Feb 1, 1989 - study of pattern formation: Rate of entropy production of random fractals. J.H. Kaufman and O. R. Melroy. IBM Almaden Research Center, 650 ...
PHYSICAL REVIEW A
FEBRUARY 1, 1989
VOLUME 39, NUMBER 3
Rate of entropy production of random fractals
study of pattern formation:
Information-theoretic
J. H. Kaufman and O. R. Melroy IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099 G. M. Dimino University
of California, Santa
Cruz, Department
of Physics, Santa
Cruz, California 95064
(Received 27 July 1988)
A model dynamical system is described which generates spatial patterns based only on connectivity and screening. The growth probability at sites on the pattern is determined by the extent to which the sites are "screened" by points on the exterior of the pattern. The scaling properties were found to depend on the strength of the imposed screening constraint. Information theory was used to measure the rate of entropy production of the different patterns. A critical point in the screening constraint was observed where the pattern geometry changed from two dimensional (compact) to one dimensional. At this critical point, random fractals are produced and the rate of entropy production is a maximum. Analogies are drawn between the patterns produced by our model system, and similar patterns generated by diffusive processes.
INTRODUCTION
The scaling properties exhibited by many spatial patterns have lead to considerable interest in naturally Notable examples are the occurring fractal patterns. random fractals produced in processes such as dielectric diffusion-controlled electrochemical deposibreakdown, and Quid flow through porous media. Theoretition, cal attempts to explain these patterns typically start with solutions to the diffusion equation. While this approach has been quite useful in the study of dendrite formalittle progress has been made towards undertion, standing why random fractals form or their peculiar scaling properties. Some experimental work has suggested that fractal growth occurs when the dynamical system responsible for the pattern formation is at a critical ' The reasons for the scaling, however, have point. not been obvious in these experiments. Currently, there is no intuitive explanation for scaling in any critical phenomenon. In a recent Letter, Bak et al. ' demonstrated that many extended dynamical systems naturally evolve to a critical point where the variables of the system begin to scale. to model the They developed a cellular automaton growth of a sand dune. Sand was added to the top of a growing dune. As the height increased, instabilities occurred on the sides of the dune when the local gradient exceeded a designated threshold value. These instabilities generated avalanches. Eventually the sand dune reached a critical height where the avalanches occurred on all scales up to the size of the system. A measurement of the sand falling off the dune revealed 1/f noise at this point. Even though the avalanches were themselves compact, the scaling properties exhibited by Bak s model dynamical system are characteristic of many critical phenomena. The onset of 1/f noise at the critical point also correof how sponds to an increase in the unpredictability much sand will fall off the dune at an instant in time.
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Bak's sand dune (which is far from equilibrium) seems to evolve to a state of higher entropy production. ' This contrasts with the suggestion of Prigogine that for selforganizing open systems in a steady state (e.g. , of flow) one can apply a principle of minimum entropy production. ' Important examples where Prigogine's principle may apply include living organisms which develop into highly ordered states that are stable for extended periods of time. The principle of minimum entropy production is valid for systems "not far from equilibriwhereas Bak's sand dune constantly produces um, configurations on the brink of collapse. One might expect that Bak's system evolves to a state of maximum entropy production since the growth is ruled by Auctuations (sand is added to the dune at random sites). In an attempt to understand the formation of fractal patterns in the context of dynamical systems, we have developed a model system which generates spatial patterns based only on connectivity and a simple screening The questions we will address are the folconstraint. lowing: (1) Are random fractals produced by this simple model (which involves no diffusion)? (2) If so, are the fractal patterns produced at a critical point of the systems? (3) Does such a critical point correspond to a state of maximum entropy production, and how (if at all) can the principle of minimum entropy production be applied. Specifically, what is the relative rate of entropy production of fractal patterns compared to other (Euclidean) patterns. (4) What is the role of fluctuations in our system?
