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Levy-flight-like particle dynamics along the stochastic web in a weakly inhomogeneous magnetic field
29 May 1995 PHYSICS ELSEVIER
LETTERS
A
Physics Letters A 201(1995) 311-318
Levy-flight-like particle dynamics along the stochastic web in a weakly inhomogeneous magnetic field B.A. Petrovichev Department of Theoretical Physics, Research School of Physical Sciences and Engineering, The Australian National Uniuersity Canberra, ACT 0200, Australia
Received 31 August 1994; revised manuscript received 7 March 1995; accepted for publication 17 March 1995 Communicated by A.P. Fordy
Abstract A numerical investigation is performed of the Hamiltonian dynamics magnetic field and the field of a finite set of electrostatic travelling perturbation of the magnetic field leads to complicated particle dynamics the existence of huge chaotic jets in the phase space of a system. Thus, transport has the nature of Levy flights and can be anomalous.
1. Introduction
Recently there has been considerable interest in the study of transport processes arising as the result of Hamiltonian chaos. This interest is connected both with the fundamental problem of the occurrence of chaos in Hamiltonian systems and with numerous applications in many areas of physics such as plasma physics and hydrodynamics [ 1,2]. The transition to chaos is accompanied by the birth of small chaotic regions known as “chaos embryos”. In Hamiltonian systems such embryos are stochastic layers in the vicinity of the destroyed separatrices of nonlinear resonances. In the case of weak perturbations, thin stochastic layers are separated from each other by invariant surfaces, and chaotic oscillations inside the layers are exponentially small. This situation can be described by the Kolmogorov-Arnold-Moser &AM) theorem [3],
of a charged particle in a weakly inhomogeneous waves. Computer simulations show that a small along the stochastic web which can reveal itself in in a well-defined range of parameters the particle
which states that most invariant surfaces in integrable systems survive under small perturbations. The essential condition for this is nondegeneracy of the unperturbed Hamiltonian. However, under certain conditions chaotic domains can produce a connected network known as a stochastic web. Penetrating the whole phase space, the stochastic web plays a significant role in the problem of global stability and a wide variation in transport possibilities can be realised between the slow and anomalously fast transport processes. A well-known example is Arnold diffusion - unlimited transport along the stochastic web in systems with more than two degrees of treedom [3]. A substantially new understanding of the problem of Hamiltonian transport appeared after the discovery of the stochastic web in the phase space of the low-dimensional Hamiltonian system that describes the interaction of a charged particle with a set of
03759601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00228-6
312
Bd. Petrovichev /Physics
electrostatic travelling waves (wavepacket) in an external uniform magnetic field [2,4-61. Such a model has various applications in plasma physics (stochastic particle acceleration and heating of charged particles). The key feature of this dynamical model, which results in a qualitatively new form of a lowdimensional Hamiltonian chaos, is degeneracy of the unperturbed Hamiltonian. In the case of an infinite number of electrostatic waves it was found [4] that, for arbitrarily small amplitudes of the wavepacket, a global stochastic web-like network exists in the phase space of this system, provided an exact resonance relationship between the cyclotron frequency and the characteristic frequency of the wavepacket is satisfied. The stochastic web has a crystalline or quasicrystalline symmetry whose order is determined by the resonant ratio between the perturbation and the cyclotron frequency. An unlimited diffusion along the channels of this web is possible, which is similar to Arnold diffusion in higher dimensional nondegenerate systems. The motion of a charged particle propagating across a uniform magnetic field and perturbed by the presence of a single plane electrostatic wave [5] and a finite number of electrostatic waves [6] has also been investigated. Similarly, it has been observed that a global web-like network with complex symmetry properties does exist, but the thickness of this stochastic web quickly decreases with increasing distance from the origin of the phase portrait. The purpose of this Letter is to consider whether any qualitatively new phenomena arise if the magnetic field is inhomogeneous. Generally speaking, this case reflects the most realistic physical situation and also is of much importance in many cases of practical interest. We will show that the presence of a very small but finite spatial perturbation of the magnetic field which is directed tranversally to the basic magnetic field leads to substantial changes of the particle dynamics along the stochastic web. We consider the classical motion of a charged particle in a weakly inhomogeneous magnetic field and the field of a finite set of electrostatic travelling waves if there exist exact resonance conditions between the unperturbed cyclotron frequency and frequencies of the waves. The unperturbed system is degenerate, as a result of which there exists a stochastic web in the phase space of the system. Computer simulations
Letters A 201 (1995) 31 I-318
show that even a very small perturbation of the magnetic field, which keeps the system close to degeneracy, leads to complicated multidimensional particle dynamics in the phase space. In this case the motion of a charged particle inside the channels of the stochastic web can reveal itself in the existence of large chaotic jets in the phase space of the system. Hence the particle transport has the nature of Levy flights and can be anomalous. The Letter is organized as follows. Section 2 presents the basic Hamiltonian for a particle in the field of a finite set of electrostatic travelling waves propagating perpendicularly to a weakly perturbed magnetic field. Section 3 describes the effect of a magnetic field inhomogeneity on the topology of phase space for a particle moving in the field of one plane electrostatic wave. In Section 4 we present numerical simulation results for the motion of a particle in the field of a finite set of electrostatic travelling waves for some particular number of harmonics in the wavepacket and different ratios of the characteristic frequency of the wavepacket to the cyclotron frequency. Our conclusions are summarized in Section 5.
