edited by JAMES
A Microcomputer Simulation of Fractal Electrodeposition F. Sagubs and J. M. Costa UniVerJitat de Barcelona Diagonal 647 Wcdona 08028. S p i n
The concept of fractals is nowadays a very fashionahle one. Since the publication of Mandelbrot's famous hook, The Fractal Geometry of Nature (I),many physicists and chemists have become more and more curious about this concept, its mathematical properties, and, more importantly, its potential utilities in describing several real forms of condensed matter. A particular category of fractal objects that has deserved a eood deal of attention durine these last vears is the so-called Fractal aggregates (2). A c t d l y , irregular aggregates built in a more or less irreversible way are quite familiar to chemists with a large variety of examples furnished by a diversity of phenomena such as sedimentation, electrodeposition, flocculation and aggregation of colloids, aerosols, dust, etc. Our motivation in preparing this paper has been to farilitate a first encounter k i t h fractal d e ~ o i i tusine s the oossihilities offered by small-size computers and, focusing on a particular phenomenon, that of electrodeposition, which we helieve can be undoubltedly of chemical interest. Electrochemical denosits occurrine on electrode surfaces have deserved the attention of ele&ochemists for many vears (3).The understandine of the intrinsic laws reeulatina their growth and morpholo~es,which can adopt, depending on the operational conditions, a rich variety of aspects ranging from smooth to dendritic, is certainly a challenge from a theoretical point of view and a t the same time a matter of great technological importance. Obviously our contribution here is by no means aimed a t discussing this intricate phenomenon. Rather we will use i t as a reference to Dresent a slightly different version of the well-known diffusion-limited aggregation model (DLA), which is known to give rise to disorderly aggregates with no apparent symmetry except for dilation. Accordine to its orieinal version ~ooularizedbv A amounts to adding Witten and ~ a n d e ;( 4 ) ,the D ~ model Brownian particles one after another to a sinale mowina cluster initiated with an immobile seed particle. i n o& simulations here, and in order more closelv to resemble electrochemical deposits, we will assume t h a t the aggregation mechanism has a simultaneous multiparticle nature and takes place initially on a regular surface instead of around a single seed particle (5).
- .
Slmulatlon's Algorithm We beein with a two-dimensional sauare lattice where a fraction of sites, which depends on a concentration parameter. c. are initiallv orcuoied with mobile particles. The botto& edge is supposed to-he the initially active area of growth, while the tor, of the grid may be, if desired, held a t a fixed concentration to sirnilate thk bulk of the solution. The mobile particles undergo simultaneous diffusion, modelled by means of an isotropic random walk, with the ohvious limitation that no two particles can occupy the same place at the same time. During a time step any of the mohile particles is examined, choosing by chance its new position from among 502
Journal of Chemical Education
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its four neighbor sites. If the selected new position is unoccupied, the particle moves to it; the particle remains fixed if i t is already occupied. If the new position reaches an aggregation site, the particle becomes aggregated with a sticking orohahilitv k. At anv time we consider as com~letelveauiva. . lent aggregation sites hoth the remaining free positions at the bottom line together with the wholesuhset of contiguous positions of the previously deposited particles. ~e;iodic boundary conditions are prescribed in the transverse direction so that particles thatleave the lattice t o the right enter from the left. Using a standard BASIC 4.0 code, our version of this algorithm was implemented in terms of a main program and several subroutines. The main program stores the set of inputs appropriate t o each particular run. Two different kinds of innut oarameters have to be considered. First. those common to any "molecular dynamics" simulation on a lattice. as for examole the dimensions of the sauare arid. the step of the random walk, and the maximumnum~eri f allowed iterations or of aameeated particles. S u ~ ~ l e m e n t a r i l v -- we have to provide the program with the "ele&ochemicai;' parameters of concentration and potential-dependent sticking. The main program is also responsible for the initial random seed of the lattice and controls the successive time steps during whirh any of the ocrupied sites is tested for motion and eventual deposition. Finally it directs the calculation of the fractal dimensions and stores the formed denosits in a matrix form. Once these output files have been created and examined on the screen, they can be saved for further manipulation or conveniently plotted. The accompanvina subroutines cover the remainine Darts of the simulatibn,ranging from those subroutines implementing the random walk and the aaereeation test to the most standard data -- correlation subroutines used t o calculate the fractal dimensions according to the procedures outlined below. All this software runs in a HP 9000, Series 200, Model 216 environment linked with a H P Color Pro plotter for graphics. Further information on the program is directly available from the authors upon request. &
&
~~
~~~~
.~ ~
~
Figure 7. Twc-dimensional simulation of diffusion-controlledeiectrodepasitian on a surtace (line). The deposit was grown from a line of 200 sites wRh concentrationand sticking parameters respectively equal to c = 0.2 and k =
1.0.
