Information-theoretic study of pattern formation: Rate of entropy production of random fractals

Kaufman, J. H.; Melroy, O. R.; Dimino, G. M.

doi:10.1103/physreva.39.1420

Cited by 14 publications

(11 citation statements)

References 17 publications

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“…14(a) and (b) illustrate. The observed distribution is similar to one calculated using an on-lattice algorithm for DLA clusters [25,44], as Fig. 14(c) shows.…”

Section: Unscreened Anglesupporting

confidence: 82%

“…Our measured distribution function of unscreened angles is qualitatively (but not quantitatively) similar to that found in an on-lattice computation for DLA [25]; it would be interesting to compare the measurements with an offlattice DLA computation. [44] and with a Gaussian distribution of the same mean and variance as the measured distribution (dotted line). Data in (a) and (b) for different forcings ∆P fall on a single curve, indicating the independence of the forcing.…”

Section: Discussionmentioning

confidence: 99%

“…We investigate the screening effect following a procedure used by Kaufman et al [44], who determined the distribution of unscreened angles for growing DLA clusters. Images of the cluster at successive time steps reveal the "active region" where growth occurs.…”

Section: Unscreened Anglementioning

confidence: 99%

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Fractal dimension and unscreened angles measured for radial viscous fingering

Praud

Swinney

2005

Phys. Rev. E

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We have examined fractal patterns formed by the injection of air into oil in a thin (0.127 mm) layer contained between two cylindrical glass plates of 288 mm diameter (a Hele-Shaw cell), for pressure differences in the range 0.25 < or = DeltaP < or = 1.75 atm. We find that an asymptotic structure is reached at large values of the ratio r/b, where r is the pattern radius and b the gap between the plates. Both the driving force and the size of the pattern, which reaches r/b = 900, are far larger than in past experiments. The fractal dimension D0 of the pattern for large r/b is 1.70 +/- 0.02. Further, the generalized dimensions D(q) of the pattern are independent of q , D(q) approximately 1.70 for the range examined, -11 < q < 17; thus the pattern is self-similar within the experimental uncertainty. The results for D(q) agree well with recent calculations for diffusion-limited aggregation (DLA) clusters. We have also measured the probability distribution of unscreened angles. At late times, the distribution approaches a universal (i.e., forcing and size-independent) asymptotic form that has mean 145 degrees Celsius and standard deviation 36 degrees Celsius. These results indicate that the distribution function for the unscreened angle is an invariant property of the growth process.

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“…14(a) and (b) illustrate. The observed distribution is similar to one calculated using an on-lattice algorithm for DLA clusters [25,44], as Fig. 14(c) shows.…”

Section: Unscreened Anglesupporting

confidence: 82%

Section: Discussionmentioning

confidence: 99%

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Fractal dimension and unscreened angles measured for radial viscous fingering

Praud

Swinney

2005

Phys. Rev. E

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“…Now, as far as 8~is concerned it must lie somewhere below 8~~z rad, and simulations suggest typically 8~= 3z/4. It is interesting to note that this angle appears to be close to the critical angle found by Kaufman and co-workers [25,26], except that here it appears as the opening angle making the maximum contribution to the growth probability.…”

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confidence: 83%

Wedge fjord model for dendritic growth

Hentschel

1992

Phys. Rev. A

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A wedge fjord model for screened growth is used to investigate the extent to which such a topology is su5cient to account for the dendritic multifractal growth characteristic of diftusion-limited aggregation. The clusters found are dendritic fractals with growth probabilities having a left-sided multifractal measure with q, =0 and a, "=.PACS number (s): 68.70.+w, 05.40.+j, 64.60.Ak, 81.10.Jt Despite the enormous importance of diffusion-limited aggregation [ll (DLA) as the basic paradigm controlling the structure of such apparently diverse phenomena as two-fluid flow in porous media [2], electrochemical deposition [3,4], dielectric breakdown [5], and retinal vasculature [6], we still remain far from a satisfactory understanding when questions concerning the relationship between growth and the geometry of the resulting dendritic aggregates are asked. We know that DLA has a stochastically self-similar structure and that fractal [7] and multifractal [8-10]concepts are required for a description of both structure, which may be weakly multifractal [11],and the growth probabilities for which the harmonic measure is strongly multifractal [12,13]. More recent theoretical work has been concerned with trying to understand the properties of the f(a) vs a and Dv vs q curves found in computer simulations of DLA and specifically whether the complete spectrum of Dq for -~& q. &~exists. Using exact enumeration techniques over the annealed ensemble of all clusters, Lee and Stanley [14] gave support to the hypothesis that the multifractal spectrum is not complete: numerical evidence suggested that for q & q, = -1.0

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“…This gives a contribution to the program length that is a comparable functional form as the particle-particle repulsion energy in the Flory formalism. The contribution to the program length will be: pairs = N pairs¯ pair (19) where N pairs is the number of hypercubes that contain pairs and¯ pair is the number of bits required to specify that an address has a pair in it. The number of pairs will be given by the number of structural Fig.…”

Section: The Complexity and Thermodynamics Of Fractal Aggregatesmentioning

confidence: 99%

Algorithmic Complexity and Thermodynamics of Fractal Growth Processes

Dewey

1997

Fractals

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A statistical mechanical formalism is developed using the computer information concepts of algorithmic complexity and Kolmogorov universal probability. This formalism provides a thermodynamic description of microstates of a system. For an isolated classical system, the algorithmic complexity is equal to the thermodynamic entropy. This approach does not rely on probabilistic ensemble concepts and can be applied to non-ergodic systems. An H-function is developed from the Kolmogorov universal probability that satisfies the properties of an entropy function. Using this approach, the thermodynamics of irreversible growth processes far from equilibrium can be investigated. Entropy functions for fractal growth processes far from equilibrium are developed from the algorithmic complexity and are seen to be similar to Flory-type equilibrium functions. This development does not require energy minimization procedures associated with equilibrium arguments. Using this approach, constraints can be put on the types of mean field models that will yield fractal structures. The algorithmic complexity of a system can also be used to explore the role of fluctuations and criticality in highly ordered systems.

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Information-theoretic study of pattern formation: Rate of entropy production of random fractals

Cited by 14 publications

References 17 publications

Fractal dimension and unscreened angles measured for radial viscous fingering

Fractal dimension and unscreened angles measured for radial viscous fingering

Wedge fjord model for dendritic growth

Algorithmic Complexity and Thermodynamics of Fractal Growth Processes

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