2. Description of the model Let the charged particle be moving in a magnetic field which is directed along the z-axis and perturbed sinusoidally along the y-direction, B = B,(l + p sin k,y)e,,
(I)
where B, is the magnetic field strength and the modulation amplitude p is supposed to be small. For instance, the magnetic field in a tokamak can be characterized by such magnetic field gradients perpendicular to the ambient magnetic field. Let a wave disturbance be propagating along the x-direction perpendicular to the magnetic field. The potential of the electrostatic wave field is represented as a number N of plane waves with amplitudes E, frequencies nw and wave numbers k,, 4(X, t) =E 5 cos(k,x-nwt). n=l
(2)
B.A. Petrovichev / Physics Letters A 201 (1995) 31 l-318
Formally the problem is reduced to the investigation of the particle dynamics with a Hamiltonian of the following kind, z-z=+;+
[py-
w,x(l + p sin k,y)12)
N +
& c
COS(k,X-ma),
n=l
where P,, PY are components of the particle momentum and o, is unperturbed cyclotron frequency. Also let the resonance condition between the unperturbed cyclotron frequency and the characteristic frequency w of the wavepacket be fulfilled in the form, 0=46&, (4) where q is an integer. The fundamental property of this system is degeneracy of the unperturbed ( /3 = 0, E = 0) Hamiltonian, which breaks the condition required for the KAM theorem to hold. The unperturbed Hamiltonian describes linear oscillations and it is not possible to use the unperturbed motion as a zero order approximation. The presence of an inhomogeneity ( /3 = 01 in the undisturbed problem removes the degeneracy. If the inhomogeneity of the magnetic field is weak enough, then the system is close to a degenerate one. In this case a small but finite amplitude of the electrostatic wave field may prove to be sufficient to form a web in the phase space of the system.
3. Case of single harmonic In the first instance, let us consider motion of a charged particle in the case when the wave consists of a single harmonic. The result of the numerical simulation for Hamiltonian (3) for the case of a small wave amplitude is given in Fig. 1. Due to the presence of a very small but finite inhomogeneity (parameter /3 = 10m7) the unperturbed system remains very close to degeneracy. On the phase plane of the transverse motion there exists a web-like network (Fig. la) which has exactly the same geometry as in the case of uniform magnetic field [5]. The skeleton of that network can be seen more clearly in Fig. lb, in which only those regular trajectories are depicted which are very close to the family of separatrices through the set of hyperbolic points. This
313
family of separatrices has the form of a web with a set of concentric circles and rays. The cells of the web form concentric belts around the coordinate origin. Inside the cells particle motion occurs along closed-type orbits, round the elliptic points, which lie in the centres of cells. Projection of the phase portrait onto the (x, y) phase plane (Fig. lc) shows that this network has intrinsically multidimensional topology. With the growth of the wave amplitude, the family of separatrix nets is destroyed and the family of web-like stochastic networks appears in the phase space of the system. These webs are networks of finite thickness, inside which the dynamics of the particles is chaotic, and outside which (i.e., in the cells of the webs) the dynamics is regular. Fig. Id depicts the set of such stochastic webs numerically obtained for three different initial conditions along the y-direction. In the 3D space (x, P,, y> there exists a family of stochastic surfaces and a particle, located at a particular level along the y-direction, executes a random walk along the corresponding surface inside the channels of the stochastic web. An example of one such web-like surface is shown in Figs. le, If. This is the projection of a single particle trajectory onto the (x, P,> and (x, y> planes obtained by the numerical calculation of the equations of motion for the time equal to lo5 periods of the wave. A particle with initial conditions sufficiently close to the separatrix of the averaged motion is captured by a stochastic layer and wanders chaotically along the channels of the stochastic web. The particle energy can vary significantly, and, thus, a universal diffusion along the stochastic web is possible, similar to Arnold diffusion. The mechanism of diffusion through the stochastic web is the same as when the wave propagates transversally to the uniform magnetic field [S]. The diffusion quickly ceases because the width of the web decreases exponentially fast as the perturbation diminishes as well as the distance from the origin of the phase portrait increases. 4. Case of several harmonics Now let us present some numerical results for the case of a particle moving in the field of a wave consisting of several harmonics.
