Dynamical properties ofa particle in a classical time-dependent potential well

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1 Available online at Physica A 33 (003) Dynamical properties ofa particle in a classical time-dependent potential well Edson D. Leonel, J. Kamphorst Leal da Silva Departamento de Fsica, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, C. P , Belo Horizonte, MG, Brazil Received 6 September 00 Abstract We study numerically the dynamical behavior ofa classical particle inside a box potential that contains a square well which depth varies in time. Two cases oftime dependence are investigated: periodic and stochastic. The periodic case is similar to the one-dimensional Fermi accelerator model, in the sense that KAM curves like islands surrounded by an ergodic sea are observed for low energy and invariant spanning curves appear for high energies. The ergodic sea, limited by the rst spanning curve, is characterized by a positive Lyapunov exponent. This exponent and the position ofthe lower spanning curve depend sensitively on the control parameter values. In the stochastic case, the particle can reach unbounded kinetic energies. We obtain the average kinetic energy as function oftime and ofthe iteration number. We also show for both cases that the distributions ofthe time spent by the particle inside the well and the number ofsuccessive reections have a power law tail. c 003 Elsevier Science B.V. All rights reserved. PACS: 05:45: a; Pq Keywords: Chaos; Lyapunov exponent; Fermi accelerator 1. Introduction The problem about energy transfers in classical systems has been extensively studied [1 7]. An interesting question is ifa particle, in a classical system that gives/takes energy to it, can have unlimited gain ofenergy? This was the fundamental question of Fermi in his work on cosmic radiation [1]. Douady [] showed that for the so-called Corresponding author. Fax: addresses: edleonel@sica.ufmg.br (E.D. Leonel), ja@sica.ufmg.br (J.K.L. da Silva) /03/$ - see front matter c 003 Elsevier Science B.V. All rights reserved. doi: /s (03)

2 18 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fermi accelerator model with suciently smooth movement ofthe wall, the response is no. This response is associated to the existence ofinvariant spanning curves observed in the phase space for high energies. Several other models involving energy transfers have essentially the same phase space characterized by invariant spanning curves at high energies and a structure ofinvariant curves like islands surrounded by an ergodic (chaotic) sea at low energies [3,4,7]. In this paper we propose a very simple time-dependent classical one-dimensional system. It consists ofa single particle inside an innite box ofpotential that contains a square well. The depth ofthe potential well oscillates in time. The dynamics is given by a two dimensional non-linear map that conserves the phase space area. For periodic oscillations, this model presents a dynamical behavior similar to the Fermi accelerator model and the particle has limited energy gain. We will evaluate numerically the Lyapunov exponent for the ergodic sea (low-energy region) for dierent values of the control parameter. The structure ofthe phase space, and thus the value ofthe Lyapunov exponent, changes, sometimes drastically, as the control parameters vary. For stochastic oscillations, the kinetic energy ofthe particle can reach unbounded values, implying that the system presents a Fermi acceleration. This behavior was also found in the study of the stochastic Lorentz gas [6]. In both, periodic and stochastic oscillations, we characterize the distributions of(i) number ofsuccessive reections in the well and (ii) the time that the particle remains inside it. Recently, a similar distribution was studied [3,4] f or a time-modulated barrier and a power law behavior was found. Here, we nd the same behavior for the periodic and the stochastic models. This paper is organized as follows. In Section we present a detailed construction ofthe model with periodic oscillations and obtain the associated two-dimensional non-linear map. We present also a briefdiscussion about the method used to evaluate the Lyapunov exponents. The numerical results for the periodic model are discussed in Section 3. In Section 4 we present the stochastic version ofthe problem and discuss the results. The conclusions and nal remarks are presented in Section 5.. The model with periodic oscillations and Lyapunov exponents.1. The model We consider the classical dynamics ofa particle inside a one-dimensional rigid box of potential that contains an oscillating symmetric square well. This problem is dened by the time-dependent Hamiltonian H(x; p; t)=(p =m)+v (x; t), where V (x; t) is given by if x 6 0 and x (a + b) ; V (x; t)= V 0 if0 x b= and (a + b=) x (a + b) ; D(t) if b= x (a + b=) : Here, V 0, a and b are constant. D(t) describes the time evolution ofthe bottom ofthe well. In the periodic case we consider D(t) = D cos(!t), with D being the amplitude

