Physica A 292 (2001) 494–508www.elsevier.com/locate/physa

E�ect of dense measurement in classical systemsD. Bar

Department of Physics, Bar Ilan university, Ramat Gan, Israel

Received 15 October 2000

Abstract

We show here, with the help of numerical simulations, that dense measurement has an e�ecton classical systems which is similar to that on quantum systems. Although we study here thecircular billiard, we believe that it can be demonstrated also for other classical systems. c© 2001Elsevier Science B.V. All rights reserved.

PACS: 07.05.Tp; 06.20.D; 42.50.Lc; 42.50.Ct

1. Introduction

In this work, we draw a strong analogy in the behaviour of a class of classicaldynamical systems and a type of behaviour typical of quantum systems, in particu-lar, the well-known Zeno e�ect. As our prototype for the classical system in whichwe want to show a Zeno-type e�ect [1–11], associated with dense measurement,we take a two-dimensional circular billiard which consists of two concentric circlesone inside the other (see Fig. 1). In this context, we mean by Zeno-type e�ect, thestabilizing e�ect of repeated measurement on the time evolution of the system. Theballs inside this billiard are point particles which are re>ected from the boundary con-sisting of the outer and inner circle perimeters. These re>ections are considered to beelastic re>ections, in which the initial angle before re>ection equals the angle afterre>ection. On the outer perimeter there is a narrow opening through which our pointparticles can be ejected from the billiard. This system may represent a classical coun-terpart of physical situations such as radioactive processes, or the situation of atomsinside a molecule, or nucleons in a nucleus [12]. Each point particle has two alternativetypes of path through which it can proceed. One type is that in which the particle isalways re>ected between the two circles perimeters, where each re>ection starts fromthe outer (inner) circle, is re>ected at the inner (outer) one, and ends at the outer(inner) circle. The second type is that in which the particle is always re>ected onlyby points on the outer circle, without touching at all the inner one. Our assumption

0378-4371/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(00)00583 -5

D. Bar / Physica A 292 (2001) 494–508 495

Fig. 1. The classical system of two concentric circles billiard.

of elastic re>ections guarantees that once the particle begins its journey on one typeof path it remains in this type all the time till its ejection from the outer circle. Weidentify these two types of path as “states”. We study such a system subject to a per-turbation (analogous to the quantum jump model of Carmichael [13,14]) which causestransitions from one state to another.

In Ref. [12] a rectangular billiard with a circular disk inside was discussed in relationto chaotic motion, without any consideration of the e�ect of dense measurement. In[15,16] the same system as that treated in Ref. [12] was also discussed from the point ofview of chaos, but these authors concentrated on the possibility of “delaying the onsetof chaos” by changing the radius of the inside circle, and thus alluding to the possibilityof preserving the initial state which is actually one of the main characteristics of theZeno e�ect. In this work, the question of chaos is not considered, but the emphasisis on a Zeno-type e�ect which is achieved through a very large number of repetitionsof the same measurement [3–10] on the point particles of state 1 or 2. In Section 2,we show that when we perform dense measurement upon all the particles of state 1 or2, in the sense to be described, then the motion, generally, remains stably associatedwith one or the other type, i.e., dense measurement do not permit a transition fromone type to the other. We also show in this section that this e�ect depends upon thevalues given to the inner radius. In Section 3, we discuss our numerical simulations interms of dense measurement along a path in the sense of Aharonov and Vardi [1,2].

2. Dense measurement for the circular billiard

As we have remarked each particle has two di�erent types of path (states 1 and 2)through which it can move in the billiard. In order to be able to di�erentiate numericallybetween these two states, we must consider the activity of each one, that is, the rate of

496 D. Bar / Physica A 292 (2001) 494–508

Fig. 2. The solid graph shows the activity of state 1, and the dashed one represents the activity of state 2.