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Letters A 201 (1995) 311-318
These figures show phase portraits of particles moving in the field of a wave consisting of several harmonics for the inhomogeneity parameter /3 = lop6 (Fig. 2). The left hand figures show phase portraits in projection onto the (x, Z’,) plane. One can see that in this case the arrangement of the
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elliptic and hyperbolic points, as well as the geometry of the’ separatrix net, is much more complicated than for the previous case. The symmetry of these phase portraits as a whole is determined by the ratio of the characteristic frequency of the wavepacket to the unperturbed cyclotron frequency. The right hand
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Fig. 1. (a) Phase portrait of a family of regular trajectories in the field of a single harmonic for p = lo-‘, E = 0.01, k, = 15.0, k, = 1.0 in projection onto the (x, P,) plane. (b), (c) Skeleton of a web-like network for the same set of parameters as in (a) in projection onto the (x, E = 0.12, k, = 15.0, k, = 1.0. (e), (f) Stochastic surface for P,) and (x, y) planes. (d) Stochastic webs in the (x, y) plane for p = lo-‘, the same set of parameters as in (d) in projection onto the (x, P,) and (x, y) plane.
BA. Petrovichev / Physics Letters A 201 (1995) 31 l-318
figures demonstrate the complicated topology of the same phase portraits in projection onto the (x, y) plane. Stochastic webs for the case of several harmonics are shown in Fig. 3. Here the amplitude of the wave packet is ten times larger than in previous figures. The right hand figures show the same particle trajec-
315
tory but in projection onto the (x, y) plane. One can observe that the diffusive process in the (x, P,> phase plane alternates with sections of long flights along the y-direction. Thus there are two types of particle behaviour in the phase space of the system: chaotic particle motion along the channels of the stochastic webs and lengthy, quasi-regular almost
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Fig. 2. Phase portrait of a family of regular trajectories in the field of several harmonics for p = 10m6, E = 0.1, k, = 1.0, k, = 1.0. (a) N = 3, 4 = 3. (b) N = 2, q = 4. (c) N = 3, q = 5. The panels on the left are projections onto the (x, P,) plane. Those on the right are projections
onto the (x, y) plane.
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BA. Petrovichev/Physics
Letters A 201 (1995) 311-318
free flights along the y-direction. The first type of motion can correspond to usual Gaussian-like diffusion in the (x, P,) plane or even slower subdiffusion due to the existence of sticking events, whereas the other one can correspond to anomalously fast transport in the y-direction. Such long excursions of particle orbits, which are known as Levy flights [7], were observed in Hamil-
-20 e -20
-10
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10
tonian dynamical systems with stochastic webs [8,9]. Extensive theoretical, numerical and experimental studies have been devoted to these statistical anomalies of chaotic motion. It was found that the correlation functions of the chaotic trajectories for this type of motion can increase much more slowly that exponentially, in particular, according to a power law [15]. Another important aspect of this effect is the
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Fig. 3. Stochastic web formed for particle motion in the field of several harmonics for /3 = lo-“, k, = 1.0. (a) N = 3, q = 3, E = 1.24, k,=1.0.(b)N=2,q=4,s=1.26,k,=5.0.(c~N=5,q=5,~=1.12,k,=1.0.Thepanelsontheleftareprojectionsontothe(x,P,~ plane. Those on the right are projections onto the (x, y) plane.
