Available online at www.sciencedirect.com
Physica A 323 (2003) 181 – 196
www.elsevier.com/locate/physa
Dynamical properties of a 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. 702 30123-970, Belo Horizonte, MG, Brazil
Received 26 September 2002
Abstract
We study numerically the dynamical behavior of a classical particle inside a box potential that
contains a square well which depth varies in time. Two cases of time 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 5rst spanning curve, is characterized by a positive Lyapunov exponent. This exponent and
the position of the 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 of time and of the iteration number. We also show for both cases that
the distributions of the time spent by the particle inside the well and the number of successive
re8ections have a power law tail.
c 2003 Elsevier Science B.V. All rights reserved.
PACS: 05:45: − a; 05.45.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 if a particle, in a classical system that gives/takes
energy to it, can have unlimited gain of energy? This was the fundamental question of
Fermi in his work on cosmic radiation [1]. Douady [2] showed that for the so-called
∗
Corresponding author. Fax: +55-31-499-5600.
E-mail addresses: [email protected] (E.D. Leonel), [email protected] (J.K.L. da Silva).
c 2003 Elsevier Science B.V. All rights reserved.
0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(03)00036-0
182
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
Fermi accelerator model with suHciently smooth movement of the wall, the response is
no. This response is associated to the existence of invariant 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 of invariant 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 of a single particle inside an in5nite box of potential that contains a
square well. The depth of the 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 diFerent values of the
control parameter.
The structure of the phase space, and thus the value of the Lyapunov exponent,
changes, sometimes drastically, as the control parameters vary. For stochastic oscillations, the kinetic energy of the 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 of successive re8ections in the well and (ii) the time
that the particle remains inside it. Recently, a similar distribution was studied [3,4] for
a time-modulated barrier and a power law behavior was found. Here, we 5nd the same
behavior for the periodic and the stochastic models.
This paper is organized as follows. In Section 2 we present a detailed construction of the model with periodic oscillations and obtain the associated two-dimensional
non-linear map. We present also a brief discussion 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 of the problem and discuss
the results. The conclusions and 5nal remarks are presented in Section 5.
2. The model with periodic oscillations and Lyapunov exponents
2.1. The model
We consider the classical dynamics of a particle inside a one-dimensional rigid box of
potential that contains an oscillating symmetric square well. This problem is de5ned by
the time-dependent Hamiltonian H (x; p; t)=(p2 =2m)+V (x; t), where V (x; t) is given by

∞
if x 6 0 and x ¿ (a + b) ;



if 0 ¡ x ¡ b=2 and (a + b=2) ¡ x ¡ (a + b) ;
V (x; t) = V0



D(t) if b=2 ¡ x ¡ (a + b=2) :
Here, V0 , a and b are constant. D(t) describes the time evolution of the bottom of the
well. In the periodic case we consider D(t) = D cos(!t), with D being the amplitude
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
183
Fig. 1. Sketch of the potential V (x; t) for the periodic case. V0 is constant. D is the amplitude of the bottom
of the oscillating well.
of oscillation and ! the frequency. In Section 4, we will consider D(t) as a stochastic
function of time. The sketch of the potential V (x; t) for the periodic case and the
lengths a and b are shown in Fig. 1. We always consider that D ¡ V0 .
The dynamics of the particle under the periodic oscillations of the well can be
described by a two-dimensional map, using the total energy E of the particle and the
phase of the bottom of the 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=2)
with initial energy E0 . At t = t0 when it arrives at x = b=2 its kinetic energy changes
from the initial value K0 = E0 − V0 to K ∗ = E0 − D cos(!t0 ). 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 = (2K )=m. After t ∗ the particle reaches the other side of the well. Its total energy
is then E1 = K ∗ + D cos(!t0 + !t ∗ ). If we de5ne 0 = !t0 as the initial phase and
1 =0 +K0 as the phase at t=t0 +t ∗ , then
we have that E1 =E0 +D[cos(1 )−cos(0 )].
The phase change is given by K0 = !a= m2 [E0 − D cos(!t0 )].
It is useful to distinguish two cases for the particle:
Case 1: If E1 ¡ V0 , then the particle does not exit the well.
Case 2: If E1 ¿ V0 , then the particle exits the well.