decay of a large number of particles all of which are in state 1 (or state 2), where aparticle is considered to have been decayed once it has been ejected out of the billiardthrough the narrow opening in the outer circle perimeter. In order to exploit this notionof activity we assume, as is done in the radioactive processes from which the termactivity is taken, that each particle has arbitrary initial conditions. That is, its initialposition can be any point on the circular surface of the inner circle, so that this circle isfunctioning as a source of our point particles, besides its main function as a re>ectionsurface for the particles of state 1. Now, since we deal here with a classical systemfor which we want to show, numerically, an e�ect that has been treated up to nowas a quantum one, we have to determine the speciFc activity that must be preservedupon performing dense measurement on the system, as is necessary in a Zeno behaviour[3–10]. We call this speciFc activity, in analogy with the quantum terminology, theinitial state. This state cannot be taken, as is usually done in discussing the quantumZeno e�ect, to be the state of the system at the initial time (at this time all the particlesare still inside the billiard and no activity has started). The state to which we comparethe states of the system with perturbation and measurement is a state which correspondsto an entire history of development without perturbation or measurement, i.e., withoutany outside interruption until the evacuation of the last particle from the billiard. Wecall this state the no-measurement state. We note that Gell-Mann and Hartle [17–19]have also deFned a “state” in terms of an entire history. The no-measurement statehas a characteristic activity of its own, and we compare with it the activities (thealtered histories of development) obtained as a result of making measurements uponthe billiard system in the sense to be described below. Now, the activities of the twostates 1 and 2 are of course di�erent as can be seen from Fig. 2, which show theactivities of these states, that is, the number of particles that escape through the holein a predetermined time interval, binned in units of 60. The velocity of each point

D. Bar / Physica A 292 (2001) 494–508 497

Fig. 3. The activity when no measurement is done.

Fig. 4. The dashed graph shows the activity that results when dense measurement is done upon all theparticles of state 2 (compare with the solid graph of Fig. 2). The solid graph shows the activity that resultswhen dense measurement is done upon the particles of state 1 (compare with Fig. 3).

particle is 2.5, and the width of the narrow opening is 0.16. We take 6.5 and 4 forthe values of the outer and inner radii, respectively. The initial number of the pointparticles was 2:5 × 105. We must note that the linear sections of the curves of thisFgure and of all the other Figs. 3–9 are interpolations (performed automatically by thecomputer) due to the fact that our time axis is binned in units of 60.

By comparing the two curves of Fig. 2 we see that the activity of state 2, wherethe inner circle is not touched by the point particles is much stronger than the activity

498 D. Bar / Physica A 292 (2001) 494–508

Fig. 5. The solid graph shows the activity of state 1, and the dashed one shows that of state 2. Both graphsare for the case of r2�r1.

Fig. 6. The solid graph shows the activity of state 1, and the dashed one shows that of state 2. Boths graphsare for the case of r2 ≈ r1.

of state 1. The reason is that for a particle in state 2 there are a great number ofpossible di�erent trajectories, connecting the points of the outer circle, through whichit can be ejected out of the billiard as compared with the number of the correspondingdi�erent trajectories connecting the points of the outer circle to those of the inner circle(characterizing the trajectories of state 1). This can be proved as follows: from Fig. 1it can easily be shown that each trajectory in state 1, that is re>ected from both circles,

D. Bar / Physica A 292 (2001) 494–508 499

Fig. 7. The graph denoted a shows the activity that results when no measurement is done upon any particleof any state when r2 ≈ r1. Note the strong activity we obtain for this case, in which in the Frst 60 timeunits 2:14 × (105) particles, out of a total of 2:5 × (105), has been ejected from the billiard. The graphdenoted b shows also the activity for the no-measurement case, this time when r2�r1. The activity for thiscase is very much weaker compared with the case r2 ≈ r1.

Fig. 8. The solid graph shows the activity that results when dense measurement is done upon all the particlesof state 1. The dashed graph shows the activity when dense measurement is done upon all the particles ofstate 2. Both graphs are for the case of r2�r1.

