The <Emphasis Type="Bold">±</Emphasis><Emphasis Type="Bold">J</Emphasis> spin glass in Migdal-Kadanoff approximation

We study the low-temperature phase of the three-dimensional ± J Ising spin glass in Migdal-Kadanoff approximation. At zero temperature, T = 0, the properties of the spin glass result from the ground-state degeneracy and can be elucidated using scaling arguments based on entropy. The approach to the asymptotic scaling regime is very slow, and the correct exponents are only visible beyond system sizes around 64. At T > 0, a crossover from the zero-temperature behaviour to the behaviour expected from the droplet picture occurs at length scales proportional to T ^-2/ds where d_s is the fractal dimension of a domain wall. Canonical droplet behaviour is not visible at any temperature for systems whose linear dimension is smaller than 16 lattice spacings, because the data are either affected by the zero-temperature behaviour or the critical point behaviour. Received 18 February 2001     

The maximal randomness principle in d-dimensional turbulence

A set of self-consistent equations in one-loop approximation in a statistical model of fully developed homogeneous isotropic turbulence, which is based on the maximal randomness principle of the incompressible velocity field with stationary energy spectral flux, is obtained. Thanks to the applied principle the model statistics becomes essentially non Gaussian. The set of equations does not possess the infrared and ultraviolet divergences near the obtained Kolmogorov spectral exponents. The solution of these equations leads to the Kolmogorov exponents, but its amplitude proportional to the Kolmogorov constantC _k is negative for Euclidean dimensiond=3. Systematic investigation is made of (inertial) steady state scaling solutions for dimensions 2<d<2.55695,where constantC _k (d) becomes positive. Considered in this way, the model stability is discussed in the context of widely studied fractal aspects of turbulence.We have greatly benefited from discussions with Dr. Altaisky from JINR Dubna. The authors (M.H. and M.S.) are grateful to D. I. Kazakov and to director D. V. Shirkov for hospitality at the Laboratory of Theoretical Physics, JINR, Dubna.This work was supported by Fundamental Research Russian Fund, International Scientific Fund (grant R-63000) and by Slovak Grant Agency for Science (grant 2/550/93).     

On the Large Time Asymptotics¶of Decaying Burgers Turbulence

The decay of Burgers turbulence with compactly supported Gaussian “white noise” initial conditions is studied in the limit of vanishing viscosity and large time. Probability distribution functions and moments for both velocities and velocity differences are computed exactly, together with the “time-like” structure functions T _n (t,τ)≡< (u(t+τ) -u(t)) ⁿ >. The analysis of the answers reveals both well known features of Burgers turbulence, such as the presence of dissipative anomaly, the extreme anomalous scaling of the velocity structure functions and self similarity of the statistics of the velocity field, and new features such as the extreme anomalous scaling of the “time-like” structure functions and the non-existence of a global inertial scale due to multiscaling of the Burgers velocity field. We also observe that all the results can be recovered using the one point probability distribution function of the shock strength and discuss the implications of this fact for Burgers turbulence in general. Received: 4 October 1999 / Accepted: 4 February 2000     

Collisionless magnetohydrodynamic turbulence in two dimensions

In this paper we give a formulation of two-dimensional (2D) collisionless magnetohydrodynamic (MHD) turbulence that includes the effects of both electron inertia and electron pressure (or parallel electron compressibility) and is applicable to strongly magnetized collisionless plasmas. We place particular emphasis on the departures from the 2D classical MHD turbulence results produced by the collisionless MHD effects. We investigate the fractal/multi-fractal aspects of spatial intermittency. The fractal model for intermittent collisionless MHD turbulence appears to be able to describe the observed k⁻¹ spectrum in the solar wind. Multi-fractal scaling behaviors in the inertial range are first deduced, and are then extrapolated down to the dissipative microscales. We then consider a parabolic-profile model for the singularity spectrum f (α), as an explicit example of a multi-fractal scenario. These considerations provide considerable insights into the basic mechanisms underlying spatial intermittency in 2D fully developed collisionless MHD turbulence.     