In the 5rst case, the particle is re8ected with the same kinetic energy and arrives
at the other side of the well after traveling a time t ∗ . The total energy is E2 = E0 +
D[cos(2 )−cos(0 )] where 2 =0 +2K0 . Again, if E2 ¡ V0 , the particle is re8ected
to the other side of the well and so on until the condition En = E0 + D[cos(n ) −
cos(0 )] ¿ V0 is ful5lled. Here, we have that n = 0 + nK0 . If En = V0 we have a
marginal situation in which the particle stays at the wall of the well with K = 0.
184
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
When the particle exits the well (case 2), it travels the distance b=2 with kinetic
energy K = En − V0 . Then it is elastically re8ected and keeps the same kinetic energy
until it arrives
at the entrance of the oscillating well. The time spent in this trajectory
is t = b= 2=m(En − V0 ). Since the total energy is constant outside the well, the particle
enters the well with energy E1 = En and phase 1 = n + !b= 2=m(En − V0 ). So the
map can be written as
E1 = E0 + D[cos(0 + nKa ) − cos(0 )] ;
1 = 0 + nKa + Kb; n mod(2) :
Here, n is the smallest integer such that
E0 + D cos(0 + nKa ) − D cos(0 ) ¿ V0
and the auxiliary variables are given by
!a
;
Ka = 2=m(E0 − D cos(0 ))
Kb; n = !b
:
2=m E0 + D cos(0 + nKa ) − D cos(0 ) − V0
After the particle enters the well, the same procedure is repeated to obtain the pair
(E2 ; 2 ) and so on.
It is convenient to work with dimensionless quantities de5ned as
m
m
D
Ek
1 = ; 2 =
!a; 3 =
!b and k =
:
V0
2V0
2V0
V0
With these de5nitions, 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 + nKa ) − cos(k )] ;
(1)
k+1 = k + nKa + Kb; n mod(2) ;
(2)
where n is the smallest integer number obtained from
k + 1 [cos(k + nKa ) − cos(k )] ¿ 1 :
(3)
Here Ka and Kb; n are de5ned as
2
;
Ka = k − 1 cos(k )
Kb; n = 3
k + 1 [cos(k + nKa ) − cos(k )] − 1
:
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
185
The coeHcients of the Jacobian matrix J are given by
3
Ka
9k+1
n1 2
sin(k + nKa )
;
=1+
j11 =
9k
2
2
j12 =
9k+1
9k
3
n21 2
Ka
;
= 1 sin(k )−1 sin(k + nKa )+
sin(k ) sin(k +nKa )
2
2
j21 =
9k+1
9k
3 3
3 n1 2
Ka
n2 Ka
3 Kb; n
sin(k + nKa )
;
1+
=−
−
2
2
3
2
2
2
j22 =
9k+1
9k
3
3
Kb; n
Ka
1 3
n1 2
sin(k )
−
sin(k )
2
2
3
2
3
3 1 3 Kb; n
n1 2
Ka
:
+
sin(k + nKa ) 1 −
sin(k )
2
3
2
2
=1−
The area of the phase space is conserved since det(J ) = 1.
2.2. Lyapunov exponents
Let us now describe brie8y the procedure used to evaluate the Lyapunov exponents.
These exponents are de5ned by [8,9]
n
1
j = lim
ln |kj |; j = 1; 2 ;
n→∞
n
k=1
n
k
where j are the eigenvalues of M = k=1 Jk (k ; k ), and Jk 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 of an orthogonal, , and a triangular, T , matrix. Let us de5ne the elements
of these matrices by
cos( ) −sin( )
T11 T12
:
; T=
=
sin( )
cos( )
0
T22
−1
Since M =Jn Jn−1 : : : J2 J1 , we can write that M =Jn Jn−1 : : : J2 1 −1
1 J1 . Here 1 J1 =T1 .
A product of J2 1 de5nes a new J˜ 2 . In a following step, M is written as M =
−1 ˜
˜
Jn Jn−1 : : : J3 2 −1
2 J 2 T1 . The same procedure is used to obtain T2 = 2 J 2 . With this
186
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
i
i
.
procedure, the problem is reduced to evaluate the diagonal elements of Ti : T11
; T22
Using the and T matrices we 5nd the eigenvalues of M , namely
j2 + j2
T11 = 112 212 ;
j21 + j11
j11 j22 − j12 j21
:
T22 = 2
2
j21 + j11
Then, we 5nd that the Lyapunov exponents are given by
n
1
j = lim
ln |Tjjk |; j = 1; 2 :
n→∞
n
k=1
Note that 1 = −2 because the map is area-preserving.
