalized Langevin equation. This equation is solved for noise correlations approximately 1/t ; the resulting velocity distribution has sharply falling tails. The system preserves memory about the initial condition for a very long time.

Solving the Chapman-Kolmogorov equation for a jumping process

A general solution to the Chapman-Kolmogorov equation for a jumping process called the "kangaroo process" is derived. A special case of algebraic dependences is discussed in detail. In particular, simple asymptotic formulas for probability distribution are presented. It is demonstrated that there are two different classes of limiting stationary distributions. An expression for the covariance is also derived.

Stochastic processes with finite correlation time: modeling and application to the generalized Langevin equation

The kangaroo process (KP) is characterized by various forms of covariance and can serve as a useful model of random noises. We discuss properties of that process for the exponential, stretched exponential, and algebraic (power-law) covariances. Then we apply the KP as a model of noise in the generalized Langevin equation and simulate solutions by a Monte Carlo method. Some results appear to be incompatible with requirements of the fluctuation-dissipation theorem because probability distributions change when the process is inserted into the equation. We demonstrate how one can construct a model of noise free of that difficulty. This form of the KP is especially suitable for physical applications.

Coupled map models for chaos in extended systems

Coupled maps with conserved quantities are introduced as models for chaos in extended systems. The long-wavelength limit of a simple one-dimensional example is investigated in detail. A Langevin equation for this system is derived, and the applicability of the fluctuation-dissipation theorem is discussed. Several numerical experiments to investigate possible thermodynamic properties are described.