F~gure2. This figurecorresponds to an enlarged view of a porton of lheemlre deposlt shown In Figure 1. The numoer sppesrlng In eachaggregated posltlon Isrelated 10 me time step at which the Mnesponding particle was added lo the deposlt lsee ten)
Simulated Deposits: Results and Discussion A typical aggregate corresponding t o r = 0.2 and k = 1.0 is shown in Figure 1. The "tenous" structure characteristic of fractal aggregates is clearly reproduced in this example. Actually, one should expect some kind of complex structure emerging from diffusive growth because a regular deposit growing basically under diffusion-limited conditions is intrinsically unstable against wrinkling. In the situation here considered this results in the treelike clusters shown in Figure 1. Since we can think of the sticking probability k as inverselv related to some sort of activation enerm that the mohile particle has to gain before getting aggregated, the limit of free sticking, k = 1,is the most approuriate one if one wants to examine the precise role p ~ a y e d b ydiffusion in the growth of these irregular deposits. Thus, restricting ourselves to this particular situation we observe that, analogous to what is found in the case of aggregation around a single seed particle, the structure that develops is rather open, specially in the limit of low concentrations. However, despite this fact, verv few oarticles are added to the inner reeions of the deposit a G t progressively grows. This fact can bk intuitively explained if we realize that as the deposition progresses the resulting "trees" get more and more ramified, and as a consequence more and more Brownian particles will intersect with
the outer parts of the deposit. We can easily visualize this hiding effect caused by the branching of the aggregate if we control the particles a t the moment of their deposition. This is precisely what is shown in Figure 2, where we assign to each particle of the deposit a digit indicating the time it was aggregated according to a timer running in some convenient units. Now we consider the effect that the basic parameters c and k have on the structure of the deuosit. In Firure 3 we show several simulations corresponding to different concentrations, keeping constant the total number of time steps over which we run the simulations. In going to larger concentrations the number of deposited particles generally increases. However, this is a rather obvious effect that hides the most important one concerning the actual influence of the concentration on the morphology or "fractality" of the deposit. In order to capture this latter effect, we present in Figure 4 the results corresponding to the same set of concentrations but now containing nearly the same number of aggregated particles. As is clearly seen, the clusters growing in a less concentrated medium are less compact and present a considerahlv lareer develoument. The role of the sticking iarameter k, is better understood bv comuarinr the different simulations shown in Fiaure 5. At low vaiues of k we observe a less tenuous and more dense structure that in some sense is reminiscent of the mossy Volume 66
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Figure3. Differentdeposttsfor ditferent concentrations: (a) c = 0.2; (b) c= 0.3 and (e) E = 0.4. Ail the deposits mnespond to an equal number of time Iterationsand were grown horn a line of 100 sites with sticking parameter k = 1.0.
deposits encountered in metal deposition a t low overpotentials. Contrarily, in going to higher values of the sticking probability, we find much more ramified deposits with noticeable side-branching effects, which speculatively could be associated with some sort of dendriticlike deposits also observed experimentally under appropriate experimental conditions (5). Fractal Dimension: Concept and Measure
Finally we analyze the question of the fractal dimension of these irregular aggregates. In general we would say that the fractal dimensions D measures, in some sense, how mass is distributed in space (6).For a compact arrangement of mass particle in a d-dimensional Euclidean space we would have D = d. On the contrary, if we consider for instance a fractal aggregate built around a central seed particle, we would find that going to larger and larger spheres centered on this central particle, the density of the cluster would not remain constant, but rather i t would fall down with a power-law relationship p(r) = rD-d.In other words, by enlarging the examined part of such an aggregate, we would find wider and wider empty regions, whereas a sort of dilation symmetry or 504
Journal of Chemical Education
Figure 4. Differentdaposltsfor different concentrations: (a) c = 0.2; (b) c = 0.3 and (c) c = 0.4. All the deposits correspond now to a practically equal number of deposited particles and were also grown from a line 01 100 sites with stlcking parameter k = 1.0.
