Competition between suppression and production of Fermi acceleration
Denis Gouvêa Ladeira1and Edson D. Leonel2 1
Campus Alto Paraopeba, Universidade Federal de São João Del-Rei, Fazenda do Cadete, CEP 36420-000, Ouro Branco, MG, Brazil 2
Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Av. 24A, 1515, Bela Vista, CEP 13506-700, Rio Claro, SP, Brazil
共Received 2 September 2009; revised manuscript received 20 January 2010; published 23 March 2010兲
The behavior of the average velocity for a classical particle in the one-dimensional Fermi accelerator model under sawtooth external force is considered. For elastic collisions, it is known that the average velocity of the particle grows unlimitedly because of the discontinuities of the derivative of the moving wall’s position with respect to time. However, and contrary to what was expected to be observed, the introduction of a friction force generated from a slip of a body against a rough surface leads to aboundaryseparating different regions of the phase space that yields the particle to either experience unlimited energy growth or suppression of Fermi acceleration. The Fermi acceleration is described by using scaling arguments. The formalism presented can be extended to two-dimensional time-dependent billiards as well as to higher-order mappings.
DOI:10.1103/PhysRevE.81.036216 PACS number共s兲: 05.45.Pq
Since the discovery of the cosmic rays in the last century
关1兴a big effort has been employed on the study and under-standing of this kind of radiation. However, the physical mechanism of the acceleration process of the high-energy cosmic rays is still to be understood. The so called cut-off energy共the upper limit on the energy that cosmic ray retains兲 is another phenomena of recent investigation 关2–5兴. More-over, it is unknown whether the cutoff occurs due to propa-gation effects or due to physical limitations of the accelera-tion process 关6兴. All these facts suggest that the subject deserves further investigation and therefore many models have been proposed and studied along last decades.
In 1949, Fermi 关7兴 launched the idea that a possible mechanism was related to scattering of a cosmic particle with a time moving magnetic field. His original idea was later investigated generating many mathematical models and producing a wide class of interesting results with applicabil-ity in tokamaks关8,9兴, waveguide关10,11兴, time-dependent os-cillating square wells 关12,13兴, chaotic dynamics and phase transition关14兴, and many other.
One of the mathematical models 关15兴 considers that the cosmic ray is replaced by a classical particle and that the time varying magnetic field is described by an infinitely heavy and periodically moving wall. The returning mecha-nism of the particle for a next collision with the moving wall is made by a fixed and rigid wall. Indeed, if the particle travels freely between the two walls and the position of the moving wall is given by xw=⑀cos共t+0兲, where ⑀ is the amplitude of oscillation, is the frequency of oscillation, and0is an initial phase, it is known that the phase space of this model presents a mixed structure. Therefore it is ob-served fixed points involved by Kolmogorov-Arnold-Moser
共KAM兲 islands, regions of nondissipative chaotic motion
共chaotic seas兲 and invariant spanning curves共also called as invariant tori兲. The first chaotic region is limited by the lowest-energy spanning curve. The presence of the spanning curves limits the values of velocity and hence the energy of the particle is bounded, consequently no Fermi acceleration
共FA兲 is observed关15,16兴. The phenomenon of FA is a pro-cess in which a classical particle acquires unbounded energy from collisions with a heavy moving wall.
The mixed structure of the phase space is totally de-stroyed if dissipation is introduced. If the particle experi-ences a fractional loss of energy upon collision with the walls, the invariant spanning curves are destroyed. The ellip-tic fixed points turn into sinks. Depending on the control parameters, boundary crisis can be characterized via crossing of stable and unstable manifolds. For the limit of high dissi-pation, it is possible to observe bifurcation cascades. For the dissipative dynamics, attractors are created in the phase space. Hence FA is not observed. If one consider the motion of the moving wall to be random, the nondissipative dynam-ics yields in the unlimited energy growth关17兴. However, the introduction of inelastic collisions is a sufficient condition to suppress FA关18–20兴. Other kind of dissipation assumes that the particle is moving in the presence of a drag force such as gas. Such dissipation acts along the full trajectory of the particle and is contrary to the inelastic collisions which act only in the instant of the impact. For the case of damping force proportional to the particle’s velocity, the phenomenon of FA is suppressed 关21兴.
