- Email: [email protected]
A universal approach to the study of nonlinear systems
Physica A 338 (2004) 1 – 6
www.elsevier.com/locate/physa
A universal approach to the study of nonlinear systems Rudolph C. Hwa∗ Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, OR 97403-5203, USA
Abstract A large variety of nonlinear systems have been treated by a common approach that emphasizes the (uctuation of spatial patterns. By using the same method of analysis it is possible to discuss the chaotic behaviors of quark jets and logistic map in the same language. Critical behaviors of quark-hadron phase transition in heavy-ion collisions and of photon production at the threshold of lasing can also be described by a common scaling behavior. The universal approach also makes possible an insight into the recently discovered phenomenon of wind reversal in cryogenic turbulence as a manifestation of self-organized criticality. c 2004 Elsevier B.V. All rights reserved. PACS: 02.50.−r; 87.15.Ya Keywords: Nonlinear systems; Fluctuations
There are many varieties of nonlinear systems and many methods of investigation that are appropriate for the di7erent types of problems. Let me classify the nonlinear systems into four categories: small, large, simple and complex. Small does not mean simple, and large does not mean complex. Some examples can illustrate what I mean. By small I have in mind quarks, hadrons and nuclei. Large systems can include magnetic material at phase transition, lasers and cryogenic turbulence. Simple systems can be the logistic map and Lorenz attractor that form the introductory topics to chaos. Finally, complex systems are exempli>ed by the human heart, human brain and evolution. All these sample problems have been studied by various methods. What I wish to discuss in this talk is a universal approach to the treatment of all of them. The basic idea ∗
Fax: +1-541-346-5217. E-mail address: [email protected] (R.C. Hwa).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.02.018
2
R.C. Hwa / Physica A 338 (2004) 1 – 6
is to examine the (uctuations of spatial patterns even for problems where temporal evolution seems more relevant than spatial distribution. Classical nonlinear dynamics that is deterministic has many familiar features, none of which are shared by small quantum systems like quarks and gluons. The dynamics of quarks and gluons is nonlinear, since the color >elds are non-Abelian and self-coupled. It is natural to ask whether the QCD dynamics is chaotic. But the conventional criteria for chaoticity, such as the positivity of the Lyapunov exponent, are useless here, since the notion of distance between nearby trajectories is meaningless for elementary quanta. If a criterion for chaos is devised for the partons (a collective term for quarks and gluons), such a criterion must also be applicable to the logistic map so that the terminology of chaoticity has a common meaning in both problems. Let us then be speci>c in our consideration of the similarities and di7erences between parton jet and the logistic map. A high-momentum quark can initiate a parton shower by successive gluon bremsstrahlung and qqE pair production. Being quantum in nature the evolution of the process cannot be traced in space or time, and the >nal products di7er from event to event even in the total number of degrees of freedom. What can be measured experimentally is the momenta of all the particles produced in a jet (actually only the charged particle). In the momentum space those particles specify a (spatial) pattern. The (uctuation of those patterns, given the same initial condition, is what we can quantify in terms of a calculable measure. The question is whether the same measure can be applied to the logistic map, which is a simple 1D problem whose time evolution is realized by repeated iteration. Since the iteration generates a series {x0 ; x1 ; : : : ; x n ; : : : ; } in space, we can extract a subseries after the exponential separation from a nearby trajectory is over. Such a subseries of a manageable length, like 30, is a spatial pattern that will di7er from the spatial pattern of another subseries initiated from a di7erent starting point, however close to x0 . Thus in both problems we have a collection of spatial patterns that (uctuate from one to another. What we need is to >nd a good measure to quantify the behaviors of those (uctuations. A pattern can always be described at any reasonable resolution, but the highest resolution is not always most desirable because too much information makes it diFcult to compare one pattern with another. In fact, if there is no speci>c scale in a problem, it is the scaling behavior that is more economical to extract. Toward that end we divide a space into many bins of equal size and count the number n of points of the pattern in a bin. De>ne the normalized factorial moment of order q Fq =
n(n − 1) · · · (n − q + 1) ; nq
(1)
where the averages are performed over all the bins of a given size . It has been shown that Fq has the virtue of >ltering out the statistical (uctuations provided that n is not too small [1]. For every pattern in any dimensional space we can determine Fq () at a resolution speci>ed by bins of size . The behavior of how the patterns (uctuate from event to event is then described by the distribution of Fq . Here, an event can correspond to a parton jet or a logistic series starting from a speci>c initial point. A large collection of events in the case of the logistic map can be just the trajectories that all start from a small neighborhood of x0 . In the case of parton jets they may
R.C. Hwa / Physica A 338 (2004) 1 – 6
3
all have precisely the same initial energy, since quantum (uctuations are suFcient to result in a wide range of possible >nal states. How to go from a distribution of Fq for a collection of patterns to a useful measure of the (uctuations of Fq involves some technicalities that are not necessary for us to delve into here. Interested readers can >nd them in Refs. [2,3]. They can be summarized by a few key equations, which will not be explained. De>ne q = Fq =Fq ;
q = q ln q ;
(2)
where · · · here denotes the average over events. Since these quantities depend on the bin size , one examines the dependence of q (). The entropy index is de>ned by q = −9 q =9 ln :
(3)