500 D. Bar / Physica A 292 (2001) 494–508

Fig. 9. The graph denoted x shows the activity that results when dense measurement is done upon all theparticles of state 1 (note that graph x is comprised of only one point denoted by x), and that denoted bshows the activity resulting from dense measurement performed upon the particles of state 2. Both graphsrepresent the case of r2 ≈ r1. The straight line is an interpolation between the initial point at 60 and theFnal point at 120. The activities shown in all the di�erent graphs are binned in units of 60, as has beenremarked.

fulFls the following condition

d162√r21 − r22 ; (1)

where r1 and r2 are the radii of the outer and inner circles, respectively. From Fig. 1we can see that the shortest trajectory in state 1 is

d1(min) = 2(r1 − r2) : (2)

The longest trajectory is

d1(max) = 2√r21 − r22 : (3)

In state 2, where the point particles do not touch the inner circle, the longest trajectoryis the same as that of state 1 (see Eq. (3)), whereas the shortest trajectory, as comparedwith that of state 1, can be arbitrarily small. This is the fundamental di�erence betweenthe two states that actually account for their di�erent activities. Now, since for anyr1 and r2 that obey the condition r2¡r1 we have d1(max) = d2(max), and since thetrajectories of state 2 can be arbitrarily short, as compared with those of state 1, wearrive at the conclusion that the number of possible paths through which the pointparticles in state 2 can decay through the narrow opening is very much larger than thenumber of the corresponding paths of state 1. This means, as we have remarked, thatthe activity of state 2 is much greater than that of state 1, as can also be seen fromFig. 2.

D. Bar / Physica A 292 (2001) 494–508 501

There are a number of numerical works [20–22] treating the problem of the closedbilliard (in which there is no opening for the billiard balls to leave). Our numericalprogram was written using Matlab and it simulates a measurement process analogousto that used in the quantum jump model developed by Carmichael [13,14,23]. Now,although this model, as its name implies, is used to describe quantum phenomena,nevertheless, the speciFc ingredients of this model, especially its unique probabilisticcharacter, makes it appropriate to describe also the classical activities discussed herewhich are strongly probabilistic processes as will be shown below. Initially, before anymeasurement and between the identical measurements discussed here, we cause thesystem to make a periodic transition (according to the number of re>ections) betweenthe two states 1 and 2. Fig. 3 shows the activity one obtains when the system wasnot disturbed at all by any measurement, but allowed to pass systematically betweenthe two states. Comparing this Fgure with Fig. 2 we see that when the system movesperiodically between the two states its activity is stronger than that of state 1 butweaker than that of state 2. The reason is that its trajectories are of the two kinds, sothat its activity strength is between the activity strengths of the two states 1 and 2.

Each measurement act, from the sequence of identical measurements, is performednumerically after a speciFed predetermined number of re>ections in which the systemproceeds periodically between the two states undisturbed. As for the quantum jumpmodel, where the mere act of measurement may cause the system to jump to the lowerstate, or leave it undisturbed [13,14,23], so also here the mere act of trying, in theprocess of measurement, to interfere with the trajectory of state 1 (or state 2) may causeit to deviate from its course to the other state 2 (or state 1). For each measurementwe generate a random number [13,14,23] and compare it with a transition parameterT which is deFned to be the length of the current trajectory divided by the longestpossible trajectory characterizing both states (see Eq. (3)). Thus, for example, if ourpoint particle was in state 1 then the transition parameter is

T1 =d1

d1(max): (4)

If it was in state 2 the transition parameter is

T2 =d2

d2(max): (5)

The transition parameters T1 and T2 are considered here as the corresponding probabil-ities to Fnd the relevant particle in states 1 or 2, respectively. The reason is that if, forexample, T2 is small this implies that d2 is also small (see Eq. (5)). That is, because ofthe specular re>ection assumption, all the trajectories of the speciFed particle betweenthe points of the outer circle are small, and in this case this particle will, essentially,in a short time, move upon a large number of points of the outer circle. The smallerthese trajectories are, a larger number of points in a shorter time will be swept bythe particle, and the result will be fast ejection from the billiard. In other words, theshorter trajectories have a shorter lifetime, as compared to the longer trajectories, andaccordingly have also smaller probabilities. The same argument applies also for T1, but

502 D. Bar / Physica A 292 (2001) 494–508

here we must take into account that the trajectories of state 1 are bounded from belowwith a minimum of 2(r1 − r2) (see Eq. (2)). This causes T1 to be also bounded frombelow with the minimum

T1 (min) =2(r1 − r2)

2√

(r21 − r22): (6)

Thus, according to our discussion, the trajectories of state 1 have generally a longerlifetime compared to those of state 2, which can have vanishing values for the lengthsof their trajectories, and so also for their T2(min). This di�erence between the minima ofT1 and T2 is the principal factor that causes us to see, in the limit of dense measurement,a corresponding di�erence in the activities of these two states as will be shown.