B_A.Petrovichev /Physics Letters A 201 (I 995) 31 l-318
E
317
6000
z 8 m 4000 5 .” c! 21 2000 a
b 0
0 2.0 1.5 Wave amplitude
2.5
1.00
d 1.20 1.30 1.10 Wave amplitude
1.10 1.20 1.30 Wave amplitude
1.40
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Fig. 4. Maximum y-displacement plotted as a function of the wav@amplitude for (a) p = 10m6, N = 2, q = 4; (b) p = 10m6, N = 3, q = 5; (~)p=lO-~, N=S,q=5;(d)p=lO+,N=5,q=S.
possibility of extremely fast diffusion [lo-121 along some degrees of freedom (in our case along the y-direction) caused by the existence of such long ballistic modes in particle orbits. The mechanism of the onset of Levy flights in this dynamical system is as follows. Flights in the y-direction correspond exactly to a lengthy sticking of the particle in well defined regions in the (x, PJ plane near the critical invariant curves which separates the regular and the chaotic motion. In that region the invariant tori are disrupted and such fine topological objects as cantori appear in the phae space [13-151. On being captured in these areas on the (x, P,> plane the particle can spend a very long time in almost regular (nonchaotic) motion in the y-direction. In our model chaotic particle transport has the nature of Levy flights in a fairly broad range of an inhomogeneity parameter as well as of wavepacket amplitude (Fig. 4). These plots show the maximum y-displacement as a function of the wave amplitude
for some cases of different structure of the wavepacket. One can see a number of peaks that correspond to the extremely long flights along the y-direction.
5. Conclusions The motion of a charged particle in a weakly inhomogeneous magnetic field and the field of a finite set of electrostatic travelling waves have been investigated numerically. Because the cyclotron frequency is independent of the gyroradius in a homogeneous magnetic field, the perturbation is singular and leads to a novel form of chaotic diffusion characterised by infrequent but large jumps (Levy flights) in the phase space of the system. We should note that chaotic diffusion of a particle propagating in the field of a finite wavepacket was found to be limited and the region of the chaotic behaviour is bounded even for the more general case of a wavepacket
318
B.A. Petrovichev /Physics
propagating obliquely to the external homogeneous magnetic field [16]. However, our observations show that the situation is quite different for the wave-particle interactions in media with an inhomogeneous magnetic field. Our numerical simulations indicate that, in the case of a single electrostatic wave, the phase space remains stratified along the y-direction, whereas, in the case of a finite number of waves, even a very small perturbation of the magnetic field leads to dramatic changes of the particle dynamics. The change reveals itself in the existence of Levy flights and results in chaotic transport in much larger regions of the phase space than in the case of a homogeneous magnetic field. On the basic mathematical side, the model that we have considered is described by a Hamiltonian with two and a half degrees of freedom and possesses the general properties intrinsic to multidimensional Hamiltonian dynamical systems. Furthermore, this system is close to degeneracy with respect to some of the degrees of freedom if the magnetic field perturbation is weak enough. Hence, phenomena occur that are similar to those considered in Refs. [8,9] - in particular, enhancement of diffusion due to Levy flights. In this sense this dynamical system can belong to the systems with adiabatic Hamiltonian chaos [9,17], where a small deviation from integrability leads to global chaotic mixing in the phase space. We would like to emphasize that our simple model does not present a full picture of a complicated particle behaviour in a weakly inhomogeneous magnetic field, since other types of inhomogenities need to be considered in many physical situations and applications.
Acknowledgments The author is very grateful to Professor R.L. Dewar for helpful suggestions and constructive re-
Letters A 201 (I 995) 31 I-318
marks. Sincere thanks also to the referees for their careful reading and valuable suggestions to improve the manuscript. This work was supported by a postdoctoral Rothmans Foundation Fellowship.
References 111A.J.