3. Numerical results of the periodic model
The map given by Eqs. (1) and (2) can be numerically iterated for a 5xed choice
of the parameters (i ; i = 1; : : : ; 3) and diFerent initial conditions (0 ; 0 ), and so the
overall structure of the phase space and its characterization can be investigated.
A typical picture of the phase space is shown in Fig. 2. Here we choose 1 = 0:5,
2 = 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 diFerent not
connected regions, implying that a particle can not have an in5nite gain of energy.
In the simulations we consider three diFerent values of the parameter 1 (1 =
0:25; 1 = 0:5 and 1 = 0:75). This parameter must be less then 1 since the bottom
can never reach the top of the oscillating well. The second parameter can be expressed
as 2 = 2(!=!c ). Here, !c is related to the time spent for a particle with kinetic
energy V0 to travel the distance a in the absence of any oscillation. In order to have
a well with a moderate frequency of oscillation (! ≈ 2:6181 : : : !c ), we assume that
Fig. 2. Phase space for 1 = 0:5, 2 = 16:45 and 3 = 500:0.
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
187
Fig. 3. Numerical convergence of the positive Lyapunov exponent for (a) double precision and (b) quadruple
precision.
2 = 16:45. The last parameter can be written as 3 = 2 (b=a). For 2 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. 2. Each one of the 5ve
diFerent initial conditions was iterated 5 × 108 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 diFerent numerical precisions, the remainder
calculations were made using double precision in order to save computational time.
A plot of the positive Lyapunov exponent versus the control parameter 3 for each
5xed pair (1 ; 2 ) is shown in Figs. 4(a) – (c). Each point in Fig. 4 was obtained by
averaging over 5ve diFerent initial conditions. An orbit is obtained by 5×108 iterations
of the map. The error bar is the standard deviation of the 5ve samples. Note that an
abrupt and sudden jump in the Lyapunov exponent behavior occurs. For 1 = 0:25 and
2 = 16:45 the jump occurs around 3 ≈ 952 as shown in Fig. 4(a). When 1 = 0:5
and 2 =16:45 the jump occurs around 3 ≈ 220 (see Fig. 4(b)). In Fig. 4(c) (1 =0:75 and
2 = 16:45) we can see that the jump occurs around 3 ≈ 135. These abrupt jumps are
associated with the sudden break oF of the 5rst spanning curve that allows a merging of
two diFerent regions of the phase space. This change in size characterizes the decrease
of the Lyapunov exponent.
188
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
Fig. 4. Linear-log plot of × 3 with 2 = 16:45, (a) 1 = 0:25, (b) 1 = 0:5 and (c) 1 = 0:75. Each point
was obtained by averaging over 5 diFerent initial conditions, each one was iterated 5 × 108 times. The error
bars is the standard deviation of the samples.
In order to investigate the abrupt change in the size of the low-energy region, we
iterate a same initial condition for two diFerent 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 = 0:25, 2 = 16:45 and (a) 3 = 951, (b) 3 = 952. 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 2 . Note that after the destruction of the 5rst
spanning curve, two regions with diFerent Lyapunov exponents merge together. After
the merging, the positive Lyapunov exponent can be obtained by an average of the
previous exponents, pondering by the relative size of the corresponding region of the
phase space. To illustrate this conjecture, we need to estimate the fraction occupied by
each region of the phase space. It is easy to see that the region I in Fig. 5(a) occupies
15% of the phase space, while region II occupies 85%. The Lyapunov exponents of
the two regions are = 2:940 ± 0:003 and 0:519 ± 0:001, respectively. Therefore, the
estimation of the Lyapunov exponent after the merging is =0:882±0:001, which is in
agreement with the observed value (see Fig. 4(a)). This naive estimation works well for
other transitions of Fig. 4. The initial oscillations in the Lyapunov exponent before these
transitions are associated to small 8uctuations in the size and form of the phase space.