In this work we revisit the dynamics of a classical particle in the one-dimensional Fermi accelerator model under an external force of sawtooth type seeking to understand and describe a competition between FA and suppression of FA. The choice of such an external perturbation of sawtooth type is because the oscillating wall always furnishes energy to the particle after each collision. Thus in the absence of any dis-sipation, FA is feasible. The main question to be answered is: does the FA remains observed under the presence of a fric-tion force generated from a slip of a body against a rough surface? The answer is not so simple and we will show it depends on the initial conditions as well as on the control parameters. Thus, our main goal in this paper is to investi-gate the consequences of dissipation due to friction in the FA process. Our results can be extendible to many other models including a wide class of two-dimensional time-dependent billiards.
regard the second situation, where the wall moves according to the expressionxw共t兲=共⑀t
⬘
+X0兲mod共2⑀兲−⑀. The symbol⑀represents the amplitude of motion, is the frequency of oscillation,t
⬘
is the time, andX0苸关0 , 2⑀兲defines the initialposition of the wallxw共t
⬘
= 0兲. According to this expression, the moving wall oscillates between the positions x= −⑀ and x=⑀. The fixed wall is at positionx=l. In terms of dimen-sionless variables we have thatX=x/landV=v/共l兲furnishthe position and velocity of both particle and wall. In addi-tion=⑀/landt=t
⬘
are the amplitude of oscillation of the moving wall and time. In terms of this set of variables the velocity of the wall is fixed as Vw= and we have that0=X0/l苸关0 , 2兲.
If we take into account nondissipative dynamics, the ve-locity and hence the kinetic energy of the particle increases at each collision with the moving wall. Therefore, the mov-ing wall always gives energy to the particle and such system represents a good prototype to produce FA. The velocity of the particle after each collision is given by
Vn+1=Vn+ 2=V0+共n+ 1兲, nⱖ0. 共1兲 On the other hand, if dissipation is introduced, the dynam-ics of the particle has a profound modification. In particular, the results might reveal presence or lack of FA, depending on the intensity of dissipation as well as on the amplitude of oscillation and also on the initial conditions. The dissipation we are considering occurs when a body slips on a rough surface. This kind of dissipation acts in a body throughout its entire trajectory. Thus, while the particle moves between the walls, a constant force F=⫾mg acts on the particle then decreasing its energy. Heremis the particle’s mass, gis the gravitational acceleration, andis the kinetic friction coef-ficient. The force F is contrary to the particle motion. The plus sign corresponds to the situation where the particle moves from the right to the left and the minus sign when it moves backward. From second Newton’s law of motion, it is easy to find the position of the particle in time as X共t兲=X共t0兲+V共t0兲t⫾
1 2␣t
2
, whereXand␣=g/共2l兲are the dimensionless position and acceleration of the particle, re-spectively. For␣= 0 all results for the nondissipative case are obtained.
Using dimensionless variables, the motion of the particle is described by a two-dimensional map of the type
T:
冦
Vn+1 =
冑
Vn 2+ 2␣共Xn+Xn+1兲− 4␣+ 2
tn+1 = tn+ 1
␣共Vn−
冑
Vn2
− 4␣兲,
冧
共2兲whereXn+1is the position of particle, and hence of the mov-ing wall, at the instant of collision共n+ 1兲th.