Lichtenberg and MA. Lieberman, Regular and chaotic dynamics (Springer, Berlin, 1992). 121G.M. Zaslavsky, R.Z. Sagdeev, DA. Usikov and A.A. Chemikov, Weak chaos and quasiregular patterns (Cambridge Univ. Press, Cambridge, 1991). (31 V.I. Arnold, Mathematical methods in classical mechanics (Springer, Berlin, 1978). [41 G.M. Zaslavsky, M. Yu. Zakharov, R.Z. Sagdeev, D.A. Usikov and A.A. Chemikov, Sov. Phys. JETP 64 (1986) 294. [51 A.A. Chemikov, M. Ya. Natenzon, B.A. Petrovichev, R.Z. Sagdeev and G.M. Zaslavsky, Phys. L&t. A 122 (1987) 39. [61S. Murakami, T. Sato and A. Hasegawa, Physica D 32 (1988) 269. 171 E.W. Montroll and M.F. Shlesinger, in: Studies in statistical mechanics, eds. J.L. Lebowitz and E.W. Montroll (NorthHolland, Amsterdam, 19841, Vol. II, p. 1. k31G.M. Zaslavsky, M. Yu. Zakharov, A.I. Neishtadt, R.Z. Sagdeev, D.A. Usikov and A.A. Chemikov, Sov. Phys. JETP 69 (1989) 885. [91 AI. Neishtadt, D.K. Chaikovsky and A.A. Chemikov, Sov. Phys. JETP 72 (1991) 423. no1 T. Geisel, A. Zacherl and G. Radons, Phys. Rev. Lett. 59 (1987) 2503. ml A.A. Chemikov, B.A. Petrovichev, A.V. Rogalsky, R.Z. Sagdeev and G.M. Zaslavsky, Phys. Lett. A 144 (1990) 127. WI J. Klafter and G. Zumofen, Phys. Rev. E 49 (1994) 4873. 1131I.C. Percival, in: Nonlinear dynamics and beam-beam interaction, eds. M. Month and J.C. Herrera (AIP, New York, 1979) p. 302 [141 S. Aubry, Physica D 7 (1983) 240. [151 R.S. MacKay, J.D. Meiss and I.C. Percival, Physica D 13 (1984) 55. [161A.A. Vasil’ev and G.M. Zaslavsky, Sov. Phys. JETP 72 (1991) 826. [171 A.A. Chemikov and G. Schmidt, Phys. Rev. E 50 (1994) 3436.
LETTERS
A
Physics Letters A 201(1995) 311-318
Levy-flight-like particle dynamics along the stochastic web in a weakly inhomogeneous magnetic field B.A. Petrovichev Department of Theoretical Physics, Research School of Physical Sciences and Engineering, The Australian National Uniuersity Canberra, ACT 0200, Australia
Received 31 August 1994; revised manuscript received 7 March 1995; accepted for publication 17 March 1995 Communicated by A.P. Fordy
Abstract A numerical investigation is performed of the Hamiltonian dynamics magnetic field and the field of a finite set of electrostatic travelling perturbation of the magnetic field leads to complicated particle dynamics the existence of huge chaotic jets in the phase space of a system. Thus, transport has the nature of Levy flights and can be anomalous.
