Unifying scaling theory for vortex dynamics in two-dimensional turbulence

We present a scaling theory for unforced inviscid two-dimensional turbulence. Our model unifies existing spatial and temporal scaling theories. The theory is based on a self-similar distribution of vortices of different sizes A. Our model uniquely determines the spatial and temporal scaling of the associated vortex number density which allows the determination of the energy spectra and the vortex distributions. We find that the vortex number density scales as n(A,t)-t(-2/3)/A, which implies an energy spectrum E-k(-5), significantly steeper than the classical Batchelor-Kraichnan scaling. High-resolution numerical simulations corroborate the model.     

The complete set of Hamiltonian intermittency scaling behaviors

Renormalization group equations describing the phenomenon of intermittency in Hamiltonian systems are presented. All solutions satisfying certain physical constraints are obtained; they are the complete set of simple singularities. Further considerations lead to precise predictions for scaling behavior at the onset of intermittency.     

Inter-cluster scaling in two-dimensional colloidal aggregation

Cluster-cluster aggregation has been studied experimentally in two-dimensional colloidal systems of moderately high area fraction. In the diffusion limited regime the system spontaneously exhibits ordering beyond the scale of the fractal aggregates, which persists into the gelled state. This ordering arises from an effective inter-cluster repulsion due to the mutually exclusive depletion zones surrounding each cluster. Various metrical and topological properties of the system indicate that this long-range order is stationary, apart from the inherent change of scale as the clusters grow. Topological correlations show that the inter-cluster self-organization is compatible with maximum entropy: the non-equilibrium system behaves as if it were in statistical equilibrium, and no physical forces are involved in the long-range order. For the reaction-limited case no such inter-cluster ordering is evident at any stage, the system is nonstationary, and the aggregation is governed by specific physical forces.     

A scaling theory of clusters in the lattice gas

It is proposed that the number ofp-particle,k-bond lattice gas cluster configurations is of the form exp{σpf(k/p)} in the limitp→t8. A simple modification permits application to finite clusters, with the consequence that asymptotically the cluster partition function is of the droplet form, i.e., Z _p =exp[kp{itμp^1?1/d }+O(ln p)]. The scaling function for two-dimensional lattices is determined numerically and is found to be qualitatively universal. The scaling theory is used to investigate the size dependence of the surface free energy. The surface tension of small clusters can significantly exceed its limiting value. For intermediate cluster sizes (～100 particles) there is a modest reduction in surface tension, in accord with Tolman's prediction and the results of recent Monte Carlo studies.     

A universal scaling theory for complexity of analog computation

We discuss the computational complexity of solving linear programming problems by means of an analog computer. The latter is modeled by a dynamical system which converges to the optimal vertex solution. We analyze various probability ensembles of linear programming problems. For each one of these we obtain numerically the probability distribution functions of certain quantities which measure the complexity. Remarkably, in the asymptotic limit of very large problems, each of these probability distribution functions reduces to a universal scaling function, depending on a single scaling variable and independent of the details of its parent probability ensemble. These functions are reminiscent of the scaling functions familiar in the theory of phase transitions. The results reported here extend analytical and numerical results obtained recently for the Gaussian ensemble.     

Anisotropic homogeneous turbulence: hierarchy and intermittency of scaling exponents in the anisotropic sectors

We present the first measurements of anisotropic statistical fluctuations in perfectly homogeneous turbulent flows. We address both problems of intermittency in anisotropic sectors and hierarchical ordering of anisotropies on a direct numerical simulation of a three dimensional random Kolmogorov flow. We achieved an homogeneous and anisotropic statistical ensemble by randomly shifting the forcing phases. We observe high intermittency as a function of the order of the velocity correlation within each fixed anisotropic sector and a hierarchical organization of scaling exponents at fixed order of the velocity correlation at changing the anisotropic sector.     

A rigorous theory of finite-size scaling at first-order phase transitions

A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetizationM _per(h, L) in cubes of sizeL ^d under periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointh _t , we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeM _per(h _t )+M tanh[ML ^d (h–h_t)] withM _per(h) denoting the infinite-volume magnetization and M=1/2[M _per(h _t +0)–M _per(h _t –0)]. Introducing the finite-size transition pointh _m (L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift ish _t –h _m (L)=O(L ^–2d ), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the pointh _t from finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.On leave from: Institut für Theoretische Physik, FU-Berlin, D-1000 Berlin 33, Federal Republic of Germany.     

Classical scaling theory of quantum resonances

The quantum resonances occurring with delta-kicked particles are studied with the help of a fictitious classical limit, establishing a direct correspondence between the nearly resonant quantum motion and the classical resonances of a related system. A scaling law which characterizes the structure of the resonant peaks is derived and numerically demonstrated.     

A single parameter scaling theory in a disordered layered system

We show that in a disordered layered system all states are localized if k_Fl<4Πg_c, irrespective of the interlayer coupling t _⊥. The crossover of dimensionality depends on t_⊥. Near the critical value t_⊥^c at which the Anderson transition occurs, the correlation length ζ^m and the localization length ζ^loc are calculated.     

A new mean field theory on critical scaling in spin-glasses

The critical exponents γ, δ in the non-linear susceptibility are given by making use of the free energy simply consisting of order parameter, magnetization and what was previously named an internal field parameter. The crossover exponent ø also is obtained. It is found that they satisfy the scaling relations proposed independently by Suzuki et al.     

A scaling theory of bifurcations in the symmetric weak-noise escape problem

We consider two-dimensional overdamped double-well systems perturbed by white noise. In the weak-noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double-well system are varied, a unique MPEP may bifurcate into two equally likely MPEPs. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. We quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our construction relies on a new scaling theory, which yields critical exponents describing weak-noise behavior at the bifurcation point, near the saddle.     

Velocity intermittency in a buoyancy subrange in a two-dimensional soap film convection experiment

In a two-dimensional soap film convection experiment, the velocity fields are found to be strongly intermittent in the buoyancy subrange, which was reported to be nonintermittent in a recent numerical simulation. The structure functions Sq(l)(= ) exhibit self-similar structures and can be described by power laws l(zetaq) for integers 8 > or = q < or = 1. By extending Kolmogorov's refined similarity hypothesis to our system, an analytical form is derived for the scaling exponent zeta(q) = q/2 + (mu/18)(3q - q2). Our measurements yield mu = 0.42, which is significantly greater than 0.2 found in high Reynolds number turbulence in wind tunnels.     

Self-consistent scaling theory for logarithmic-correction exponents

Multiplicative logarithmic corrections frequently characterize critical behavior in statistical physics. Here, a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific-heat critical exponent vanishes. Also, the theory is widened to encompass the correlation function. The new relations are then confronted with results from the literature, and some new predictions for logarithmic corrections in certain models are made.     

Exact solution and scaling theory of superradiance

The general scaling theory of transient phenomena near the instability point, which has been proposed by one of the present authors (M.S.), is applied to investigate the fluctuation and relaxation of superradiance near the complete inversion (or instability point). An exact solution for a simple model of superradiance has been obtained to study the relaxation and fluctuation of it near the complete inversion and to confirm the validity of the scaling theory. It is found that this asymptotic evaluation method yields very good results for a large system size Ω. The Ω-expansion method by van Kampen and by Kubo et al. is also discussed in this model in order to clarify the connection of it with the scaling theory.