self-similarity property would appear more and more inherent to the mass distribution of the fractal object (2,6). In order to compute the fractal dimension of deposited clusters, one usually invokes a geometric scaling law exoressed bv X = h"I(l-d+DJ. where X is some measure of the deposit tl;ickness, N the number of aggregated particles, and D,is the aoorooriate fractal dimension (7). In our presentation here x i s chosen as an averaged height ( h ) ,bf the socalled "upper surface", i.e., the surface that an observer would see looking straight down a t the deposit. Apart from this technique of evaluating D,, we have also used alternatively ageneralization of amethod based on the use of thesocalled "radius of gyration", originally proposed in the context of radial aggregates (8).T o this end we assume a relationship S 4 = ( N ( S ) ) dbetween the examined area of the deposit and number of aggregated particles contained in it.
1.5
2
2.5
3
log N
(b) 3
-
2
-
1
-
0
--
0.85 2~0.85 1.70
slops:
D,, --L-.I-
L
2
1
0
log S
3
5
4
Figure 6. Power law relationships used to compute the fractal dimensions of
ttw deposit reproduced in Figure 1. (a) Plot of log ( h ) vs. log Nand (b) plat of log YS)vs. Sgiving D., and D... respectively. Figure 5. Diffsrentdeposhsfor different values of ttw sticking parameter:(a) k = 1.0;(b) k = 0.7 and (c)k = 0.5. All the deposns were grown horn a line of 100 Sites wlth concentration c = 0.3.
In practically implementing the firvt method, to give a fractal dimension noted D.I. we calculate oairs of values (. (. h.).. M. a t different stages of the simulation,-whereas in the second urocedure we wait until the denosit is larae enough t o he statistically consistent and t h e k e stop the simulation and we comnute N as a function of Sfor laraer and laraer squares horizonkdly centered in the lattice andall of themtangent to the hasal surface. The fractal dimension calculated in this way will he noted. D.2. Examples of such calculations are presented in Figure 6 for the particular case c = 0.2, k = 1.0. The effect of these parameters on D, is summarized in the table. In s ~ i t of e the fact that our simulations were oerformed a t the level of standard microcomputers with the subsequent limitations of ha\,ine - statistics uro" to relv on the middline vided by the small-size aggregates generated in this way, the results ohtained for D, are in reasonably good agreement with the value D, = 513 predicted for the two-dimensional version of the DLA model (8).This is soecialls true for D.1 with some more appreciable deviation; for D.%, this latter one being nrobahlv more intrinsically dependent on the statistical consistency of the examined deposits. In what respects to the influence of the parameters c and k, and focusing specifically on D,1, we observe that they slightly modify the results, with positive deviations in the limits of large concentrations and small sticking parameters. Obviously a more detailed discussion of those effects is completely out of our purposes here.
Computed F r a d a l Dimensions and Number 01 Deposited Particles Corresponding to the Deposit Shown in Flgures 1, 3, 4, and 5
c
k
N
4 1
D.2
0.2
1.0
408 1303
1.29 1.71
0.3
1.0 0.7 0.5 1.0
1.72 1.75 1.74 1.78 1.77 1.81 1.83 1.82
2853
0.4
1324 894 608 3845 1336
.