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
189
Fig. 5. (a) Iteration of 3 diFerent initial conditions bellow, above and inside the spanning invariant curve
with parameters 1 = 0:25, 2 = 16:45 and 3 = 951:0. (b) Iteration of one initial condition with 1 = 0:25,
2 = 16:45 and 3 = 952:0.
Other two quantities of interest are the re8ection time and re8ection number that
the particle has inside the well. The re8ection time is de5ned as the time spent by
the particle inside
the well due to successive collisions with the walls. It is given by
t = (n − 1)tc = k − 1 cos(k ). Here, tc is the time spent by the particle to travel the
distance a in the absence of oscillations with kinetic energy K =V0 . Note that (n−1) is
the number of re8ections of the particle. When n=1, the number of successive collisions
with the walls of the well is 0 and the particle is transmitted. If the particle has total
energy k ¿ 1+21 at the entrance of the well, it will be transmitted without re8ections
inside the well. The re8ection time (Pt ) and re8ection number (Pn ) distributions are
shown, respectively, in Figs. 6(a) and (b). The data in a log–log plot indicate a power
law behavior: Pn ∼ n$ , Pt ∼ t % . A good 5t 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 5rst invariant spanning curve.
4. The stochastic model
Let us consider now D(t) as a stochastic function of time. We de5ne D(t) = Dg(t),
with g(t) taking random values equidistributed in the interval [ − 1; 1]. The variables
of the non-linear map are the energy and time. It is easy to obtain the equations of
190
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
Fig. 6. Log–log plot of the re8ection (a) number and (b) time distributions. Here 1 = 0:5, 2 = 16:45 and
3 = 500:0. The best 5t gives us the exponents $ = % ≈ − 3.
the map, namely
k+1 = k + 1 [g(tk + nKta ) − g(tk )] ;
tk+1 = tk + nKta + Ktb :
Now, n is the smallest integer number obtained from
k + 1 [g(tk + nKta ) − g(tk )] ¿ 1 ;
(4)
and the auxiliary variables are given by
2
;
Kta = k − 1 g(tk )
Ktb = 3
k + 1 [g(tk + nKta ) − g(tk )] − 1
:
The kinetic energy of the particle is always changing due to the interaction with the
time-dependent potential. We consider the average kinetic energy when the particle is
outside of the well (b=2 ¿ x ¿ a + (b=2)) as a function of time t and of the 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 of time. The average is
evaluated in an ensemble of particles with the same initial energy 0 = 1:1. We 5nd
that the average kinetic energy grows as k ∼ n' , with ' ≈ 1=2, for a broad range of
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
191
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 2 = 3 = 16:45. The initial energy is 0 = 1:1 and 500 samples are
considered in the average. The best 5t 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 2 =3 =16:45.
The average was evaluated over 500 samples and the map was iterated 105 times. We
obtain ' = 0:50 ± 0:01 with a very good 5t. We also 5nd that k ∼ t ( , with ( ≈ 2=3.
This result seems not depend of the parameter values and it implies that a particle,
even with very low initial energy, can have unlimited gain of energy. A plot of k × t
is shown in Fig. 7(b). Again we consider 500 samples and 105 iterations of the map.
The best 5t 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 of oscillation
0 . We consider 0 equidistributed in the interval [0; 2]. For an ensemble of particles
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 re8ection number distribution Pn for a long orbit in the
stochastic case. Consider a particle outside the well with energy 0 ¿ 1. If 0 ¿ 1+21 ,
the particle passes through the well without any re8ection. When 1 ¡ 0 ¡ 1 + 21 , we
have two cases for the particle entering the well depending on the kinetic energy
inside the well k1 = 0 − g1 1 , where g1 = g(t1 ) is a random number evaluated at the
time that the particle enters the well. If k1 ¿ 1 + 1 , the particle can not be re8ected.