3 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 1. Sketch ofthe potential V (x; t) for the periodic case. V 0 is constant. D is the amplitude ofthe bottom ofthe oscillating well. ofoscillation and! the frequency. In Section 4, we will consider D(t) as a stochastic function oftime. The sketch ofthe potential V (x; t) for the periodic case and the lengths a and b are shown in Fig. 1. We always consider that D V 0. The dynamics ofthe particle under the periodic oscillations ofthe well can be described by a two-dimensional map, using the total energy E ofthe particle and the phase ofthe bottom ofthe oscillating well evaluated when the particle enters the well as variables. Suppose that the particle is in the left side of the well (x 6 b=) with initial energy E 0.Att = t 0 when it arrives at x = b= its kinetic energy changes from the initial value K 0 = E 0 V 0 to K = E 0 D cos(!t 0 ). Inside the well, the kinetic energy remains constant because there are no forces acting on the particle. Therefore, the time spent by the particle to travel the distance a is t = a=v where v = (K )=m. After t the particle reaches the other side ofthe well. Its total energy is then E 1 = K + D cos(!t 0 +!t ). Ifwe dene 0 =!t 0 as the initial phase and 1 = 0+ 0 as the phase at t=t 0 +t, then we have that E 1 =E 0+D[cos( 1 ) cos( 0)]. The phase change is given by 0 =!a= m [E 0 D cos(!t 0 )]. It is useful to distinguish two cases for the particle: Case 1: If E 1 V 0, then the particle does not exit the well. Case : If E 1 V 0, then the particle exits the well. In the rst case, the particle is reected with the same kinetic energy and arrives at the other side ofthe well after traveling a time t. The total energy is E = E 0 + D[cos( ) cos( 0)] where = Again, if E V 0, the particle is reected to the other side ofthe well and so on until the condition E n = E 0 + D[cos( n) cos( 0 )] V 0 is fullled. Here, we have that n = 0 + n 0.IfE n = V 0 we have a marginal situation in which the particle stays at the wall ofthe well with K =0.

4 184 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) When the particle exits the well (case ), it travels the distance b= with kinetic energy K = E n V 0. Then it is elastically reected and keeps the same kinetic energy until it arrives at the entrance ofthe oscillating well. The time spent in this trajectory is t =b= =m(e n V 0 ). Since the total energy is constant outside the well, the particle enters the well with energy E 1 = E n and phase 1 = n +!b= =m(e n V 0 ). So the map can be written as E 1 = E 0 + D[cos( 0 + n a) cos( 0 )] ; 1 = 0 + n a + b;n mod() : Here, n is the smallest integer such that E 0 + D cos( 0 + n a) D cos( 0 ) V 0 and the auxiliary variables are given by a = b;n =!a =m(e0 D cos( 0 )) ;!b =m [ E 0 + D cos( 0 + n a) D cos( 0 ) V 0 ] : After the particle enters the well, the same procedure is repeated to obtain the pair (E ; ) and so on. It is convenient to work with dimensionless quantities dened as 1 = D V 0 ; = m V 0!a; 3 = m V 0!b and k = E k V 0 : With these denitions, the two-dimensional non-linear map that gives the total energy and phase when the particle is entering the well at the k iteration is given by k+1 = k + 1 [cos( k + n a ) cos( k )] ; (1) k+1 = k + n a + b;n mod() ; () where n is the smallest integer number obtained from k + 1 [cos( k + n a ) cos( k )] 1 : (3) Here a and b;n are dened as a = k 1 cos( k ) ; b;n = 3 k + 1 [cos( k + n a ) cos( k )] 1 :