1.70
1.60 1.50 1.29 1.87 1.75
Fig. 3a Fig. 4a Fig. 1 Figs. 3b. 4b. Fig. 5b Fig. 5c Fig. 3c Fig. 4c
5a
Concluslons After this presentation our hope is that we have aided a little bit to spread among the community of chemistry professors and students the fascination by fractal aggregates and their intriguing properties concerning growth, shapes, and dimensions. Much of the work that has been presented here may he directly reproduced using standard personal compuc&, and in this sense we believe it might headopted as one of the most interesting and rewarding computeraided chemistrv~ractices.Our exoerience leadsus to believe that although %dents must he k n e d about the obvious fact that such a simulation merelv serves as a caricature of any real electrodeposition process, i t may well give them a feel of how microscopic processes are expressed in macro-
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sconic shanes in the context of nattern-forming phenomena of chemical interest. The possibility of runningthe program under different concentration and sticking conditions, apart from favoring the discussion on the cen&al role played by concentration and potential on electrochemical processes, facilitates the c o n t k t of such students with real experiments on electrode~ositionrecently performed (9).In addition, a diversity of aspects that can he of interest t o a wide range of chemistry students may be examined under the spirit of such simulations. From a formal or mathematical point of view the question itself of the definition of fractality, its quantitative determination in terms of fractal dimensions, or even the practical implementation of the different nrocedures t o evaluate those dimensions, depending on thkgeometry and characteristics of the aggregate, are worth examinina carefullv. Apart from that, other questions with more physjcal chemist4 incidence, like ~rownian-motion-simulated diffusion, electrochemical processes on electrode surfaces, or even more generically diffusion-controlled reactions taking place on surfaces or lines could he approached in thecontext of such simulations. Acknowledgment We thank M. Vilarrasa for helping us in preparing some preliminary versions of the simulation routines, and F. Mas for a critical reading of the manuscript.
Figure 1. Number of cwmers on endl aqusre (random placement).
LRerature Cited 1. Mandclbwt,H H 'lh~Fror~olGeomelryo/Notun;Pr~rman San Frannwn. 1982. 2. Wincn. T A In (lance ond Aforlx. Prw ill lhc NATO AS1 Lcs Hovthsr h a o n XI.VI. 1 9 s . Snulet:c., I . Vannlmcnur. J: Smra. H FA- Nlmh-Holland: Amaurdam. 1987. 3. B"dersk1.E. B.:Depie,A.R.InCompreh.~iw neati*eo/Eloetmeh.miatry;Conway, B.E.;Bockris, J.WM.'Yewer,E.:Khan,S. U.M.; White,R.E.,Eds.;Pleaum: New York, 1983: Vol. 7, Chapter 7, Partp A and B. Coafa, J. M. Fundomntoa de Elscfrodica;ALhambra: Madrid, 1981. Southampton EleNochemiatry Gmup.lnrfrumntal Method8 i n Electmchamialry; Chicheater, 1985. 1981.47.14W-1403:Phy~.Re~.BI983.27, 4. Witten,T.A.;Sander,L.M.Phya.R~~.L~tt. 56865697. 5 Vass,R. F.; Tmkievic~.M.J.Ekrtmrkm. Soc. 1985,132.371-375. 6. Jui1ien.R. Com-nts C a d . Mot. Phys. 1987.13.117-205. 7. Meskin. P. Phya. R w . A 1983,27.26162623: Phys. Em. B 1984,30,420'4214. 8. Meakin. P. Phys.Reu. A 1983.27.M)"607:J. Chem. Phys. 1984.80.2115-2122. 9. Mstsushita, M.; Sam. M.; Heyskawa, Y.: Honjo. H.: Saruada, Y. Phys. Re". Loft. 1984, 53,286;Sawada. Y.:Dougherty,A.:Gollub.J.P.Phys.Rm.LeLf. 1986,56,12MI,Grier, D.; BenJamb, E.; Clarke, R.; Sander, L. M. Phys. Re". k t f . 1986.56 1264: Aqoui, F.;Arneodo. A,: Craaseau,G.; Sluinney, H. L. Phya. Re". Lett. 1988.61.2558.
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Counters on Grids Ben Sellnger and Ralph Sutherland me AvaWalian National Universiw GPO Box 4 Canberra, ACT.
Ausbatia 2601 Statistical thermodynamics is often illustrated with the use of colored balls1, cards, or dice games2. We have chosen the latter approach with an example generally used to illustrate thedistribution of energy (as well asmolecules) asin an Einstein solid and exploited i t further and linked the results to more familiar experiences. We have called our game Counters on Grids3. You proceed as follows: Set up a grid, say 6 X 6, and on it place any number of counters, say 108 (to give a ratio of 3 counters per square). You can place them on squares a t random or on squares evenly, or all the counters on one square. Alternatively you can choose your own unique scheme. You then plot out a histogram of the number of squares with0,1,2, etc., counters versus the number of counters 0,1,2, etc. For placement of the counters on squares a t random, one set of results is seen in Figure 1. Now you useapair of dice @sided for a 6 X 6 grid, or 4-, 8-, 10-sided etc. for other-sized grids), to specify a particular square on the grid, like in a city map reference. One die 506
Journal of Chemical Education
Five 2. Numbers 01 squares whh 0.1.2.