Since we are looking at conditions to suppress or not FA, the natural observable to be characterized is the average ve-locity. It is shown in Fig.1 the behavior ofV¯⫻n for V0= 1,
␣= 9⫻10−4, and different values of. The average velocity is obtained as V¯= 1/共n+ 1兲兺
i=0 n
Vi. A striking result that emerges from Fig. 1 is that FA might either be observed or not for␣⫽0. The curves ofV¯ depend on the control
param-eters and initial velocities. In Fig. 1共a兲it is clear the
depen-dence on the control parameter. The dependepen-dence on the ini-tial velocity will be discussed latter. Basically, we observe that the average velocity stays almost constant for a long time and then it might rises or decreases. If the velocity decreases the particle reaches the rest. On the other hand, if FA is observed, after a characteristic changeover from a con-stant velocity to a regime of growth, the curve of V¯ grows with a law of the type V¯⬀n. We see that different control parameters generate different curves. However, a rescaling in the horizontal axis is sufficient to merge all the curves onto a single curve, as it is shown in Fig.1共b兲.
Let us discuss the dependence of the FA on the initial velocity. It is expected that there must exist some relation between the initial velocity, V0, and parameters and ␣, which characterizes the transition from unlimited energy gain to completely energy dissipation. We now describe this transition.
The limit situation occurs when the amount of dissipated energy is exactly furnished to the particle by the moving wall. In this case the velocities after each collision satisfy the relation Vn+1=Vn=Vc. Because Xn苸关−,兲 we have that 2␣共Xn+Xn+1兲苸关−4␣, 4␣兲. Therefore, regarding the veloc-ity expression in map 共2兲 and considering the extreme pos-sibilities of 2␣共Xn+Xn+1兲, we obtain the velocitiesV−andV+
V−= 2
+␣共1 −兲
, V+=
2+␣共1 +兲
. 共3兲
In this sense we observe that for V0⬍V−the energy fur-nished by the moving wall to the particle at each collision is smaller than the energy dissipated across the trajectory. Therefore after some time the particle losses all its energy, stopping between the two walls.
100 102 104 106 108
n
10−2 100 102 104 106
V
α=9×10−4,ε=10−2
α=9×10−4,ε=5×10−3
α=9×10−4,ε=2×10−3
α=9×10−4,ε=10−4
10−3 10−1 101 103 105
nε 100
102 104 106
V
(a)
(b)
FIG. 1. 共Color online兲 共a兲 Average velocity curves forV0= 1, ␣= 9⫻10−4, and different values of⑀, as labeled in the figure.共b兲
On the other hand, forV0⬎V+ the amount of energy fur-nished by the wall at each collision with the particle is larger than the energy dissipated due to the friction between two impacts with the moving wall. Therefore, the system presents a net increase in particle’s velocity after each collision, re-sulting in FA.
For the intermediate situation where the initial velocity of the particle lies in V−⬍V0⬍V+, the energy of the particle might grows without limits or not. Basically it depends on the combination of initial conditions 0 and V0. Figure 2 allows us to understand the general behavior of the velocity curves for this situation. We used = 10−3, ␣= 10−3, and defined a grid of 600⫻600 initial conditions with V−⬍V0⬍V+and 0ⱕ0⬍2. For each initial condition, map
共2兲was iterated and the asymptotic behavior of the trajectory of the particle was analyzed. If the velocity of the particle becomes greater thanV+, then the particle presents FA. If the velocity becomes smaller thanV−then the particle loses all its energy. In this way the white color in Fig. 2共a兲 corre-sponds to the initial conditions for which FA occurs. The blue共dark兲color in Fig.2共a兲represents the initial conditions for which the iteration of map 共2兲 results in total energy dissipation. Hence the particle stops between the walls after colliding against the moving wall nmax times. The value of maximum collisions numbernmaxdepends on the initial con-ditions. The color共gray兲patterns used in Fig.2共b兲illustrates the maximum collision numbernmaxbefore the particle’s ki-netic energy is completely absorbed. The black line in Fig.
2共b兲corresponds to the border that separates the suppression from the FA phenomena and corresponds to the numerical approximation ofVcfor those parameters values.