5 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) The coecients ofthe Jacobian matrix J are given by j 11 = 9 k+1 =1+ n [ ] 3 1 a sin( k + n a ) ; 9 k j 1 = 9 k+1 9 k = 1 sin( k ) 1 sin( k + n a )+ n 1 [ ] 3 a sin( k ) sin( k +n a ) ; j 1 = 9 k+1 9 k = n [ ] 3 a 3 [ b;n 3 ] { 3 1+ n [ ] } 3 1 a sin( k + n a ) ; j = 9 k+1 9 k =1 n [ a sin( k ) [ b;n 3 ] sin( k) [ ] 3 b;n ] { 3 sin( k + n a ) 1 n [ ] } 3 1 a sin( k ) 3 : The area ofthe phase space is conserved since det(j )=1... Lyapunov exponents Let us now describe briey the procedure used to evaluate the Lyapunov exponents. These exponents are dened by [8,9] n 1 j = lim n n ln k j ; j =1; ; k=1 where k j are the eigenvalues of M = n k=1 J k( k ; k ), and J k is the Jacobian evaluated on the orbit k ; k. In order to evaluate the eigenvalues of M we note that J can be written as a product J = T ofan orthogonal,, and a triangular, T, matrix. Let us dene the elements ofthese matrices by ( ) ( ) cos() sin() T11 T 1 = ; T = : sin() cos() 0 T Since M =J n J n 1 :::J J 1, we can write that M =J n J n 1 :::J J 1. Here 1 1 J 1=T 1. A product of J 1 denes a new J. In a following step, M is written as M = J n J n 1 :::J 3 1 J T 1. The same procedure is used to obtain T = 1 J. With this

6 186 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) procedure, the problem is reduced to evaluate the diagonal elements of T i : T i 11 ;Ti. Using the and T matrices we nd the eigenvalues of M, namely T 11 = j 11 + j 1 ; T j 1 + j11 = j 11j j 1 j 1 : j 1 + j11 Then, we nd that the Lyapunov exponents are given by j = lim n n k=1 1 n ln T k jj ; j =1; : Note that 1 = because the map is area-preserving. 3. Numerical results of the periodic model The map given by Eqs. (1) and () can be numerically iterated for a xed choice ofthe parameters ( i ; i=1;:::;3) and dierent initial conditions ( 0 ; 0 ), and so the overall structure ofthe phase space and its characterization can be investigated. A typical picture ofthe phase space is shown in Fig.. Here we choose 1 =0:5, =16:45 and 3 = 500. It is easy to see the ergodic sea, the KAM islands and the spanning curves. The spanning curves separate the phase space in dierent not connected regions, implying that a particle can not have an innite gain ofenergy. In the simulations we consider three dierent values ofthe parameter 1 ( 1 = 0:5; 1 =0:5 and 1 =0:75). This parameter must be less then 1 since the bottom can never reach the top ofthe oscillating well. The second parameter can be expressed as =(!=! c ). Here,! c is related to the time spent for a particle with kinetic energy V 0 to travel the distance a in the absence ofany oscillation. In order to have a well with a moderate frequency of oscillation (! :6181 :::! c ), we assume that Fig.. Phase space for 1 =0:5, =16:45 and 3 = 500:0.