1. Introduction
Recently there has been considerable interest in the study of transport processes arising as the result of Hamiltonian chaos. This interest is connected both with the fundamental problem of the occurrence of chaos in Hamiltonian systems and with numerous applications in many areas of physics such as plasma physics and hydrodynamics [ 1,2]. The transition to chaos is accompanied by the birth of small chaotic regions known as “chaos embryos”. In Hamiltonian systems such embryos are stochastic layers in the vicinity of the destroyed separatrices of nonlinear resonances. In the case of weak perturbations, thin stochastic layers are separated from each other by invariant surfaces, and chaotic oscillations inside the layers are exponentially small. This situation can be described by the Kolmogorov-Arnold-Moser &AM) theorem [3],
of a charged particle in a weakly inhomogeneous waves. Computer simulations show that a small along the stochastic web which can reveal itself in in a well-defined range of parameters the particle
which states that most invariant surfaces in integrable systems survive under small perturbations. The essential condition for this is nondegeneracy of the unperturbed Hamiltonian. However, under certain conditions chaotic domains can produce a connected network known as a stochastic web. Penetrating the whole phase space, the stochastic web plays a significant role in the problem of global stability and a wide variation in transport possibilities can be realised between the slow and anomalously fast transport processes. A well-known example is Arnold diffusion - unlimited transport along the stochastic web in systems with more than two degrees of treedom [3]. A substantially new understanding of the problem of Hamiltonian transport appeared after the discovery of the stochastic web in the phase space of the low-dimensional Hamiltonian system that describes the interaction of a charged particle with a set of
03759601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00228-6
312
Bd. Petrovichev /Physics
electrostatic travelling waves (wavepacket) in an external uniform magnetic field [2,4-61. Such a model has various applications in plasma physics (stochastic particle acceleration and heating of charged particles). The key feature of this dynamical model, which results in a qualitatively new form of a lowdimensional Hamiltonian chaos, is degeneracy of the unperturbed Hamiltonian. In the case of an infinite number of electrostatic waves it was found [4] that, for arbitrarily small amplitudes of the wavepacket, a global stochastic web-like network exists in the phase space of this system, provided an exact resonance relationship between the cyclotron frequency and the characteristic frequency of the wavepacket is satisfied. The stochastic web has a crystalline or quasicrystalline symmetry whose order is determined by the resonant ratio between the perturbation and the cyclotron frequency. An unlimited diffusion along the channels of this web is possible, which is similar to Arnold diffusion in higher dimensional nondegenerate systems. The motion of a charged particle propagating across a uniform magnetic field and perturbed by the presence of a single plane electrostatic wave [5] and a finite number of electrostatic waves [6] has also been investigated. Similarly, it has been observed that a global web-like network with complex symmetry properties does exist, but the thickness of this stochastic web quickly decreases with increasing distance from the origin of the phase portrait. The purpose of this Letter is to consider whether any qualitatively new phenomena arise if the magnetic field is inhomogeneous. Generally speaking, this case reflects the most realistic physical situation and also is of much importance in many cases of practical interest. We will show that the presence of a very small but finite spatial perturbation of the magnetic field which is directed tranversally to the basic magnetic field leads to substantial changes of the particle dynamics along the stochastic web. We consider the classical motion of a charged particle in a weakly inhomogeneous magnetic field and the field of a finite set of electrostatic travelling waves if there exist exact resonance conditions between the unperturbed cyclotron frequency and frequencies of the waves. The unperturbed system is degenerate, as a result of which there exists a stochastic web in the phase space of the system. Computer simulations
Letters A 201 (1995) 31 I-318
show that even a very small perturbation of the magnetic field, which keeps the system close to degeneracy, leads to complicated multidimensional particle dynamics in the phase space. In this case the motion of a charged particle inside the channels of the stochastic web can reveal itself in the existence of large chaotic jets in the phase space of the system. Hence the particle transport has the nature of Levy flights and can be anomalous. The Letter is organized as follows. Section 2 presents the basic Hamiltonian for a particle in the field of a finite set of electrostatic travelling waves propagating perpendicularly to a weakly perturbed magnetic field. Section 3 describes the effect of a magnetic field inhomogeneity on the topology of phase space for a particle moving in the field of one plane electrostatic wave. In Section 4 we present numerical simulation results for the motion of a particle in the field of a finite set of electrostatic travelling waves for some particular number of harmonics in the wavepacket and different ratios of the characteristic frequency of the wavepacket to the cyclotron frequency. Our conclusions are summarized in Section 5.
