Biomedical subjects

Itamar Procaccia

Publications and source records attributed to Itamar Procaccia.

42 records · Page 3Linked to original sources

PubMed2002

Inverse cascade regime in shell models of two-dimensional turbulence.

We consider shell models that display an inverse energy cascade similar to two-dimensional turbulence (together with a direct cascade of an enstrophylike invariant). Previous attempts to construct such models ended negatively, stating that shell models give rise to a "quasiequilibrium" situation with equipartition of the energy among the shells. We show analytically that the quasiequilibrium state predicts its own disappearance upon changing the model parameters in favor of the establishment of an inverse cascade regime with Kolmogorov scaling. The latter regime is found where predicted, offering a useful model to study inverse cascades.

Journal Article↗

PubMed2002

Quasi-Gaussian statistics of hydrodynamic turbulence in 4 / 3 +epsilon dimensions.

The statistics of two-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian-like in equilibrium. The skewness S identical with S3(R)/S(3/2)(2)(R) was measured as S (exp) approximately 0.03. This contradiction is lifted by understanding that two-dimensional turbulence is not far from a situation with equipartition of enstrophy, which exists as true thermodynamic equilibrium with K41 exponents in space dimension of d= 4 / 3. We evaluate the skewness S( d) for 4 / 3 < or =d< or =2, showing that S(d)=0 at d= 4 / 3, and that it remains as small as S (exp) in two dimensions.

Journal Article↗

PubMed2002

Iterated conformal dynamics and Laplacian growth.

The method of iterated conformal maps for the study of diffusion limited aggregates (DLA) is generalized to the study of Laplacian growth patterns and related processes. We emphasize the fundamental difference between these processes: DLA is grown serially with constant size particles, while Laplacian patterns are grown by advancing each boundary point in parallel, proportional to the gradient of the Laplacian field. We introduce a two-parameter family of growth patterns that interpolates between DLA and a discrete version of Laplacian growth. The ultraviolet putative finite-time singularities are regularized here by a minimal tip size, equivalently for all the models in this family. With this we stress that the difference between DLA and Laplacian growth is not in the manner of ultraviolet regularization, but rather in their deeply different growth rules. The fractal dimensions of the asymptotic patterns depend continuously on the two parameters of the family, giving rise to a "phase diagram" in which DLA and discretized Laplacian growth are at the extreme ends. In particular, we show that the fractal dimension of Laplacian growth patterns is higher than the fractal dimension of DLA, with the possibility of dimension 2 for the former not excluded.

Journal Article↗

PubMed2002

Quasistatic fractures in brittle media and iterated conformal maps.

We study the geometrical characteristic of quasistatic fractures in brittle media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern, it crosses over from about 0.5 for small scales to about 0.75 for large scales. We propose that this crossover reflects the increased ramification of the fracture pattern.

Journal Article↗

PubMed2002

Multifractal structure of the harmonic measure of diffusion-limited aggregates.

The method of iterated conformal maps allows one to study the harmonic measure of diffusion-limited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as 10(-35), and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions D(q) are infinite for q<q*, where q* is of the order of -0.2. In the language of f(alpha) this means that alpha(max) is finite. The f(alpha) curve loses analyticity (the phenomenon of "phase transition") at alpha(max) and a finite value of f(alpha(max)). We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of q* and f(alpha(max)). We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.

Journal Article↗

PubMed2002

Statistically preserved structures in shell models of passive scalar advection.

It was conjectured recently that statiscally preserved structures underlie the statistical physics of turbulent transport processes. We analyze here in detail the time-dependent (noncompact) linear operator that governs the dynamics of correlation functions in the case of shell models of passive scalar advection. The problem is generic in the sense that the driving velocity field is neither Gaussian nor delta correlated in time. We show how to naturally discuss the dynamics in terms of an effective compact operator that displays "zero modes," which determine the anomalous scaling of the correlation functions. Since shell models have neither a Lagrangian structure nor "shape dynamics," this example differs significantly from standard passive scalar advection. Nevertheless, with the necessary modifications, the generality and efficacy of the concept of statistically preserved structures are further exemplified. In passing we point out a bonus of the present approach, in providing analytic predictions for the time-dependent correlation functions in decaying turbulent transport.

Journal Article↗