7 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 3. Numerical convergence ofthe positive Lyapunov exponent for (a) double precision and (b) quadruple precision. =16:45. The last parameter can be written as 3 = (b=a). For constant, the increasing of 3 means that the ratio b=a increases. In order to evaluate the Lyapunov exponents, two numerical precisions were used and the same results were obtained. In Fig. 3 it is shown the convergence for the positive Lyapunov exponent. The parameters are the same as used in Fig.. Each one ofthe ve dierent initial conditions was iterated times. Note that both graphs, obtained with numerical double (Fig. 3a) and quadruple precision (Fig. 3b), present the same asymptotical behavior. In this case, the positive Lyapunov exponent is =0:949±0:008. Since the exponents are the same using dierent numerical precisions, the remainder calculations were made using double precision in order to save computational time. A plot ofthe positive Lyapunov exponent versus the control parameter 3 for each xed pair ( 1 ; ) is shown in Figs. 4(a) (c). Each point in Fig. 4 was obtained by averaging over ve dierent initial conditions. An orbit is obtained by iterations ofthe map. The error bar is the standard deviation ofthe ve samples. Note that an abrupt and sudden jump in the Lyapunov exponent behavior occurs. For 1 =0:5 and =16:45 the jump occurs around 3 95 as shown in Fig. 4(a). When 1 =0:5 and =16:45 the jump occurs around 3 0 (see Fig. 4(b)). In Fig. 4(c) ( 1 =0:75 and =16:45) we can see that the jump occurs around These abrupt jumps are associated with the sudden break o ofthe rst spanning curve that allows a merging of two dierent regions ofthe phase space. This change in size characterizes the decrease ofthe Lyapunov exponent.

8 188 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 4. Linear-log plot of 3 with =16:45, (a) 1 =0:5, (b) 1 =0:5 and (c) 1 =0:75. Each point was obtained by averaging over 5 dierent initial conditions, each one was iterated times. The error bars is the standard deviation ofthe samples. In order to investigate the abrupt change in the size ofthe low-energy region, we iterate a same initial condition for two dierent values of 3, both near the transition. These orbits are shown in Figs. 5(a) and (b). Note that the parameters have the values 1 =0:5, =16:45 and (a) 3 =951, (b) 3 =95. The same mechanism characterizes the jumps showed in Figs. 4(b) and 4(c). The same behavior for the Lyapunov exponent is also observed for other values of. Note that after the destruction of the rst spanning curve, two regions with dierent Lyapunov exponents merge together. After the merging, the positive Lyapunov exponent can be obtained by an average ofthe previous exponents, pondering by the relative size ofthe corresponding region ofthe phase space. To illustrate this conjecture, we need to estimate the fraction occupied by each region ofthe phase space. It is easy to see that the region I in Fig. 5(a) occupies 15% ofthe phase space, while region II occupies 85%. The Lyapunov exponents of the two regions are = :940 ± 0:003 and 0:519 ± 0:001, respectively. Therefore, the estimation ofthe Lyapunov exponent after the merging is =0:88±0:001, which is in agreement with the observed value (see Fig. 4(a)). This naive estimation works well for other transitions offig. 4. The initial oscillations in the Lyapunov exponent before these transitions are associated to small uctuations in the size and form of the phase space.

9 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 5. (a) Iteration of3 dierent initial conditions bellow, above and inside the spanning invariant curve with parameters 1 =0:5, =16:45 and 3 = 951:0. (b) Iteration ofone initial condition with 1 =0:5, =16:45 and 3 = 95:0. Other two quantities ofinterest are the reection time and reection number that the particle has inside the well. The reection time is dened as the time spent by the particle inside the well due to successive collisions with the walls. It is given by t =(n 1)t c = k 1 cos( k ). Here, t c is the time spent by the particle to travel the distance a in the absence ofoscillations with kinetic energy K =V 0. Note that (n 1) is the number ofreections ofthe particle. When n=1, the number ofsuccessive collisions with the walls ofthe well is 0 and the particle is transmitted. Ifthe particle has total energy k 1+ 1 at the entrance ofthe well, it will be transmitted without reections inside the well. The reection time (P t ) and reection number (P n ) distributions are shown, respectively, in Figs. 6(a) and (b). The data in a log log plot indicate a power law behavior: P n n, P t t. A good t is obtained with = 3. This power law is observed for a large range of the parameters and occurs only for a chaotic orbit bellow the rst invariant spanning curve. 4. The stochastic model Let us consider now D(t) as a stochastic function of time. We dene D(t)=Dg(t), with g(t) taking random values equidistributed in the interval [ 1; 1]. The variables ofthe non-linear map are the energy and time. It is easy to obtain the equations of