The analogue of the quantum jump model of Carmichael [13,14] is as follows.Suppose that our point particle was in state 1 at the time of the measurement. Ashas been remarked, the measurement corresponds to comparing T1 with a generatedrandom number. If we Fnd that T1 is smaller, then our particle is transmitted to state2, and if T1 is bigger our particle is allowed to continue in state 1 undisturbed. Itis found that if the rate of repeating the measurement upon the particles of state 1becomes very large, in comparison with the rate of the undisturbed transition betweenthe two states, then the initial no-measurement activity is actually obtained, that is, weobtain a Zeno-type e�ect. In our numerical simulations the rate of the regular transitionsbetween the two states is taken as 1000 re>ections. That is, any particle that begins inany state is transferred by regular perturbation to the other state after it has completed1000 re>ections from the outer perimeter. To simulate the Zeno e�ect we perform ouridentical measurement after every three re>ections. In other words, for each regulartransition (that is performed after 1000 re>ections) we disturb the system, or repeatthe same measurement, 333 times. This is our dense measurement.

The dashed graph in Fig. 4 shows the activity we obtain when we make densemeasurement upon all the particles in state 2, without disturbing those in state 1. Thee�ect we obtain upon interfering with all the particles in state 2 is actually causingthem to deviate from the corresponding paths of state 2 to those of state 1, and sincewe leave the routes of state 1 undisturbed the corresponding activity we obtain is, ofcourse, only that of state 1. Indeed, by comparing this graph with the continuous graphof Fig. 2, which shows the activity of state 1, we realize that both represent the sameactivity.

The solid graph in Fig. 4 shows the activity we obtain when we make dense mea-surement upon all the particles in state 1, without interfering with those in state 2. Butsince T1 cannot take on the very small values (see (6)) that T2 can, the probability toFnd it larger than an arbitrarily generated random number in its range is bigger thanthat of T2, so the particles in state 1 have a big chance to be ejected from the billiardwhile they are still in state 1, before they will be transferred (as a consequence of somemeasurement) to state 2. In other words, performing the algorithm of dense measure-ment upon the particles of state 1 has no e�ect upon them, and the activity we obtainfor them is essentially that of Fig. 3, which is the activity for the no-measurement

D. Bar / Physica A 292 (2001) 494–508 503

case. That is, we have a Zeno-type e�ect. Comparing the solid graph in Fig. 4 withFig. 3 we see that this is indeed the case. We, therefore, see that for the normal caseof r1¿r2, where r2 is not too much smaller than r1 and not approximately equal to it,doing dense measurement upon the particles of state 2 results in a general transfer ofthem to state 1 with the corresponding activity. On the other hand, performing densemeasurement upon the particles of state 1 does not result in such an outcome, andthe activity we obtain, generally, is the same as for the no-measurement case. That is,these particles exhibit a behaviour typical to that of the Zeno e�ect.

As we have remarked all our numerical simulations thus far were conducted for thespeciFc case of assigning the values of 6:5 and 4 to the radii of the outer and innercircles, respectively. As can be seen from Eqs. (1)–(6) the activities, just described, aresigniFcantly a�ected upon changing the value of the inner radius. It has been shown,through our numerical simulations, that as the value of the inner radius becomes moreand more large (remaining always smaller than r1) the activities of the states 1 and2, in the limit of dense measurement, become similar to each other, and when r2 ≈ r1they will be both similar also to the no-measurement case. That is, in the limit r2 ≈ r1doing dense measurement upon either the particles of state 1 or 2 results in no e�ectat all. In other words, both kinds of these particles demonstrate a Zeno-type behaviour.On the other hand, when the inner radius becomes more and more small, the activitiesof the two states 1 and 2 continue to show the same characteristics discussed for thecase where r2 is not very much smaller than r1. That is, the activity obtained whenperforming dense measurement upon the particles of state 2 is that of state 1, andthat obtained when performing dense measurement upon the particles of state 1 is theno-measurement one.