Local multifractal thermodynamics of 3D turbulence

Using results of a direct numerical simulation (DNS) of 3D turbulence we show that the observed generalized scaling (i.e. scaling moments versus moments of different orders) is consistent with a lognormal-like distribution of turbulent energy dissipation fluctuations with moderate amplitudes for all space scales available in this DNS (beginning from the molecular viscosity scale up to largest ones). Local multifractal thermodynamics has been developed to interpret the data obtained using the generalized scaling, and a new interval of space scales with inverse cascade of generalized energy has been found between dissipative and inertial intervals of scales for sufficiently large values of the Reynolds number. Received 21 July 2000     

The axisymmetric equivalent of Kolmogorov's equation

A type of turbulence which is next to local isotropy in order of simplicity, but which corresponds more closely to turbulent flows encountered in practice, is locally axisymmetric turbulence. A representation of the second and third order structure function tensors of homogeneous axisymmetric turbulence is given. The dynamic equation relating the second and third order scalar structure functions is derived. When axisymmetry turns into isotropy, this equation is reduced to the well-known isotropic result: Kolmogorov's equation. The corresponding limiting form is also reduced to the well-known isotropic limiting form of Kolmogorov's equation. The new axisymmetric and theoretical results may have important consequences on several current ideas on the fine structure of turbulence, such as ideas developed by analysis based on the isotropic dissipation rate or such as extended self similarity (ESS) and the scaling laws for the n-order structure functions. Received 20 October 2000 and Received in final form 25 May 2001     

Second-order moving average and scaling of stochastic time series

Long-range correlation properties of stochastic time series y(i) have been investigated by introducing the function σ² _MA = [y(i) - (i)]², where (i) is the moving average of y(i), defined as 1/n y(i - k), n the moving average window and N_max is the dimension of the stochastic series. It is shown that, using an appropriate computational procedure, the function σ _MA varies as n^H where H is the Hurst exponent of the series. A comparison of the power-law exponents obtained using respectively the function σ _MA and the Detrended Fluctuation Analysis has been also carried out. Interesting features denoting the existence of a relationship between the scaling properties of the noisy process and the moving average filtering technique have been evidenced. Received 31 December 2001     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Nonperturbative renormalization group approach to turbulence

We suggest a new, renormalization group (RG) based, nonperturbative method for treating the intermittency problem of fully developed turbulence which also includes the effects of a finite boundary of the turbulent flow. The key idea is not to try to construct an elimination procedure based on some assumed statistical distribution, but to make an ansatz for possible RG transformations and to pose constraints upon those, which guarantee the invariance of the nonlinear term in the Navier-Stokes equation, the invariance of the energy dissipation, and other basic properties of the velocity field. The role of length scales is taken to be inverse to that in the theory of critical phenomena; thus possible intermittency corrections are connected with the outer length scale. Depending on the specific type of flow, we find different sets of admissible transformations with distinct scaling behaviour: for the often considered infinite, isotropic, and homogeneous system K41 scaling is enforced, but for the more realistic plane Couette geometry no restrictions on intermittency exponents were obtained so far. Received: 28 December 1997 / Accepted: 6 August 1998     

Energy Transfer in Spectra of the d-Dimensional Past Grid Turbulence

Free decay theory of the homogeneous and isotropic developed turbulence isconsidered in the d-dimensional case. The basic quantities under our consideration are the kinetic energy spectrum E(k,t) and energy transfer spectrum T(k,t) as functions of wave number k and decay time t. Starting point for studying E and T represents their adaptation from the stationary model which predicts the Kolmogorov spectrum which is multiplicatively dependent on an unknown scaling function F. In order to study the spectra of decaying turbulence both parameters l and εɛ are supposed to be dependent on t. Formerly derived basic integro–differential equation for F (by Adzhemyan, et al., 1998) has been here solved numerically in the dimension interval d∈(2, 3) for two cases of the Saffman invariant and the Loitsyansky integral fixing an arbitrary theor parameter α (α ⩵ 2 and 4, correspondingly). The energy transfer spectrum T(k) has been analyzed for several dimensions d≤3 showing the presence of integration regions in the wavenumber space where an inverse energy cascade can occur. PACS numbers: 47.27.ef, 47.11.-j, 47.27.er     

Beyond Scaling and Locality in Turbulence

An analytic perturbation theory is suggested in order to find finite-size corrections to the scaling power laws. In the frame of this theory it is shown that the first order finite-size correction to the scaling power laws has following form , where η is a finite-size scale (in particular for turbulence, it can be the Kolmogorov dissipation scale). Using data of laboratory experiments and numerical simulations it is shown shown that a degenerate case with α ₀=0 can describe turbulence statistics in the near-dissipation range r > η, where the ordinary (power-law) scaling does not apply. For moderate Reynolds numbers the degenerate scaling range covers almost the entire range of scales of velocity structure functions (the log-corrections apply to finite Reynolds number). Interplay between local and non-local regimes has been considered as a possible hydrodynamic mechanism providing the basis for the degenerate scaling of structure functions and extended self-similarity. These results have been also expanded on passive scalar mixing in turbulence. Overlapping phenomenon between local and non-local regimes and a relation between position of maximum of the generalized energy input rate and the actual crossover scale between these regimes are briefly discussed. PACS: 47.27.-i, 47.27.Gs.     

Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic ɛ-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as r ^{- (d + σ)}, where 2 < σ < 4, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature T _c the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems. Received 8 August 2001     

Model glasses coupled to two different heat baths

In a p-spin interaction spherical spin-glass model both the spins and the couplings are allowed to change with time. The spins are coupled to a heat bath with temperature T, while the coupling constants are coupled to a bath having temperature T_J. In an adiabatic limit (where relaxation time of the couplings is much larger that of the spins) we construct a generalized two-temperature thermodynamics. It involves entropies of the spins and the coupling constants. The application for spin-glass systems leads to a standard replica theory with a non-vanishing number of replicas, n=T/T _J . For p>2 there occur at low temperatures two different glassy phases, depending on the value of n. The obtained first-order transitions have positive latent heat, and positive discontinuity of the total entropy. This is an essentially non-equilibrium effect. The dynamical phase transition exists only for n<1. For p=2 correlation of the disorder (leading to a non-zero n) removes the known marginal stability of the spin glass phase. If the observation time is very large there occurs no finite-temperature spin glass phase. In this case there are analogies with the non-equilibrium (aging) dynamics. A generalized fluctuation-dissipation relation is derived. Received 12 July 1999 and Received in final form 8 December 1999     

Scaling in large Prandtl number turbulent thermal convection

We study the scaling properties of heat transfer Nu in turbulent thermal convection at large Prandtl number Pr using a quasi-linear theory. We show that two regimes arise, depending on the Reynolds number Re. At low Reynolds number, NuPr ^-1/2 and Re are a function of RaPr ^-3/2. At large Reynolds number NuPr ^1/3 and RePr are function only of RaPr ^2/3 (within logarithmic corrections). In practice, since Nu is always close to Ra ^1/3, this corresponds to a much weaker dependence of the heat transfer in the Prandtl number at low Reynolds number than at large Reynolds number. This difference may solve an existing controversy between measurements in SF₆ (large Re) and in alcohol/water (lower Re). We link these regimes with a possible global bifurcation in the turbulent mean flow. We further show how a scaling theory could be used to describe these two regimes through a single universal function. This function presents a bimodal character for intermediate range of Reynolds number. We explain this bimodality in term of two dissipation regimes, one in which fluctuation dominate, and one in which mean flow dominates. Altogether, our results provide a six parameters fit of the curve Nu(Ra, Pr) which may be used to describe all measurements at Pr≥0.7. Received 27 February 2002 / Received in final form 29 May 2002 Published online 31 July 2002     

Finite-size scaling in the theory above the upper critical dimension

We derive exact results for several thermodynamic quantities of the O ( n ) symmetric field theory in the limit in a finite d-dimensional hypercubic geometry with periodic boundary conditions. Corresponding results are derived for an O ( n ) symmetric model on a finite d-dimensional lattice with a finite-range interaction. The leading finite-size effects near T_c of the field-theoretic model are compared with those of the lattice model. For 2 < d < 4, the finite-size scaling functions are verified to be universal. For d > 4, significant lattice effects are found. Finite-size scaling in its usual simple form does not hold for d > 4 but remains valid in a generalized form with two reference lengths. The finite-size scaling functions of the field theory turn out to be nonuniversal whereas those of the lattice model are independent of the nonuniversal model parameters. In particular, the field-theoretic model exhibits finite-size effects whose leading exponents differ from those of the lattice model. The widely accepted lowest-mode approach is shown to fail for both the field-theoretic and the lattice model above four dimensions. Received: 20 October 1997 / Accepted: 5 March 1998     

Phase transition and random-field induced domain wall response of SrTi O 3

The dielectric permittivity ε^′ - i of SrTi ¹⁸O ₃ (STO18) is studied under a dc electric field E as a function of the temperature, T. In ε^′ vs. T, a double-peak is found when 0 < E < 30 KV/m. While the peak at high-T is attributed to the smeared ferroelectric phase transition, the low-T one is induced by domain wall motion. The transverse Ising model including an external homogeneous and quenched random-fields is successfully used to describe both the smeared phase transition and the domain wall response in the low-T domain state. The calculations are in good agreement with the experimental results. Received 4 January 2002 / Received in final form 25 March 2002 Published online 19 July 2002