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THEORY As previously suggested by a number of authors, information theory may be useful in understanding pattern '' Our simulation is designed to make the formation. ' application of information theory straightforward. Using
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INFORMATION-THEORETIC information theory, the entropy production associated with the evolution of a spatial pattern can be measured. Note that the rate of "entropy production" is determined here by direct measurement of the change in entropy associated with adding particles to a pattern. In other works, the term describes an indirect measure of some other thermodynamic quantity such as the generation of waste heat. To understand the correspondence between information and entropy production in relation to pattern formation, it is first convenient to consider a process which adds points to a plane at random. This models, for example, a totally inept dart thrower. The information produced by the system is the final location of each dart thrown. If multiple darts can occupy the same site, the expectation of where the next dart will land is completely flat. This corresponds to a minimum production of information and a maximum production of entropy. If, on the other hand, darts are constrained to one per site, then every dart which hits the plane destroys information in the form of possible sites where future darts can land. For any set of constraints, the process which destroys inwill correspond to formation at a minimum rate' the maximum state of entropy production. Unlike the inept dart thrower, a typical growing system adds new particles adjacent to existing parts of the pattern Thus, to model two-dimensional pattern. formation, in addition to limiting one point per site, an must be imposed which conadditional requirement strains new points to be added adjacent to occupied sites. At any instant in time the possible locations for growth are limited to some subset of the surface area of the pattern. For purposes of discussion, the surface area sites are defined as the nearest-neighbor unoccupied sites adjacent to occupied sites. For example, for a single point, on a square lattice, the surface area would be four while for two adjacent points, the surface area would be six. Active sites are defined as the subset of the surface area sites where future growth is possible. It is important to note that the pattern of maximum total surface area is not necessarily the maximum entropy producer. For example, on a square lattice, the maximum surface area is produced by growth in a straight line. For every site (possibility) destroyed by adding a point to the end of the line, the net surface area is increased by two. One surface site is destroyed by the new point, awhile two new sites are added to the side and one ahead of the new tip. Any other operation produces new surface area at a slower rate. However, in constraining growth to a straight line, the vacancies on the sides are not active. The active area is constant (two). One expects the next point to be added at one of the two tips so that the rate of information production and destruction is also constant. Knowledge of both the "active area" and total surface area are necessary to determine the rate of entropy production of a gro~ing pattern. The relationship between the change in total and active area, and the entropy follows from the information theory of a discrete noiseless system. ' If P~, is the probability that a symbol(s) r. will be received, given that symbol(s) r; have been received, the unpredictability (W, ) of a sequence of symbols is
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= —gP;logP, , l
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average information produced is this quantity weighted by the average probability P; that the sequence ' Knowledge of these probability r; has been received. distributions requires some knowledge about the system For example, in the English producing information. language, if one receives the symbol Q, the probability of the next symbol being U is very high. Having received the two symbols TH, the unpredictability of the next character is fairly high. The combinations THA and THE, for example, are both possible with some probability. Knowledge of the probabilities for the possible letters following TH determines the unpredictability of those future events. The entropy of the sequence TH is determined by weighting the unpredictability [Eq. (I)] by the average probability of the sequence TH. TH would, for example, have a higher weight than the sequence CZ. Note that, in general, the P; may represent higher-order processes. Having received the two symbols TH, it is meaningful to ask whether the fourth letter will be a T. The probability of THXT will, of course, change as the pattern grows. For example, THA may still be likely to evolve into THAT whereas, THR will never evolve into THRT if it is not a sequence in the English language. In pattern formation, the information produced is the possible locations for future growth. Real systems which produce spatial information (e.g. , electrochemical deposition or fiuid liow through porous media) often produce patterns of various geometries. In general, growth which results in the greatest number of distinguishable patterns of the same fractal dimension corresponds to the state of maximum entropy production. To measure the rate of entropy production, one can apply information theory to discrete growth on a lattice. Such a measure can, in principle, be applied to real systems with suitable digitization The probability that a specific and renormalization. unoccupied site will become active (or inactive) is the unpredictability for growth at that site. If one has absolute knowledge about the change in active area of a growof the ing pattern, one can determine the unpredictability next growth event. Given some state i described by a