Otherwise, the particle has a probability of be re8ected by the opposite wall, which
can be de5ned as follows. When the particle hits the opposite wall, its total energy is
given by 1 = k1 + g2 1 , with g2 being a random number between [ − 1; 1]. The total
192
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
energy 1 varies from k1 − 1 up to k1 + 1 . Since the particle has a collision only if
1 ¡ 1, the probability p1 that the particle be re8ected is given by the fraction of the
total energy allowing a collision, namely
p1 =
1 − (k1 − 1 )
1 + 1 − 1 + g1 1
:
=
k1 + 1 − (k1 − 1 )
21
(5)
The kinetic energy is constant while the particle is inside the well suFering successive
re8ections. Then, each one of these collisions occurs with the same probability. Moreover, these collisions are statistically independents, implying that the probability f1 (n)
of n successive re8ections is given by
f1 (n) = p1n (1 − p1 ) :
After n (n = 0; 1; 2 : : :) collisions the particle leaves the well. Then, it can pass through
the well sometimes (the probability of re8ection is zero in all these passages) and
eventually it will enter the well for the second time with a non-zero probability p2
of being again re8ected in the walls of the well. De5ning N as the number of times
that the particle has entered the well with non-zero probability of being re8ected, a
set of successive re8ections can be characterized by probabilities p1 ; p2 ; : : : ; pN and by
f1 (n1 ); : : : ; fN (nN ). If the elements of {fj } are statistically independents, the frequency
of n successives re8ections F(N; n) in N enterings can be written as
F(N; n) = N
N
fj (n) + (N − 1)(1 − f1 (n))
j=1
+ · · · + fN (n)
N
fj (n)
j=2
N
−1
(1 − fj (n)) =
j=1
N
fj (n) :
j=1
Since pj depends explicitly on the value of random variable gj , pj should be considered
as a random variable with a distribution ,j (p). Therefore, the distribution of the
re8ection number Pn (N ) can be de5ned as the average of F(N; n), namely
Pn (N ) =
N
(pjn − pjn+1 ) :
(6)
j=1
Here, : : :, means average with the distribution ,j (p).
Two limits cases can be easily solved. First consider that all pj are equal and
constant (,j (p) = ((p − p∗ )). Then, we 5nd that Pn decays exponentially as
Pn (N ) = N (1 − p∗ ) exp[n ln(p∗ )] :
In the second case, we consider that pj is equally distributed in the interval [0; 1] for
any j. Using that pn = (n + 1)−1 we obtain that
N
Pn (N ) = 2
;
n + 2n + 2
implying that asymptotically Pn ∼ n−2 .
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
193
Fig. 8. Discrete distribution -j (j = 5; 10 and 20) as a function of the probability p of the particle be
re8ected by the walls of the well in the stochastic model. Here we have 1 = 0:25 and 2 = 3 = 16:45. In
the average, 106 samples with the same initial energy 0 = 1:01 are considered. A very good 5t give us that
-j = 0:020(1 − p) for any j.
Let us return to the original problem. We do not know ,j (p) and neither if the
elements of {fj (n)} are statistically independents. However, we can use Eq. (5) as
de5nition of the probability p and determine the discrete version of ,j (p), -j from
simulations. In fact, we divide the interval [0; 1] in 100 subintervals of size (p = 1=100
and evaluate frequency histograms. In Fig. 8, it is shown the discrete distribution
-j (j = 5; 10 and 20) for 106 iterates of the map, averaged in 106 samples. In this
case we have considered 0 = 1:01, 3 = 2 = 16:45 and 1 = 0:25. These histograms are
equals and similar to ones obtained for other values of the parameters. In fact, only
the histogram for j = 1 is diFerent when the initial energy is low. In this case, the
5rst entering is equidistributed. When the initial energy is high, even the 5rst 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) = 2(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 pn as pn = 2=(n2 + 3n + 2).
Using Eq. (6) we obtain that the re8ection number distribution is given by
4N
:
(7)
Pn (N ) = 3
n + 6n2 + 11n + 6
194
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
Fig. 9. Log–log plot of the normalized re8ection number distribution Pn∗ = Pn =N for an orbit of 108 iterations
of the stochastic map obtained by simulation and by the equation (7) of text. The parameters are the same
as in Fig. 8, but there is no sample average.