In the following we show, numerically, the e�ects of the changes of the inner radiusupon the activities of states 1 and 2, in the limits r2�r1 and r2 ≈ r1. Fig. 5 showsthe activities of the states 1 and 2 when r2�r1, and Fig. 6 shows the activities ofthese states when r2 ≈ r1. For these graphs we take the values r2 = 0:2 and r1 = 6:5 tosimulate the case r2�r1, and r2 = 6:3, r1 = 6:5 for the case of r2 ≈ r1.

When r2 is very much smaller than r1, the longest trajectories of the two statesremain equal to each other. As for the speciFc trajectories of state 1, we can easilysee that each such path is approximately equal to d1(max) (compare Eq. (2) with (3)in the limit of r2 being very much smaller than r1). In other words, when r2�r1 allthe trajectories of state 1 are approximately equal to each other and to d1(max). Thus,

d1 ≈ d1(max) : (7)

Upon comparing the last equation with Eq. (4) we see that we obtain for the transitionparameter T1, when r2�r1

T1 =d1

d1(max)≈ 1 : (8)

This shows that a Zeno behaviour can be obtained, for the particles of state 1, alsofor the case of r2�r1. The reason is that since in each repetition we compare T1

with a random number, then, of course, we would have T1 generally greater, and this

504 D. Bar / Physica A 292 (2001) 494–508

outcome, according to the conditions of our numerical simulations, guarantees that allthe particles that began in state 1 will continue in their undisturbed motion until theirejection from the billiard. In other words, making dense measurement upon all theparticles of state 1, when r2�r1, has the same e�ect as in the no-measurement case.Indeed, by comparing the solid graph in Fig. 8 (which shows the activity resultingwhen making dense measurement upon all the particles of state 1, without disturbingthose of state 2, for the r2�r1 case), with the graph denoted b in Fig. 7 (which showsthe no-measurement case when r2�r1) we actually see the same activity. These twographs appear to be di�erent only because they are not plotted for the same range ofthe ordinates.

As for the particles in state 2 we can see that their longest trajectory becomeslarger than the corresponding one when r2 is not so much smaller than r1. Sincethe trajectories of state 2 are not bounded above zero and can be inFnitesimal, as wehave remarked, we can see that by frequently repeating, in the dense measurement limit,the comparison of T2 with a generated random number we will certainly obtain a resultwhich will transfer these particles to the characteristic trajectories of state 1, beforethey leave the billiard. Moreover, since the trajectories of state 1 are approximately themaximal ones (see Eq. (7)), then according to the previous discussion their lifetimeis also maximal and they will not be transferred to state 2 during their undisturbedmotion before their ejection from the billiard. In other words, all the initial particles ofstates 1 and 2 will end their journeys in the billiard in state 1. Indeed, by comparingthe dashed graph in Fig. 8 (which shows the activity when dense measurement is madeupon all the particles of state 2 for the r2�r1 case), with the solid graph in Fig. 5(which shows the activity of state 1 when r2�r1) we have almost the same activity.Again the two graphs look di�erent only because they are not plotted for the samerange of the ordinates. The graph denoted b in Fig. 7 shows the activity we obtain forthe case r2�r1 when no measurement is done upon any particle. The graph denoted ain Fig. 7 shows the activity for the opposite case of r2 ≈ r1 for the no-measurementcase. If we compare the two graphs of Fig. 7 to Fig. 3, which also shows the activityfor the no-measurement case when r1 and r2 take values of 6.5 and 4, respectively,then we see that the graph a has a much stronger activity than that shown in Fig. 3,and the graph b has a much weaker activity.