. . .countero.
defmes the horizontal coordinate and the other the vertical coordinate. On the first throw of the dice, you pick up a counter, if there is one, from the square selected by the throw. If you hit a blank square you throw again. On the next throw you place the counter down again, onto the square selected by the throw. You repeat this hundreds of thousands of times and note the changing shape of the histogram along the way. We programmed the game for the Macintosh4, but it can he run on the tiniest of computers. What happens is very interesting. No matter how you start, you end un with the same result. a fluctuatine. -. annroximatelv .. expon~ntiallyfalling distribution. Let us start with 108 counters placed randomlv on a 6 X 6 mid. You usuallv obtain a histogram with a hump;n the middie, Figure 2 (approaches normal for a high ratio of counters t o squares). The random experiment of dice throwing (say 1000 times),
' Porter. George. "The Laws of Disorder"; BBC Fllms.
School Sci. Rev. 1976. 57. 654. Henderson. C. N.: ~elinge;.6. K. "Microprocessors in Chernir try". In Bringing Computers info College and Universify Teaching: HERSA, Tertiary Education Centre UNSW: 1981: pp 33-42; paper and videotape. Program on disk available from the authors for $25 lncl postage.
'
Figure 3.Histogram alter 1000 iterations.
Figve 4. Hismgam averaged for 500 distributiom, after 1000 nsrstions Numbw of Squarr with O , I , Z . . .Counten q
N u m b s of throws
%
n?
m
k?
n4
5
%
W
Variance
~Morm
0 5 13 18 16 15 15 19 0
36 26 12 9 9 11 10 9 0
0 5 9 4 7 6 8 3 0
0 0 2 2 3 3 2 2 0
0 0 0 2
0 0 0
1
0 0 0
0 0 0 0 0 0 0
1
1
I 6 X 10" 2 x lo" 1.67 X 10" 1.62X 10" 1.65 X 10'O 9.72X lo" 7 X 10'' 36
0 0.29 0.57 1.83 1.26 1.17 1.14 2.29 36
5 10 20 40 60 80 100 all corner0
1
I 1 1
...m s = l
on one square
causes the histogram to change shape, see Figure 3, to an exponential distribution. This becomes more obvious if you smooth out the fluctuations and display the average of a number of distributions (say 500), after the 1000 iterations. See Figure 4. The explanation for the changes can be seen if we calculate the number of possibilities for each arrangement5, see the table. Consider a starting position that is even, and calculate the possihilities of each consequent position. N is the number of squares, while the no,nl, etc., are the number of squares with 0,1, etc,, counters, is the number of possibilities of distributing counters among squares without changing the histogram. The general expression for asample variance (which is a measure of spread of the distribution) is given by S2= Z h - .WAN - 1) = [Ey:
- (EyJ2INll(N-
1)
where yi is the value for x = i, and 7 is the mean. Thus for no = 16, nl = 9, nz = 7, ns = 3, nr = 1,
9 = [[(I6X 0')
+ (9 X 12)+ (7 X 2') + (3 X 3%)
The maximum number of possibilities means keeping the product of the numbers on the bottom line of the expression for Woas small as possible, i.e., keeping the numbers themselves as small as possible, subject to two constraints:
+ (n3X 3) + ... = no. of counters = no. of squares ( 2 ) N o + n l + n l + n s + ... +
(1)(n, X 1) (n, X 2)
The algebraic way of doing this uses the Lagrangian Method of undetermined multipliers. Note the following: (a) Very few changes in the original distribution cause a rapid increase in W. (b) , . W is the value of W for the moat probable configuration and is of the order of 1018.