We now discuss a scaling present for the average velocity as function of the control parameters and initial conditions, then leading to FA for the limit ofV0ⰇV+. First it is impor-tant to understand the dependence ofV¯ as a function of vari-ablen, parameters,␣, and initial velocityV0. We initially kept fixed = 10−3,V0= 10 and performed some simulations of V¯⫻nfor different values of ␣as shown in Fig.3共a兲. We note that different values of ␣ seem not to affect the V¯ curves. Therefore␣is not considered in the scaling analysis. Figure3共b兲shows four curves ofV¯ for␣= 10−5and different values of andV0. For small values ofnwe observe thatV¯
is constant and does not depend on. Moreover, in this ini-tial plateau, V¯⬇V0. For large values of n,V¯ depends on n and according toV¯⬀ng共兲, where is the growth
expo-nent and g is a function of. The changeover from the re-gime of constant value to the growth rere-gime of V¯ is marked by a characteristic crossovernxgiven bynx⬀ z1V0
z2, wherez 1 andz2 are dynamical exponents.
The exponent  is obtained via a nonlinear fit in the growth regime. The average value we obtained for 55 V¯ curves with 苸关10−5, 10−2兴 and V0苸关100, 9⫻102兴 is
= 0.9972共7兲. Figure 3共c兲 shows a log-log plot of g共兲=V¯/n as function of. Performing power-law fittings to the numerical data we obtaing共兲⬀ with= 0.969共5兲. In order to obtain the exponentsz1andz2we performed two sets of simulations. Figure 3共d兲 shows the crossover nx for different values of with fixedV0= 10. Applying power law fittings to the numerical data we obtainz1= −0.991共1兲. Simi-larly, we performed a set of simulations with fixed = 10−3 and different values of V0. Figure 3共e兲 displays the results. The best fit to the numerical data furnishes z2= 1.00共2兲.
With the above hypotheses we suppose that V¯ obeys a homogeneous function of type V¯共n,V
0兲=lV¯共lan,lbV0兲, wherelis a scaling factor andaandbare scaling exponents. We show thatis related to the critical exponents,, and z1. Choosingl=V0−1/bwe obtain¯V共n,V0兲=V0
−1/bf共V 0 −a/bn
兲. From this expression we find nx⬀ −V0
a/b
. Hencez1= −and z2=a/b. ConsideringnⰆnx, where V¯ is constant, we obtain (b)
(a)
FIG. 2. 共Color online兲 共a兲Combinations of initial conditions for which the particle experiences FA, white region, and for which the particle losses energy until to reach the rest, blue共dark兲region.共b兲 The white region corresponds to combinations of initial conditions that results in Fermi acceleration and the regions with different color 共gray兲 patterns illustrate the maximum collision number before the total energy dissipation. The values of parameters are
= 10−3and␣= 10−3.
100 102 104 106 108
n 100 102 104 106 V α=10−6 α=10−5 α=10−4 ε=10−3
V0=10
100 102 10n4 106 108
101 103 105 107 109 V
ε=10−5,V0=10
ε=10−2,V0=10
ε=10−3,V0=103
ε=10−3,V0=102
10−5 10−4 10−3 10−2
ε 10−5 10−4 10−3 10−2 10−1 g ( ε ) Numerical data Best fit
g(ε)∝εη
η=0.969(5)
10−5 10−4 10−3 10−2
ε 102 103 104 105 106 nx Numerical data Best fit
nx∝εz1
z1=−0.991(1)
100 101 102 103
V0 103 104 105 106 nx Numerical data Best fit
z2=1.00(2)
nx∝V0z2
10−6 10−3 100 103
nV0−a/bεζ
10−1 101 103 105 107 V / V0 −1/ b
ε=10−5,V0=10
ε=10−2,V0=10
ε=10−3,V0=103
ε=10−3,V0=102
(b) (a)
(c) (d)
(f) (e)
FIG. 3. 共Color online兲 共a兲 Behavior of V¯⫻n for = 10−3, V0= 10, and different values of ␣, 共b兲 behavior of V¯⫻n for ␣= 10−5 and different values of and V
0. 共c兲 Log-log plot of g共兲=V¯/nas function of.共d兲The best fit to the numerical data on thenxversusplot furnishesz1= −0.991共1兲.共e兲Log-log plot of nx as function ofV0.共f兲 Collapse ofV¯ curves of共b兲 onto a single
b= −1. For nⰇnx we have f=共V0 −a/bn
兲. Thus we find
=, a= −1/, and =/. Therefore we have a= −1.0028共7兲and= 0.972共6兲. Applying appropriate scaling transformations along the axis of Fig. 3共b兲we show that all curves merge together onto a single and universal curve, as shown in Fig.3共f兲. This result confirms that, for any combi-nation of and V0 above the critical region where FA is suppressed shall experience basically the same general be-havior.