particular pattern of labeled points and area sites, the probability that a labeled site a will become active is P, o: d A id% (where X is the number of points on the pattern). By analogy to information theory, the total event is for the next unpredictability growth P.;logP; The . weight o. f the state i is the total surface area produced (on average) per growth event, P; ~dA'"'/dX. Patterns which produce total area at a minimal rate (active area at a relative maximal rate) are compact and have low weight. Patterns which produce total surface area quickly ean evolve into many di6'erent and distribute active area in many configurations more ways. The change in entropy is then 5S ~ — P, P;logP;, where the sum is taken over every possible pattern at every stage in its growth. Obviously, it is not practical to take this sum even if the patterns are constrained to be finite subsystems of a finite medium. However, if the surface area and the active area both
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J. H. KAUFMAN, O. R. MELROY,
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scale, then the corresponding probabilities for very large patterns (all of size N) will not vary much between patterns of the same fractal dimension. Such scaling also implies that the change in area A = N" per particle added tends to dA/dN= A/N for large N. If there exists a value of the fractal dimension for which the number of distinct ways to grow is a maximum for every N, then the number of possible patterns will also be a maximum at that fractal dimension. Self averaging of this kind allows one to compare the number of choices open to particular patterns of different fractal dimension without summing over many configurations. The change in entropy, 5S, per particle is then given by the approximate relation,
5S = —( A '"' /N)(
"',
A
"'/N)log(
A
"'/N),
the number of active sites, and A '"', the towhere A tal surface area, both scale with N to some fractal dimension. We note that this "assumption of scaling" is an anticipation of the coming result, and does not influence whether or not the patterns produced will scale. Equation (2) can be used to measure the rate of entropy production of an arbitrary pattern if one has knowledge of, or can estimate, the active and total surface area. If a particular pattern does not scale, one should substitute derivatives for the ratio's in Eq. (2). It is not obvious from Eq. (2) that a unique pattern exists which maximizes the entropy production. If, for example, the entropy scaled with the fractal dimension, then the pattern which maximizes 6S could depend on the value of N. To see how the entropy production is influenced by the geometry of the pattern, it is useful to examine two limiting cases; the filled circle and the straight line. For the circle, A"'/N and A'"' /N are identical. They both scale as 2(m IN) . The rate o'f enfrom Eq. (2), is proportional to tropy production, (1/N)(logN+C) where the constant C is the log of the coeScient of the active area. For a straight line, A'"' /N is constant (two) while A'"/N scales as N '. The rate of is production to again proportional entropy (1/N)(logN+C). The more general case of spatial patterns whose properties do scale can now be considered. For all values of the fractal dimension, d, the total surface area per particle added scales as N" "'/N. If A"'/N scales as N" ' "'/N, the entropy production does not scale with d, and is proportional to (1 —1/d)(1/N)[logN +C] for all values of N and d. The relative rate of entropy production then depends only on the fractal dimension and the appropriate prefactors of the area terms (which may themselves be functions of d) and is independent of the size of the pattern. To test the validity of these relations, and to find the maximum state of entropy production (if it exists), it is necessary to produce patterns with various scaling properties. Spatial patterns produced only under the constraints of connectivity and one point per site are compact and have a fractal dimension of 2. Such patterns are relatively uninteresting for the purposes of this paper since the scaling properties cannot be varied. Screening is known to be an important factor in the formation of random fractals. Points on the outer parts of a growing fractal determine the growth probability at sites on the inside of
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the pattern. In general, as the fractal gets very large, area sites deep inside the pattern become inactive for future growth. Stated another way, points on the outside of the pattern transmit information to points on the inside about their growth probability (e.g. , whether or not they are "screened"). Determining to what extent a point inside a pattern is screened and how this affects the growth probability is di%cult and dependent upon the physical process one is attempting to model We have adopted a simplified measure of both the extent of the screening and its relationship to the growth probability. The unscreened angle of a site is defined as the solid angle in which that point "sees" the outside world. The growth probability has been assigned as a step function. All surface area sites with an unscreened angle less than a designated threshold value are inactive. All surface area sites with an unscreened angle greater than the threshold are active and have equal probabilities. This screening constraint is primitive and not intended to model any experimentally found pattern. It does, however, elucidate the physics essential to answering the questions posed earlier. The simulation is performed for thresholds between 0' and 300', for 50000 points or until the pattern reaches the border of a 2000X2000 grid. A flowchart of the simulation, which we will refer to as the threshold screening model, is given below. There is no diffusion or random walk involved. (1) A five-point cross is placed at the center of the lattice as the initial seed. (2) The active growing sites