2. Description of the model Let the charged particle be moving in a magnetic field which is directed along the z-axis and perturbed sinusoidally along the y-direction, B = B,(l + p sin k,y)e,,
(I)
where B, is the magnetic field strength and the modulation amplitude p is supposed to be small. For instance, the magnetic field in a tokamak can be characterized by such magnetic field gradients perpendicular to the ambient magnetic field. Let a wave disturbance be propagating along the x-direction perpendicular to the magnetic field. The potential of the electrostatic wave field is represented as a number N of plane waves with amplitudes E, frequencies nw and wave numbers k,, 4(X, t) =E 5 cos(k,x-nwt). n=l
(2)
B.A. Petrovichev / Physics Letters A 201 (1995) 31 l-318
Formally the problem is reduced to the investigation of the particle dynamics with a Hamiltonian of the following kind, z-z=+;+
[py-
w,x(l + p sin k,y)12)
N +
& c
COS(k,X-ma),
n=l
where P,, PY are components of the particle momentum and o, is unperturbed cyclotron frequency. Also let the resonance condition between the unperturbed cyclotron frequency and the characteristic frequency w of the wavepacket be fulfilled in the form, 0=46&, (4) where q is an integer. The fundamental property of this system is degeneracy of the unperturbed ( /3 = 0, E = 0) Hamiltonian, which breaks the condition required for the KAM theorem to hold. The unperturbed Hamiltonian describes linear oscillations and it is not possible to use the unperturbed motion as a zero order approximation. The presence of an inhomogeneity ( /3 = 01 in the undisturbed problem removes the degeneracy. If the inhomogeneity of the magnetic field is weak enough, then the system is close to a degenerate one. In this case a small but finite amplitude of the electrostatic wave field may prove to be sufficient to form a web in the phase space of the system.
3. Case of single harmonic In the first instance, let us consider motion of a charged particle in the case when the wave consists of a single harmonic. The result of the numerical simulation for Hamiltonian (3) for the case of a small wave amplitude is given in Fig. 1. Due to the presence of a very small but finite inhomogeneity (parameter /3 = 10m7) the unperturbed system remains very close to degeneracy. On the phase plane of the transverse motion there exists a web-like network (Fig. la) which has exactly the same geometry as in the case of uniform magnetic field [5]. The skeleton of that network can be seen more clearly in Fig. lb, in which only those regular trajectories are depicted which are very close to the family of separatrices through the set of hyperbolic points. This
313
family of separatrices has the form of a web with a set of concentric circles and rays. The cells of the web form concentric belts around the coordinate origin. Inside the cells particle motion occurs along closed-type orbits, round the elliptic points, which lie in the centres of cells. Projection of the phase portrait onto the (x, y) phase plane (Fig. lc) shows that this network has intrinsically multidimensional topology. With the growth of the wave amplitude, the family of separatrix nets is destroyed and the family of web-like stochastic networks appears in the phase space of the system. These webs are networks of finite thickness, inside which the dynamics of the particles is chaotic, and outside which (i.e., in the cells of the webs) the dynamics is regular. Fig. Id depicts the set of such stochastic webs numerically obtained for three different initial conditions along the y-direction. In the 3D space (x, P,, y> there exists a family of stochastic surfaces and a particle, located at a particular level along the y-direction, executes a random walk along the corresponding surface inside the channels of the stochastic web. An example of one such web-like surface is shown in Figs. le, If. This is the projection of a single particle trajectory onto the (x, P,> and (x, y> planes obtained by the numerical calculation of the equations of motion for the time equal to lo5 periods of the wave. A particle with initial conditions sufficiently close to the separatrix of the averaged motion is captured by a stochastic layer and wanders chaotically along the channels of the stochastic web. The particle energy can vary significantly, and, thus, a universal diffusion along the stochastic web is possible, similar to Arnold diffusion. The mechanism of diffusion through the stochastic web is the same as when the wave propagates transversally to the uniform magnetic field [S]. The diffusion quickly ceases because the width of the web decreases exponentially fast as the perturbation diminishes as well as the distance from the origin of the phase portrait increases. 4. Case of several harmonics Now let us present some numerical results for the case of a particle moving in the field of a wave consisting of several harmonics.
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These figures show phase portraits of particles moving in the field of a wave consisting of several harmonics for the inhomogeneity parameter /3 = lop6 (Fig. 2). The left hand figures show phase portraits in projection onto the (x, Z’,) plane. One can see that in this case the arrangement of the
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BA. Petrovichev / Physics Letters A 201 (1995) 31 l-318
figures demonstrate the complicated topology of the same phase portraits in projection onto the (x, y) plane. Stochastic webs for the case of several harmonics are shown in Fig. 3. Here the amplitude of the wave packet is ten times larger than in previous figures. The right hand figures show the same particle trajec-
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tory but in projection onto the (x, y) plane. One can observe that the diffusive process in the (x, P,> phase plane alternates with sections of long flights along the y-direction. Thus there are two types of particle behaviour in the phase space of the system: chaotic particle motion along the channels of the stochastic webs and lengthy, quasi-regular almost
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Fig. 2. Phase portrait of a family of regular trajectories in the field of several harmonics for p = 10m6, E = 0.1, k, = 1.0, k, = 1.0. (a) N = 3, 4 = 3. (b) N = 2, q = 4. (c) N = 3, q = 5. The panels on the left are projections onto the (x, P,) plane. Those on the right are projections
onto the (x, y) plane.