10 190 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 6. Log log plot ofthe reection (a) number and (b) time distributions. Here 1 =0:5, =16:45 and 3 = 500:0. The best t gives us the exponents = 3. the map, namely k+1 = k + 1 [g(t k + nt a ) g(t k )] ; t k+1 = t k + nt a +t b : Now, n is the smallest integer number obtained from k + 1 [g(t k + nt a ) g(t k )] 1 ; (4) and the auxiliary variables are given by t a = k 1 g(t k ) ; t b = 3 k + 1 [g(t k + nt a ) g(t k )] 1 : The kinetic energy ofthe particle is always changing due to the interaction with the time-dependent potential. We consider the average kinetic energy when the particle is outside ofthe well (b= x a+(b=)) as a function of time t and ofthe number of iterations n. Note that n and t are not proportional because a fast particle is described with more iterations that a slow one in the same interval oftime. The average is evaluated in an ensemble ofparticles with the same initial energy 0 =1:1. We nd that the average kinetic energy grows as k n, with 1=, for a broad range of

11 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 7. The average kinetic energy k as a function of (a) the iteration number n and (b) time t for the stochastic model with 1 =0:5, and = 3 =16:45. The initial energy is 0 =1:1 and 500 samples are considered in the average. The best t gives us =0:5 and =0:66. parameter values. In Fig. 7(a), it is shown a plot of k n for 1 =0:5 and = 3 =16:45. The average was evaluated over 500 samples and the map was iterated 10 5 times. We obtain =0:50 ± 0:01 with a very good t. We also nd that k t, with =3. This result seems not depend ofthe parameter values and it implies that a particle, even with very low initial energy, can have unlimited gain ofenergy. A plot ofk t is shown in Fig. 7(b). Again we consider 500 samples and 10 5 iterations ofthe map. The best t furnishes =0:66 ± 0:01. The unbounded grow of k does not happen in the periodic case. In this case, we do the average in the initial phase ofoscillation 0. We consider 0 equidistributed in the interval [0; ]. For an ensemble ofparticles with low initial energy ( 0 =1:1), the average kinetic energy reaches an steady value, after a very brief transient, for both dependent variables n and t. Let us now discuss the reection number distribution P n for a long orbit in the stochastic case. Consider a particle outside the well with energy 0 1. If , the particle passes through the well without any reection. When ,we have two cases for the particle entering the well depending on the kinetic energy inside the well k 1 = 0 g 1 1, where g 1 = g(t 1 ) is a random number evaluated at the time that the particle enters the well. If k , the particle can not be reected. Otherwise, the particle has a probability ofbe reected by the opposite wall, which can be dened as follows. When the particle hits the opposite wall, its total energy is given by 1 = k 1 + g 1, with g being a random number between [ 1; 1]. The total

12 19 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) energy 1 varies from k 1 1 up to k Since the particle has a collision only if 1 1, the probability p 1 that the particle be reected is given by the fraction of the total energy allowing a collision, namely 1 (k 1 1 ) p 1 = k (k 1 1 ) = g 1 1 : (5) 1 The kinetic energy is constant while the particle is inside the well suering successive reections. Then, each one ofthese collisions occurs with the same probability. Moreover, these collisions are statistically independents, implying that the probability f 1 (n) of n successive reections is given by f 1 (n)=p n 1(1 p 1 ) : After n (n =0; 1; :::) collisions the particle leaves the well. Then, it can pass through the well sometimes (the probability ofreection is zero in all these passages) and eventually it will enter the well for the second time with a non-zero probability p ofbeing again reected in the walls ofthe well. Dening N as the number oftimes that the particle has entered the well with non-zero probability ofbeing reected, a set ofsuccessive reections can be characterized by probabilities p 1 ;p ;:::;p N and by f 1 (n 1 );:::;f N (n N ). Ifthe elements of{f j } are statistically independents, the frequency of n successives reections F(N; n) inn enterings can be written as N N F(N; n)=n f j (n)+(n 1)(1 f 1 (n)) f j (n) j=1 N f N (n) (1 f j (n)) = j=1 j= N f j (n) : Since p j depends explicitly on the value ofrandom variable g j, p j should be considered as a random variable with a distribution j (p). Therefore, the distribution of the reection number P n (N ) can be dened as the average of F(N; n), namely P n (N)= N j=1 j=1 (p n j p n+1 j ) : (6) Here, :::, means average with the distribution j (p). Two limits cases can be easily solved. First consider that all p j are equal and constant ( j (p)=(p p )). Then, we nd that P n decays exponentially as P n (N)=N(1 p ) exp[n ln(p )] : In the second case, we consider that p j is equally distributed in the interval [0; 1] for any j. Using that p n =(n +1) 1 we obtain that N P n (N)= n +n + ; implying that asymptotically P n n.