are determined and their coordinates placed in a list. (3) One active site is chosen at random from the list, removed, and a point placed on the lattice at the corresponding coordinates. (4) Sites adjacent to the new point are checked to determine if they have become active sites. If so, they are added to the active list. (5) A11 previously active sites are checked to determine if the addition of the new point has decreased their unscreened angle below the threshold value. If so, they are removed from the active list. (6) Return to step (3). The key element in this simulation is the screening check made in step (5). Since the pattern is connected, exactly two points on the pattern determine the unscreened angle of an area site. It is possible to keep track of the coordinates of these points for every potential growth site. After each growth event, one can then check each active area site to determine if the newly added particle changed one of its two limits. If it has, the new point is substituted for the appropriate limit. ~
RESULTS AND DISCUSSION Patterns obtained for various screening conditions are shown in Figs. 1(a)-1(f). For small thresholds, the screening condition is weak and compact structures with small holes are formed. As the minimum unscreened angle is increased, the holes increase in size until holes as large as the pattern radius form. At this point the pattern is made up of numerous thick branches. Larger
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KAUFMAN, O. R. MELROY, AND G. M. DIMINO
screening angles result in thinning of the branches. Eventually, branches begin to screen themselves. Above 200, patterns with fractal dimension near one are formed. These patterns are roughly linear. Short branches form at the tip, but as the tip advances past the branch point the branch quickly becomes screened. For each pattern, the active surface area, and the total surface area was measured. The active area was, indeed, found to scale (approximately as N ' '~"') as discussed above. The change in active and total surface area (normalized to N) as a function of the screening angle threshold are shown in Fig. 2. Almost all of the data were obtained for patterns of 50000 points. The compact structures produced at small screening angles have the lowest total surface area, but the highest active area, as expected. All of the surface sites on a filled circle are active. However, the final pattern is independent of the order in which the perimeter is filled in. Large screening angles resulted in the highest total surface area and smallest active area. The minimum solid angle specified for each pattern corresponds only to the threshold above which surface area sites are active. For every pattern, there is a distribution of unscreened angles (above the threshold) corresponding to the state of each active area site when it was chosen for growth. Two such distributions are shown in Fig. 3. To obtain the mean unscreened angle, this distribution was measured for every pattern and fit to a Gaussian. At low angles, the distributions were fairly broad. For example, at a threshold of 40, the full width at half maximum (FWHM) was 100'. This narrowed to a minimum of 32 at a 160 threshold. It is possible to calculate the rate of entropy production of the various patterns from the measured active and total surface areas using Eq. (2). The results for N =50000 are shown in Fig. 4(a). The data are plotted as a function of the mean unscreened angle, ( 0 ), instead of the designated threshold, in order to consistently cornpare the current simulation to other random aggregation experiments (vide infra). The maximum rate of entropy production occurs at a mean unscreened angle of about 160'. The fractal dimension of each pattern was computed from the two-dimensional autocorrelation and is plot0.04 ~ ~~0 ~ ~ ~ ~0 ~ ~
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ted in Fig. 4(b). It is clear from Figs. 1(c) and 1(d) that the patterns formed near the peak in entropy production are random fractals. It is also evident that a phase transition between compact and one-dimensional structures occurs at this point. The peak in entropy seems to occur just before the drop in the fractal dimension. However, this may be an artifact of the autocorrelation function measurement which systematically underestimates the fractal dimension of all of the finite compact patterns. As a result, it is not possible from the data in Fig. 4 to assert that there is (or is not) a special value of the fractal di0.06 (a) CA
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INFORMATION-THEORETIC mension which corresponds to the maximum entropy producing state. If the transition in Fig. 4 is second order, finite-size scaling measurements (perhaps of much larger patterns) should determine whether or not there is a special fractal dimension in the limit of infinite systems. To the extent that the growth of the compact and onedimensional (1D) patterns are more stable with respect to variations in the screening constraint, one might consider the steady-state growth of compact or 1D patterns to be "closer to equilibrium" than the growth of a fractal pattern. That these patterns produce entropy more slowly than the fractals is consistent with Prigogine's minimum entropy principle. This observation does not prove that principle in a general sense. In fact, Landauer' proved that while many open systems (which satisfy the "not far from equilibrium" condition) evolve to ordered In states, they are extremely sensitive to fluctuations. particular, the lifetime of the ordered state can depend quite sensitively on the history of the system as it developed. It is interesting to consider the two "more stable" pattern geometries our system is capable of proBoth the compact patterns and the oneducing. dimensional patterns produce entropy more slowly than the fractals which form at the critical point. Of the two, the compact geometry produces slightly more