Let us now discus the situation where one of the walls exhibit stochastic motion. It is well known that FA is also observed for the nondissipative stochastic Fermi based sys-tems关17,22,23兴. Someone can ask about the presence or lack of FA when the frictional dissipation for the stochastic Fermi-Ulam model is considered. Thus instead of the wall to furnish an increment 2 in velocity at each collision 关Eq.
共2兲兴, we regard the situation where the moving wall furnishes an increment 2z to the velocity of the particle, where z as-sumes a random value in the interval关−,兲. In this case the velocity of the particle after the collision 共n+ 1兲th is Vn+1=兩
冑
Vn2
+ 2␣共Xn+Xn+1兲− 4␣+ 2z兩 and the expression of tn+1is the same as shown in Eq.共2兲. The absolute value bars in the velocity expression prevent the particle to leave the region between the walls.
Figure 4 shows the velocity after each collision as func-tion ofnfor two initial conditions and for both sawtooth and stochastic motion forV0greater thanV+共expression共3兲兲. We considered = 10−2 and␣= 10−5. When the wall moves ac-cording to the sawtooth expression the particle experiences FA, as discussed before. When the wall moves stochastically we observe for some intervals of n that the velocity decreases while it increases for other intervals of n. After n=i collisions with the moving wall the particle’s velocity matches the condition Vi⬍Vmin, where Vmin =
冑
2␣共2 −Xn−Xn+1兲. Then the particle stops between the walls and the time to the collision i+ 1 diverges. For the stochastic case if the particle’s velocity has a finite value, then the particle stops after some collisions.Note that a similar discussion applies to the situation
where the wall moves according to a sine function
关15,24,25兴. Similarly to the stochastic case, the velocity of the particle eventually becomes small and the particle loses all its energy.
As a final remark, we have considered a dissipative ver-sion of the one-dimenver-sional Fermi accelerator model with sawtooth external perturbation in the presence of a dissipa-tive force generated from a slip of a body against a rough surface. We showed that there exists a critical line that sepa-rates a region of total energy dissipation from a region where all initial conditions produce FA even in the presence of dis-sipation. Moreover, the behavior of the velocity curves is scaling invariant. Our results confirm that FA is sensible to the type of dissipation considered. In particular, a dissipation introduced via a friction force generated from a slip of a body against a rough surface is not a sufficient condition to suppress FA generically, but then it does happen under spe-cific conditions.
D.G.L. thanks FAPEMIG and CNPq. E.D.L. thanks FAPESP, CNPq, and FUNDUNESP, Brazilian agencies.
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1×100 1×102 1×104 1×106 1×108 n
1×10−3 1×10−1 1×101 1×103 1×105 1×107
V
V
0=10 2
, stochastic V
0=10, stochastic
V
0=10 2
, saw-tooth V
0=10, saw-tooth
ε=10−2 α=10−5
FIG. 4. 共Color online兲 Velocity of the particle after each collision with the oscillating wall as function ofnfor the situation
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