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free flights along the y-direction. The first type of motion can correspond to usual Gaussian-like diffusion in the (x, P,) plane or even slower subdiffusion due to the existence of sticking events, whereas the other one can correspond to anomalously fast transport in the y-direction. Such long excursions of particle orbits, which are known as Levy flights [7], were observed in Hamil-
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tonian dynamical systems with stochastic webs [8,9]. Extensive theoretical, numerical and experimental studies have been devoted to these statistical anomalies of chaotic motion. It was found that the correlation functions of the chaotic trajectories for this type of motion can increase much more slowly that exponentially, in particular, according to a power law [15]. Another important aspect of this effect is the
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B_A.Petrovichev /Physics Letters A 201 (I 995) 31 l-318
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possibility of extremely fast diffusion [lo-121 along some degrees of freedom (in our case along the y-direction) caused by the existence of such long ballistic modes in particle orbits. The mechanism of the onset of Levy flights in this dynamical system is as follows. Flights in the y-direction correspond exactly to a lengthy sticking of the particle in well defined regions in the (x, PJ plane near the critical invariant curves which separates the regular and the chaotic motion. In that region the invariant tori are disrupted and such fine topological objects as cantori appear in the phae space [13-151. On being captured in these areas on the (x, P,> plane the particle can spend a very long time in almost regular (nonchaotic) motion in the y-direction. In our model chaotic particle transport has the nature of Levy flights in a fairly broad range of an inhomogeneity parameter as well as of wavepacket amplitude (Fig. 4). These plots show the maximum y-displacement as a function of the wave amplitude
for some cases of different structure of the wavepacket. One can see a number of peaks that correspond to the extremely long flights along the y-direction.
5. Conclusions The motion of a charged particle in a weakly inhomogeneous magnetic field and the field of a finite set of electrostatic travelling waves have been investigated numerically. Because the cyclotron frequency is independent of the gyroradius in a homogeneous magnetic field, the perturbation is singular and leads to a novel form of chaotic diffusion characterised by infrequent but large jumps (Levy flights) in the phase space of the system. We should note that chaotic diffusion of a particle propagating in the field of a finite wavepacket was found to be limited and the region of the chaotic behaviour is bounded even for the more general case of a wavepacket
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propagating obliquely to the external homogeneous magnetic field [16]. However, our observations show that the situation is quite different for the wave-particle interactions in media with an inhomogeneous magnetic field. Our numerical simulations indicate that, in the case of a single electrostatic wave, the phase space remains stratified along the y-direction, whereas, in the case of a finite number of waves, even a very small perturbation of the magnetic field leads to dramatic changes of the particle dynamics. The change reveals itself in the existence of Levy flights and results in chaotic transport in much larger regions of the phase space than in the case of a homogeneous magnetic field. On the basic mathematical side, the model that we have considered is described by a Hamiltonian with two and a half degrees of freedom and possesses the general properties intrinsic to multidimensional Hamiltonian dynamical systems. Furthermore, this system is close to degeneracy with respect to some of the degrees of freedom if the magnetic field perturbation is weak enough. Hence, phenomena occur that are similar to those considered in Refs. [8,9] - in particular, enhancement of diffusion due to Levy flights. In this sense this dynamical system can belong to the systems with adiabatic Hamiltonian chaos [9,17], where a small deviation from integrability leads to global chaotic mixing in the phase space. We would like to emphasize that our simple model does not present a full picture of a complicated particle behaviour in a weakly inhomogeneous magnetic field, since other types of inhomogenities need to be considered in many physical situations and applications.
Acknowledgments The author is very grateful to Professor R.L. Dewar for helpful suggestions and constructive re-
Letters A 201 (I 995) 31 I-318
marks. Sincere thanks also to the referees for their careful reading and valuable suggestions to improve the manuscript. This work was supported by a postdoctoral Rothmans Foundation Fellowship.
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