entropy than the 1D case. This is evident in Fig. 4(a), or if one simply considers the limits of filling in an area or growing in a straight line. Linear growth never produces more than two options. Is it meaningful to consider the stability of these two limits? Certainly one could conceive an automata to produce 1D and 2D patterns of arbitrarily large size. However, for all of our patterns, though selfconsistent, the choice of the location for the next growth event is determined by fiuctuations (a random number generator). These fiuctuations put no limit on the size of the compact (or the fractal) patterns, but they can limit the lifetime of the lower entropy 1D state. At very large values of the screening angle thresholds, there is a finite probability that a linear pattern will "die" (i.e., it can grow to a configuration with zero active area sites available for future growth without violating its screening constraint). To completely study the kinetics of this what fraction of the patterns eventually die and process on what time scale it is necessary to generate a large This number of patterns at each screening threshold. work is in progress and will be reported in a future publication. However, it is already evident that the only states of our system which die are the states of lowest entropy production. These states (the 1D patterns) are the most sensitive to the fluctuations which occur during growth. There are some artifacts of performing the simulation on a square lattice which can be seen in Fig. 1. For example, the compact structures fill in a roughly square area. To make sure that these artifacts did not impact the results in Figs. 2 —4, similar simulations were performed using a polar coordinate construction. In this case, a single point was used as the seed and the adjacent sites defined as the two radial and two azimuthal points. It is dificult using this construction to keep track of the unscreened angle with the technique used in the square lattice simulation. Instead of keeping track of the two
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STUDY OF PATTERN. . .
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points which define the unscreened angle, a search was performed on 360 rays (1 increments) originating from the area site in question. For each site, a set of 360 bits were stored to indicate which directions were unscreened. If a particular direction was screened by some point on the pattern, the corresponding bit was set to zero. It is unnecessary to check a screened direction more than once. After each growth event, only the known clear directions of the active sites need be checked. The sum of the bits then determines the unscreened angle. The patterns obtained in this simulation are shown in Figs. 5(a) —5(h). The overlying symmetry of these patterns is now circular as expected. With the exception of this the patterns observed exhibit the same symmetry, changes in geometry as those in Fig. 1, although in the one-dimensional patterns, the growth on the polar lattice allows some symmetries [Fig. 5(g)] not possible on a square lattice. The subsequent analysis of the patterns in Fig. 5 produced results which were substantially the same as those presented above. The patterns formed at the critical point in Figs. 1 and 5 are similar to the fractals observed in random aggregation experiments. In fact, the information theory applied to our patterns is meaningful for other processes. In a diffusion-limited aggregation (DLA) simulation, for example, one can measure the unscreened angles of (active) area sites when they are chosen for growth. The average of the resulting distribution for a DLA simulation (nearest-neighbor growth on a square lattice) of 10000 points is 160 with a FWHM of about 75 (Fig. 6). This distribution is analogous to the histograms shown in Fig. 3. In the simulation described above, the screening constraint can be varied by choosing a different cutoff for the active sites. In DLA, of course, one has no control over the strength of the screening. While the selection of a threshold is a simplification, even in more realistic simulations such as DLA parts of the pattern transmit information to the interior about the growth probability at active area sites. In real systems such as electrochemical deposition, these probability distributions are determined by the Laplace equation and the geometry of the pattern. Our choice of a step function to model the effect of screening was motivated by our studies of DLA. When the histogram in Fig. 6 is integrated and normalized to the distribution for total area sites, one obtains the "activity" probability as a function of screening angle for DLA. The probability of being active rises smoothly from zero to one at about 160. The transition is not a sharp step function. It has a width of +30'. From this distribution, we can estimate the number of active and inactive area sites in a DLA simulation. The rate of entropy production of DLA at 10000 points was about than the maximum entropy production in 30%%uo higher our simulation for patterns of the same size. One could use a fit to the DLA growth probability distribution in a simulation similar to the one described here. Area sites could be given a variable weight (growth probability) determined by this or any other function. Such an approach would produce patterns which more closely mimic aggregation simulations such as DLA. The advantage to using a step function is that it provides
J. H. KAUFMAN, O. R. MELROY,
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AND G. M. DIMINQ
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FIG. 5. (a) —(h). The patterns formed in a polar coordinate simulation. The screening angle thresholds are 10, 60', 100, 110, 140, 210', 230', and 250, respectively. Note that the pattern in (h) (250' threshold) died after only a few points. Though each growth event was allowed within the model, it evolved to a state with zero active area sites.