13 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 8. Discrete distribution j (j =5; 10 and 0) as a function of the probability p ofthe particle be reected by the walls ofthe well in the stochastic model. Here we have 1 =0:5 and = 3 =16:45. In the average, 10 6 samples with the same initial energy 0 =1:01 are considered. A very good t give us that j =0:00(1 p) for any j. Let us return to the original problem. We do not know j (p) and neither ifthe elements of {f j (n)} are statistically independents. However, we can use Eq. (5) as denition ofthe probability p and determine the discrete version of j (p), j from simulations. In fact, we divide the interval [0; 1] in 100 subintervals ofsize p =1=100 and evaluate frequency histograms. In Fig. 8, it is shown the discrete distribution j (j =5; 10 and 0) for 10 6 iterates ofthe map, averaged in 10 6 samples. In this case we have considered 0 =1:01, 3 = =16:45 and 1 =0:5. These histograms are equals and similar to ones obtained for other values of the parameters. In fact, only the histogram for j = 1 is dierent when the initial energy is low. In this case, the rst entering is equidistributed. When the initial energy is high, even the rst entering in the well occurs after several interates of the map and the histogram for j =1 is similar to the ones shown in Fig. 8. Noting that the continuous distribution is related to the discrete one by j (p)= j =p, we obtain a normalized continuous distribution, namely j (p) = (1 p) : This distribution is valid for any j and any initial energy, except when j = 1 and the initial energy is close to 1. Then, we can evaluate p n as p n ==(n +3n + ). Using Eq. (6) we obtain that the reection number distribution is given by 4N P n (N)= n 3 +6n +11n +6 : (7)

14 194 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 9. Log log plot ofthe normalized reection number distribution P n =Pn=N for an orbit of 108 iterations ofthe stochastic map obtained by simulation and by the equation (7) oftext. The parameters are the same as in Fig. 8, but there is no sample average. The quantity Pn = P n (N )=N can be directly compared with the one obtained by simulation. Ifboth agree, then the hypothesis about the statistical independence ofthe elements of {f j } is correct. Moreover, we can anticipate that is hard to obtain the asymptotical behavior P n n 3 in direct simulations, because we must have n 6 in order to have n 3 6n. So, we must have n 30 or greater. The problem is that these kind ofsuccessive collisions happen with very low probabilities and is hard to obtain a good estimation for Pn. On the other hand, we have a good numerical precision for Pn if n 10. In Fig. 9 we can see that the simulation distribution agrees very well with the one given by Eq. (7), except for large n. The parameters have the same values as described in Fig. 8, excepting that now there is no sample average. It is worth mentioning that the rst simulation values (P1 =0:167, P =0:0659, P 3 =0:039, P4 =0:0195, P 5 =0:0116) agree well with the ones ofequation (7), namely P 1 =0:1667, P =0:0667, P 3 =0:0333, P 4 =0:0191 and P 5 =0:0119. Similar results are obtained for other values of the parameters. We also determined directly from simulation the exponents and, which characterize the reection number (P n ) and reection time (P t ) distributions for a long orbit. These distribution are shown in Figs. 10(a) and (b) for iterates ofthe map. Again we have a power tail with the same exponent :75 for both distributions. The value of is not 3 because we need a more long run. We stopped at iterates ofthe map due to the period ofour random number generator. For shorter runs, we obtain lower values for this exponents. Therefore, we have no doubt that the exponent is the one obtained from Eq. (7), namely = 3. The new result obtained from