(c) The distribution can fluctuate at equilibrium provided the value of W stays near ta W,,. W- is the sum of all the possibilities for all the configurations and is given by6: (N W,, = (N - l)!q! = s u m of w over all configurations +
For N = 36, q = 36, W,, = 1 X 1018, W b u = 2 X loz0. Some idea of the magnitude of these number of possibilities can be obtained when you consider the number of possibilities for setting Rubik's cube. Hofstadter7 estimates this as 4.3 X lOI9, (well, actually 43,252,003,274,489,856,000, to be precise). You can imagine that throwing a die to select a move will "never" bring a random cube configuration back to the start. This then is the rationale for the second law; systems drift toward configurations that are made up from the greatest number of possibilities or indistinguishable microstates. There are some other interesting observations to make about the distributions. When you start placing counters a t random one a t a time on the squares you produce a binomial distribution. For 36 squares-and 36 counters, (using the usual statistical notation), n = 36 and p = 1/36. The mean is n p = 1, and the expected variance is np(l - p). Under these conditions, the binomial is approximately Poisson where the mean is p and equals the expected variance, because n p ( 1 p ) np. When, in addition, fl is small, the Poisson distribution becomes a geometric distribution (i.e., a discrete exponential, the parameters of which are discussed in the appen-
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(a) School Sci. Rev. 1976, 57, 654. (b) "A Picture of Shuffling Quanta". in NuffieM Advanced Science Chemistry, Teachera' Guide I; Longman: England. 1984:Topics 1-11. pp 75-79. Bent. H. A. TheSecondLaw: OxfordUniversity: New York, 1965:
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chanter 21.. -. -. -
Hofstadter. D. Metarnagical Themas: Questingfor the Essence of MindandPattern:Penguin: 1985; p 305. Volume 66
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dix). This is the condition almost used in the Nuffield text, but without explanation."On the other hand, keepingp the same but increasing n t o give a large np (>>I), we find that the binomial = Poisson, tends to a normal or bell-shaped error distribution (still with equal mean and variance). With 108 counters on 36 squares we are on the way to approaching this limit. When we start the game with 108 on 36,we quickly see that the random starting distribution does not have the largest number of possibilities W , and thus the distribution changes in shape with play. Seeing that we started with a random placement, to say that the drive is now dominated by a "the tendency to randomness" is not helpful unless we exolain how the constraints have changed when we change thk rule. We started by depositing the counters binomial&. This fixed the expected variance a t np(1 - p ) , (=3l. However, as the game broceeds, the variance i s no longer constrained but increases, and another distribution takes over. For aficionados, you can actually derive the equilibrium to which this distribution tends. For i counters per square the probability distribution n ( i ) is given by
- -
where C is the combinatorial symbol. -, q m (or taking an average of a In the limit as N large numher of runs), the distribution becomes a geometric distribution. The expected equilihrium probability ll(iJ,for this limit is now given as n(i) = { M I+ dl x ld(1t d where 0 = q/N. The variance = ( d ( 1 + @)I + I1/(1 + @)'I. Thus for 36 counterson 36 squares theequilibrium distribution n l i ) iseiven hv 11/21 , . . X (112)'. . . . ,while for 108counters it is 114 X ($4);: (The expected equilibrium variances are 2 and 12. resoectivelv. Trv some exoeriments to show that this is distribitions are discrete exponential so:) ~ 6 e s e in the limit of laree (or averaeed) samdes. This game illustrates the fundamental concept of the second law of thermodvnamics. the understandine of which C. P. Snow declared, & his ~ e d lecture e (later p;blished as a books), as the test of scientific literacy, the equivalent to being able to appreciate a play by Shakespeare. Interestingly enough, he later drew hack from this test. "This law", he later said,
.
is one of the greatest depth and generality; it has its awn sombre beauty; like all major scientificlaws it evokes reverence. There is of course no value in the non-scientistknowing it by the rubric in an encyclopaedia. It needs understanding which eannot he attained unless one has learned some of the laneuaee of ohvsies. That understanding ought to he part of the cokgon tGedtieth century culture. Nevertheless I wish I had chosen another example. He eoes on t o sav that he would now have chosen molecular biokgy but added "theideas in this branch of science are not as phvsicallv deer, or of such universal significance as those of the secon-d law. The second law is a generalisation which covers the cosmos!' Counters on grids is a game that provides a small-scale simulation of the probability interpretation of entropy, the Einstein solid. Bose-Einstein statistics with the Boltzmann distrihution limit. In fact the Holtemann limit is found for any large-scale distribution for which tho only constraint, or sole information, is the constant value of the mean. Examples include the intensity of light passing throughan
Snow, C. P. The Two Cultures and a Second Look; Cambridge University: 1969.