We check now the case r2 ≈ r1. From Eq. (3), which holds also for the tra-jectories of state 2, we see that d2(max) becomes a very small number, and sincethis tells us that the length of each other trajectory of state 2 becomes also a verysmall number, we can approximately assign the following value for the transition para-meter T2,

T2 =d2

d2(max)≈ 1 : (9)

Now the same argument that follows Eq. (8), which shows us how the particlesin state 1 are strongly a�ected when r2�r1, applies here for the particles in state2 when r2 ≈ r1. That is, a Zeno-type behaviour will be demonstrated for these

D. Bar / Physica A 292 (2001) 494–508 505

particles in the limit of dense measurement. As for the particles of state 1 we alsosee here from Eqs. (2) and (3) that both d1(min) and d1(max) become very small whenr2 ≈ r1. That is, we have almost the same external conditions as those for the par-ticles of state 2. This can be seen from a comparison of the graphs x and b ofFig. 9, which show the activities we obtain for the case of r2 ≈ r1 when perform-ing dense measurement upon the particles of states 1 and 2, respectively, with thegraph a of Fig. 7, which shows the no-measurement case for r2 ≈ r1. These threegraphs show a very much strong activity in which about 2:3 × 105 from the initial2:5 × 105 has been ejected from the billiard during the Frst 60 time units (we havebinned the activities in units of 60). Note that the graph x of Fig. 9 is comprisedof only one point, which means that for this case about 95% of the initial parti-cles leave the billiard during the Frst 60 time units. We thus see that in the densemeasurement limit both kinds of particles demonstrate a Zeno-type behaviour whenr2 ≈ r1.

In summary, we see from this discussion that for reasonable values of r1 and r2,where r1¿r2 we obtain a Zeno-like e�ect for the particles of state 1 when we dodense measurement upon them as described. That is, performing dense measurementupon these particles has no e�ect upon them, and the resulting activity we obtain isthe same as that obtained for the no-measurement case. This state of a�airs remainsthe same either when r2 becomes smaller all the way down to the limit of r2�r1, orwhen r2 becomes bigger all the way up to the limit of r2 ≈ r1. As for the particlesof state 2, performing dense measurement upon them causes them to be transferred tostate 1, and the activity obtained is that of state 1. This state of a�air remains when r2becomes smaller all the way to the limit r2�r1. When r2 becomes bigger then, in thelimit of dense measurement performed upon the particles of state 2, the activity beginsto deviate from that of state 1 and approaches that of the no-measurement case. In thelimit of r2 ≈ r1 the activity is actually that of the no-measurement case. That is, wehave a Zeno behaviour in this limit for the particles of state 2 also.

3. Dense measurement along a path

From the previous section we saw that our Zeno-type behaviour is not that of pre-serving the initial state of the billiard system (which is characterized by a zero activity),but, as we have remarked, that of obtaining the no-measurement state of it (which is,in fact, a complete history of the development of the system without perturbation ormeasurement). This seems at Frst sight to contradict the conventional perception ofthe Zeno e�ect which is understood to be that dense measurement upon some system,being initially in some deFnite state, causes this state to be preserved in time. But ascan be seen from the description of this measurement in Section 2 not only this doesnot contradict the notion of the Zeno e�ect, but strengthens and enhances it in twoaspects: Frst, it generalizes it from its previous signiFcance as being purely a quan-tum e�ect, and second, of showing that this e�ect also includes the transference of the

506 D. Bar / Physica A 292 (2001) 494–508

system from an initially deFnite state to another determined state. This second aspect isno other than the dense measurement along a path in the sense of Aharonov and Vardi[1,2]. In other words, the results we have obtained in our numerical simulations of theZeno behaviour for the di�erent cases discussed in the previous section are, actually,performances of dense measurement along paths that begin in the same initial state ofzero activity (when all the initial particles are still in the billiard system), and end inthe same Fnal state of no-measurement activity. Each time we repeat the simulationswe always begin from the same initial state and end in the same Fnal state, but theroute between these two states will never be the same for any two such repetitions,even if the number of these repetitions is very large. The reason for this is that for eachrunning of our program we actually made dense measurement along each trajectory ofeach particle, and since the result of each such measurement depends randomly uponthe speciFc random number generated for this experiment and upon the length of thecurrent path between the last two points on the outer circle (see Section 2), then it isobviously clear that although the system ends in the same state, the routes to this stateare di�erent from one another, upon a multiple running of the program.