The quantity Pn∗ = Pn (N )=N can be directly compared with the one obtained by simulation. If both agree, then the hypothesis about the statistical independence of the
elements of {fj } is correct. Moreover, we can anticipate that is hard to obtain the
asymptotical behavior Pn ∼ n−3 in direct simulations, because we must have n6 in
order to have n3 6n2 . So, we must have n ∼ 30 or greater. The problem is that these
kind of successive 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 5rst simulation values (P1∗ = 0:1672, P2∗ = 0:0659, P3∗ = 0:0329,
P4∗ =0:0195, P5∗ =0:0116) agree well with the ones of equation (7), namely P1∗ =0:1667,
P2∗ = 0:0667, P3∗ = 0:0333, P4∗ = 0:0191 and P5∗ = 0:0119. Similar results are obtained
for other values of the parameters.
We also determined directly from simulation the exponents $ and %, which characterize the re8ection number (Pn ) and re8ection time (Pt ) distributions for a long orbit.
These distribution are shown in Figs. 10(a) and (b) for 1010 iterates of the map. Again
we have a power tail with the same exponent $ ≈ % ≈ −2:75 for both distributions. The
value of $ is not −3 because we need a more long run. We stopped at 1010 iterates
of the map due to the period of our 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
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
195
Fig. 10. Log–log plot of the re8ection (a) number and (b) time distributions for an orbit of 1010 iterations
of the stochastic map. The initial energy is 0 = 1:1. The values of the parameters are 1 = 0:5, 2 = 16:45
and (a) 3 = 500:0 and (b)3 = 16:45. The slopes of the best 5t are $ = 2:76 ± 0:03 and % = 2:73 ± 0:03.
these simulations is that the asymptotical behavior of Pt is the same as the one of
Pn (% = $).
5. Conclusions
We studied the classical dynamics of a particle in a box of potential that contains
a square well with time-dependent depth. For the periodic case, the dynamics of this
problem is given by a two-dimensional non-linear map that conserves the phase space
area. For high energies, the existence of invariant spanning curves prevents the particle
to gain in5nite energy. The low-energy region is chaotic and limited by a 5rst invariant
spanning curve. The Lyapunov exponents of this region were obtained numerically. The
plot of the positive Lyapunov exponent versus the parameter 3 shows an abrupt jump
which is related to the destruction of the 5rst spanning curve, which allows a merging
of two large regions of the phase space. After the merging, the Lyapunov exponent
can be obtained by a ponderable average of the previous Lyapunov exponents of the
regions below and above the 5rst spanning curve. The distributions of the re8ection
time and re8ection number have a power law tail with the same exponent $ = % ≈ − 3.
These distributions characterize the chaotic orbit bellow the 5rst invariant spanning
196
E.D. Leonel, J.K.L. da Silva / Physica A 323 (2003) 181 – 196
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 5nd it for the Fermi accelerator
[1]. This system consists of a particle inside two walls, one 5xed 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 of successive
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 of the problem, we found exponents describing the growth of
the particle average kinetic energy with time and iteration number. The distribution of
re8ection number was determined by analytical and simulation arguments. The exponent
$ = −3 obtained is probably the exact one. The distribution of the re8ection time has
a power law tail with exponent $ ≈ %. It is worth mentioning, that these value of the
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
CientTU5co e TecnolTogico (CNPq) and FundaVcão de Amparo aW Pesquisa do Estado de
Minas Gerais (Fapemig), Brazilian agencies. Part of the numerical results were obtained
in CENAPAD-MG/CO.
References
[1] E. Fermi, Phys. Rev. 75 (6) (1949) 1169–1174.
[2] R. Douady, Applications du thTeorWeme des tores invariants, ThWese de 3Weme Cycle, University of Paris
VII, 1982.
[3] J.L. Mateos, J.V. JosTe, Physica A 257 (1998) 434–438.
[4] J.L. Mateos, Phys. Lett. A 256 (1999) 113–121.
[5] S. OliFson Kamphorst, S. Pinto de Carvalho, Nonlinearity 12 (1999) 1–9.
[6] A. Loskutov, A.B. Ryabov, L.G. Akinshin, J. Phys. A 33 (2000) 7973–7986.
[7] G.A. Luna-Acosta, G. Orellana-Rivadeneyra, A. Mendonza-GalvTan, C. Jung, Chaos Solitons Fractals 12
(2) (2001) 349–363.
[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.