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Journal of Chemical Education
absorbing solution (constant absorbance), first-order kinetics (conscant rate constant), radioactive or fluorescence decav (constant lifetime), radial electron density in an s orbital (fixed Bohr radius), etc. However, what we believe is incredibly interesting is that the thinking behind the second law is by no means limited to these physical examples. Just call the counters "money" and the squares "people." You then discover a truism. In a laissez-faire economy in which monev is exchaneed freelv between individuals for goods and services without any restrictions, (in particular redi~tributionvia taxation and social henefits: the model fits authoritarian economies hetter than democratic ones), most peopleend up with few dollars and a few end up with most of the-dollars.-~venif you increase the mean numher of counters per square from less than one to greater than one, the final distribution does not change significantly in shape, so that people's relativities hardly change, even though in absolute te&s thev are better off. A closer model i s where hundreds of people enter a casino with savan eoualmodest stake and a t the end of the evenine leave with their pockets emptied or filled. As a group the$ winnings follow very closely the exponential distribution. I t is extremely important to note that we cannot predict who will he rich and who will be poor. The more possibilities for distributing the money between people without effecting the overall distribution, the more probable that distribution. Even more interesting is that even where we ourselves control the result, as, for example, in the way the organizers deliberately distribute prizes in a lottery, there are few large prizes, moderate numbers of moderate prizes, and many small prizes. With a fixed amount of prize money and number of prizes, the exponential distribution is the most natural way to do things. You say that you are not into gambling. Well, you are, you know! As a policy holder in any insurance scheme (car, health, house, etc.), you contribute a small premium t o protect vourself against the certaintv that natural events will dictate that a few people will neeb large payouts and more will need smaller ones. The small premium monies will redistribute themselves as claims, roughly along the lines of the exponential distribution. From the nrohabilitv " of occurrence of individual letters in this article to the sire of cities in a country or oil wells in the world or stars in the ealaxv, the distrihution as determined by the second law of ~ h e ~ o d y n a m i (with c s minor reservations). eives the eist of the answer. Further examdes can he seen in the loss of control of a private company when i t distributes shares t o a large public (look a t the share distribution of shareholders in a large company). If we move slightlv laterally to the random exchange of power and influence in an organization, then its distribition follows the same pattern; the number of people a t any level decreases with the height of the level-hierarchies behave just like the real atmosphere, and for the same reason; they become rarer the higher you go. The thinkine behind the second law has other evervdav analogues. ~ o r k x a m ~ lwhat e , is the fundamental reason fo; it beine so much harder t o park a car in a tieht soot than to drive itout again afterwards? (Consider thenumher of possibilities for being parked cornoared to beinp-"unoarked".) . The reason fori&k of suppok for teaching innovation can he seen as a consequence of the second law. I t is much more natural to distribute funds to provide a small spectrum of more spectacular items to fill the granting body's annual report and for parliamentarians & make-speeihes about than to uniformly improve the whole field. Laying foundation stones for hieh-tech hosoitals heats small and unnoticed public health improvements'across the board. For the same reason i t is very difficult t o replace the concept of a large, centralized power-generation system (be it coal-fired or nu-
-
~
~.
~~~~
-
clear) with many small distributed solar units. Finally, no lotterv would attract custom if. for the same total amount of prize money, it offered only e q h , relatively smallprizes and no outstandinn ones. quantum theory, Einstein once said "God In regard does not play dice." Einstein was probably wrong. Anyway, in thermodynamics He definitely does play dice. And so do we, all the time.
Supposethere are to he i counterson a certain fired square j. The total numher of arrangemento of our n vectors with i counters in box is obtained from the ahwe formula on replscmg N hy N - I and n
to
Therefore the exact probability of i counters in our special hox j , equals
Acknowledgment We acknowledge the continuous help and encouragement offered t o us by Peter Hall of the Department and Research School of Statistics a t ANU. Appendix: Analysls ol the Dlstrlbutlon Suooose there are N counters on o sauares. Reoresent the occupane;brob~emas a vector of length ,; w'hose jth ciement equals the number of counters on square j. These vectors are distributed uniformly tall microstates are equally probahle-Hose-Einstein statistics). The number of such vectors may be shown to be equal to
-
IfN- -, n then
+
(N n - i - 2)!n!(N - l)! (n - l)!(N - i - l)!(N + n - l)!
--
andNIn- r (= mean number of counters per box), Pi
where p = r/(l
-
f ( l + rY+' = (1- p)pi
+ p). This is a geometric progression.
Volume 66
Number 6 June 1989
509