15 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) Fig. 10. Log log plot ofthe reection (a) number and (b) time distributions for an orbit of10 10 iterations ofthe stochastic map. The initial energy is 0 =1:1. The values ofthe parameters are 1 =0:5, =16:45 and (a) 3 = 500:0 and (b) 3 =16:45. The slopes ofthe best t are =:76 ± 0:03 and =:73 ± 0:03. these simulations is that the asymptotical behavior of P t P n ( = ). is the same as the one of 5. Conclusions We studied the classical dynamics ofa particle in a box ofpotential that contains a square well with time-dependent depth. For the periodic case, the dynamics ofthis problem is given by a two-dimensional non-linear map that conserves the phase space area. For high energies, the existence ofinvariant spanning curves prevents the particle to gain innite energy. The low-energy region is chaotic and limited by a rst invariant spanning curve. The Lyapunov exponents ofthis region were obtained numerically. The plot ofthe positive Lyapunov exponent versus the parameter 3 shows an abrupt jump which is related to the destruction ofthe rst spanning curve, which allows a merging oftwo large regions ofthe phase space. After the merging, the Lyapunov exponent can be obtained by a ponderable average ofthe previous Lyapunov exponents ofthe regions below and above the rst spanning curve. The distributions ofthe reection time and reection number have a power law tail with the same exponent = 3. These distributions characterize the chaotic orbit bellow the rst invariant spanning

16 196 E.D. Leonel, J.K.L. da Silva / Physica A 33 (003) curve, because the system in this orbit eventually enter the well and remains inside having successive collisions. Moreover, this exponent seems to have the same value for several one-dimensional models. This exponent has been found for the transversal-time distribution for a time-modulated barrier [3,4] and we nd it for the Fermi accelerator [1]. This system consists ofa particle inside two walls, one xed space and the other oscillating in time. Now, the particle can have, for low kinetic energy, successive collisions with the moving wall. The distributions of(a) the number ofsuccessive collisions and (b) the time spent for the particle in theses collisions have a power law with the same exponent ( = 3). In the stochastic version ofthe problem, we found exponents describing the growth of the particle average kinetic energy with time and iteration number. The distribution of reection number was determined by analytical and simulation arguments. The exponent = 3 obtained is probably the exact one. The distribution ofthe reection time has a power law tail with exponent. It is worth mentioning, that these value ofthe exponents and is equal to the one obtained for oscillating models. Finally, let us point out that the potential proposed here can be easily quantized. Quantum results will allow a direct comparison with regions that presents chaotic and regular behavior on the classical model. We are presently working in this problem. Acknowledgements This research was supported in part by the Conselho Nacional de Desenvolvimento Cientco e Tecnologico (CNPq) and Fundacão de Amparo a Pesquisa do Estado de Minas Gerais (Fapemig), Brazilian agencies. Part ofthe numerical results were obtained in CENAPAD-MG/CO. References [1] E. Fermi, Phys. Rev. 75 (6) (1949) [] R. Douady, Applications du theoreme des tores invariants, These de 3eme Cycle, University ofparis VII, 198. [3] J.L. Mateos, J.V. Jose, Physica A 57 (1998) [4] J.L. Mateos, Phys. Lett. A 56 (1999) [5] S. Olison Kamphorst, S. Pinto de Carvalho, Nonlinearity 1 (1999) 1 9. [6] A. Loskutov, A.B. Ryabov, L.G. Akinshin, J. Phys. A 33 (000) [7] G.A. Luna-Acosta, G. Orellana-Rivadeneyra, A. Mendonza-Galvan, C. Jung, Chaos Solitons Fractals 1 () (001) [8] J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57 (1985) 617. [9] J.-P. Eckmann, S. Ollifson Kamphorst, D. Ruelle, S. Ciliberto, Phy. Rev. A 34 (1986) 4971.