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(Continued). unambiguous information about the exact number of active area sites at all stages of growth. Since this is the information required to measure the relative rate of entropy production, for the purpose of this study it was more important to have absolute knowledge of the active area than to use arbitrary parameters to make the pictures more "realistic. Several analogies can be drawn between the critical screening condition and other critical phenomena such as percolation. In particular, the distance to the farthest screening point diverges at the critical point, as do the range of the correlations in the case of percolation. It is evident from Figs. 1(a) and 1(b) and (5a) and (Sb) that the separation of the most remote screening point from the site it screens is on the order of the largest hole in the pattern. When the holes grow to the pattern radius [e.g. ,
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STUDY OF PATTERN. . .
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FIG. 5.
Fig. 1(d)] this separation diverges with the radius of the pattern. awhile the inAuence of a remote screening point will decay with distance, the location of every point on the pattern stores information which determines the activity of every area site. As in the case of other critical phenomena, the exact screening angle where scaling occurs may depend on the size of the system. However, as discussed above, the rate of entropy production does self-average. In Fig. 7, the rate of entropy production as a function of N is shown. As one would expect, the rate of information loss (number of distinct ways to grow) decreases as the pattern gets larger. The fact that the decrease is monatonic and has the same functional dependence for all screening threshof 5S(N) for olds makes possible the comparison difFerent patterns. Note that 5S(N) does not (and should
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J. H. KAUFMAN, O. R. MELROY,
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not) scale as discussed above. In conclusion, we have used information theory to study pattern formation governed only by connectivity and screening. As the screening constraint is made more severe, the pattern changes from compact to that of a random fractal and eventually to a one-dimensional structure. The rate of entropy production is found to reach a maximum at a screening angle near that observed in a true DLA simulation. At this point the pattern exhibits self-similarity and scaling. The only mechanism in our simulation which can transmit information from the smallest scales to the largest scales of the system is screening. While many critical phenomena do not involve screening, it is characteristic of virtually all real dynamical systems which produce random fractal patterns. It is not unreasonable that in those systems the scaling is a consequence of the screening constraint on the pattern formation. Intuitively, the scaling occurs be-
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Phys. Rev. A 32, 2364 (1985).
(1987); and (unpublished)
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cause the addition of a single particle (an operation on the smallest possible scale) can transmit information, screen, parts of the pattern as far away as the pattern radius. The formation of fractal patterns and maximization in the entropy production at the same structural phase transition reAect the fact that the dynamical system responsible for the pattern formation has reached a critical point. The changes in the pattern as the screening constraint is shifted away from criticality are analogous to those observed in other dynamical systems such as percolation.
ACKNOWLEDGMENT
We thank Richard useful discussions.
Voss and Per Bak for some very
E. Skannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Chicago, 1962). '5I. Prigogine, C. Nicolis, and A. Babloyantz, Phys. Today 25 (11), 23 (1972). ' I. Prigogine, Science 201, 777 (1978}. ' S. Wolfram, Scaling Phenomena in Disordered Systems, Vol. 133 of NATO Advanced Study Institute, Series B (Plenum, New York, 1985), p. 249. R. Shaw, The Dripping Faucet as a Model Chaotic System (Aerial, Santa Cruz, CA 1984), Chaps. 6 —9. 9R. Landauer, Phys. Today 31(11),23 (1987). 2oR. Landauer, Phys. Rev. A 12, 636 (1975). 'R. Landauer, J. Stat. Phys. 13, 1 (1975). An increase in the statistical error is evident above the 200' screening threshold in Fig. 4(a). For large screening thresholds, the pattern becomes essentially one dimensional and a pattern of 50000 points will not fit on a 2000X2000 grid. For screening thresholds above 260', the rate of destruction of active growth sites can exceed the rate of production of new ways to grow. As a result of this information loss, it is possible for such patterns to cease growing after only a few thousand points. For these patterns it was simple to extrapolate the active and total surface areas to 50000 point, because the active area was constant and the total area was linear. ~~C.