We can understand each such running that leads our system from the initial state tothe Fnal state as closely analogous to some deFnite Feynman path [24,25] (we referby this term to the classical conFguration space analog). From what we have just said,we see that there is very large number of such Feynman paths in conFguration spaceall beginning at the same unique state and end at another unique state. Each time werun our numerical simulations we actually realize one deFnite Feynman path from allthe possible ones in the sense of Aharonov and Vardi [1,2]. This realization of eachFeynman path can be done, as has been remarked in Ref. [1], only through densemeasurement along this speciFc Feynman path.

We can prove this as follows: The no-measurement state we want to obtain has theproperty that performing a measurement upon a moving particle results in no e�ect atall as if we did no measurement upon it. That is, it remains in its initial state. Initially,each point particle moves periodically between states 1 and 2 till the measurementtimes when we compare Eq. (6) or (7) with a random number such that if the value(6) or (7) is greater than this number, then the relevant point particle continues in itsformer state; otherwise, it is transmitted to the second state. Now, the probability ofobtaining any speciFc result from these two possibilities is of course 1

2 . We submit tothese measurements only one kind of particles (either of state 1 or 2). After the Frstmeasurement of these particles, assuming that their number was n, the probability ofFnding all of them still in their initial state is ( 1

2 )n. For a large n we can assume that

the number of particles that remain still in their state is approximately n=2. The largeris n the more accurate this number will be. The second measurement is conducted nowupon these n=2 particles, and the probability to Fnd all of them in their initial state is( 1

2 )n=2. Again for a very large n we can assume that we remain with n=4 of the initial

particles. Thus, if we perform k experiments, we would be left with n=2k particles.Each such experiment in each stage is performed on all the remaining particles (fromthe previous stage) in the relevant initial state. The result of each measurement done

D. Bar / Physica A 292 (2001) 494–508 507

on any one of these particles does not depend, of course, on the results obtainedfrom any other measurement performed on any other particle. That is, we deal herewith independent probabilities. Now, if we take into account only the kth experimentperformed on the approximately n=2k particles (in the relevant initial state) that remainsfrom the previous (k − 1) experiments, then the probability of Fnding these particlesin this initial state is

pk ≈(

12

)n=2k: (10)

When we consider all these k experiments we obtain the probability of Fnding all theinitial n particles still in their initial state:

Pk ≈ 1k(p1 + p2 + · · · + pk−1 + pk)

≈ 1k

((12

)n+(

12

)n=2+ · · · +

(12

)n=2k−1

+(

12

)n=2k)

=1k

k∑m=0

(12

)n=2m: (11)

The last result tends to the limit of 1 when k becomes very much large. In otherwords, when the number of these measurements becomes large enough, the probabilityof Fnding all these particles in the same state they began with is 1, that is, we havea Zeno-type e�ect.

4. Concluding remarks

We have shown that Zeno-type behaviour may be considered as more general thana purely quantum e�ect. Using numerical simulations it has been demonstrated thateven a simple classical system like the concentric billiard can show a Zeno-like e�ectin the limit of dense measurement, as has been described in Sections 2 and 3. Thise�ect has been shown directly without having to use chaotic notions as was done inRefs. [15,16]. By this we have proved, numerically, the Simonius assumption [11] thatactually many physical phenomena, even classical and macroscopic ones, are the resultsof the Zeno-type e�ect. We show also through our numerical simulations that the notionof Aharonov and Vardi [1,2] of doing dense measurement along some Feynman path,from a large number of possible ones, and thus realizing it, is demonstrated here fora classical system. Our results are exclusively the consequence of dense measurementalong some deFnite path of the system. In other words, without this dense measurementwe will not obtain the results of the graphs shown in Section 2.

We have also shown that this Zeno behaviour depends strongly upon the values takenby the radius of the inner circle. Our numerical work simulates only the classical systemof the two concentric circles billiard, but as we have remarked we believe that our

508 D. Bar / Physica A 292 (2001) 494–508

results hold generally for other classical systems also and are not conFned only to thecircular billiard.

Acknowledgements

I wish to thank L.P. Horwitz for discussions on this subject, and for his reviewof the manuscript. I also wish to thank J. Levitan for discussions on the subject ofbilliards at an early stage of this research.

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