If a linear dependence of q on ln cannot be found, a more general form of q can be considered [3]. The index q is our measure of the (uctuation property of the spatial patterns. For the logistic map the index 2 has been calculated and is found to agree well with the Lyapunov exponents for values of the control parameter r where ¿ 0. For
¡ 0 the value of 2 is zero, since 2 cannot be negative. The same type of result is obtained by the Lorenz attractor. Thus we have proven that 2 ¿ 0 is also a criterion for chaotic behavior [3]. With this result we can now apply the method to the parton jets whose chaoticity can then be similarly determined. We >nd that q for quark jets are always larger than those for gluon jets at any coupling strength S [2]. If we vary S by hand in an attempt to >nd the threshold for the onset of chaos, we discover that q increase with decreasing S . The implication is that there is no threshold and that parton jets are always chaotic with quark jets being more so than gluon jets. With that result we come to the conclusion that the concept of chaoticity is not very useful. One can only say that large q means large (uctuations in the >nal states from event to event. If indeed q quanti>es spatial (uctuations, then surely the measure should be applied to the study of the Ising model in 2D, for which the (uctuation of spatial patterns is the hallmark of the critical phenomenon. That application has been carried out as an example of how the method can be useful in the study of large systems exhibiting critical behavior [3]. We have found that 2 has a sharp peak as a function of temperature, situated at Tc . More recently, the same approach has been applied to the study of heavy-ion collisions at relativistically high energies. Those are large systems by the standard of individual partons or even nucleons. The goal of heavy-ion collisions is to create quark–gluon plasma, for which many signatures have been proposed [4]. We have emphasized the phase transition from the plasma phase to the hadron phase near the end of the evolution of the many-parton system. The observables proposed are variations of the entropy index; they are focused on the voids and gaps in the momentum space of the particles detected. Limitation of space here prevents us from any discussion of the subject. Interested readers are referred to Ref. [5]. The studies of the (uctuations of spatial patterns described so far above have been done by computer simulations. It is also possible to investigate analytically certain similar problems in phase transition and determine the corresponding scaling exponents.
4
R.C. Hwa / Physica A 338 (2004) 1 – 6
One problem, in particular, is the critical behavior of systems that can be described by the Ginzburg–Landau theory. If, as in quark-hadron phase transition, the temperature is not a quantity that can be directly measured, we use the (uctuation of spatial pattern as a manifestation of the critical behavior that can be calculated. We consider the normalized factorial moments Fq as de>ned in Eq. (1) but with the averages performed over the distribution 1 DPn [] exp − d d r F[] ; Pn = (4) Z where Z is the integral above without Pn [], and is the order parameter in terms of which the free energy has the Ginzburg–Landau form F[] = a|(r)|2 + b|(r)|4
(5)
without derivative term. It turns out that Fq can be calculated analytically, and that it satis>es the scaling law [6]
Fq ˙ F2 q ;
(6)
where q = (q − 1) ;
= 1:304 :
(7)
This result is general enough so that it should be applicable to any critical system that has multiplicity n as an observable. The veri>cation of the above scaling law for heavy-ion collisions has been sti(ed by statistical (uctuations at SPS energies, and has not been carried out at RHIC. However, there is a system that can readily be applied to, and that is the photocount at the threshold of lasing in single-mode lasers. It has been known for a long time that a laser at the threshold has the dynamics of a second-order phase transition [7]. An experiment was designed to measure the (uctuation of the number of photons produced at threshold, and Eqs. (6) and (7) have been veri>ed to a high degree of accuracy [8]. Thus in a large system such as lasers the method of studying the (uctuation of spatial patterns has been found to be fruitful. Another large system that we have investigated is cryogenic turbulence. An experiment using liquid helium in a large cylinder with a larger temperature di7erence between the top and the bottom plates has shown that at high Rayleigh number there exists a rotation of the whole (uid, called mean wind [9]. The wind direction, however, can change abruptly at irregular intervals that seem random. Suspecting that it is the manifestation of a critical behavior, we analyzed the distribution of the intervals between wind reversals by use of a moment analysis developed in Ref. [5] and then compared the result with that obtained by applying the same method to the Ising system. Amazingly, we found that the results agree remarkably well [10]. Thus the wind phenomenon can be understood in the picture where the wind is the ordered motion of the whole system, while the plumes that are emitted randomly are the disordered motion. The latter can disrupt the former and cause the wind to reverse direction at intervals of all lengths as if the system is at a critical point. What makes this description plausible is the use of the moment analysis that can be applied to both the turbulence and the Ising problems, which are two systems usually studied by totally
R.C. Hwa / Physica A 338 (2004) 1 – 6
5
di7erent methods. Since the results of our analysis on the two systems are similar, and since there is no control parameter in the turbulence problem that has been tuned, we come to the conclusion that the phenomenon of wind reversal is a manifestation of self-organized criticality. Finally, for complex systems such as the human heart or the human brain we have also made progress in the use of our method, suitably adapted for individual problems. We have studied the electrocardiogram (ECG) of a person entering into cardiac >brillation. As is usual for a physicist working in the medical >eld, it is the acquisition of data that is more diFcult than the analysis. I was fortunate to have a piece of data to examine, and the result is not surprising, since one can even see visually the change of the nature of heartbeat pulses after the onset of >brillation. The point is then to >nd a good measure to quantify the change. Using wavelet decomposition an entropy-like measure is de>ned, and di7erent scaling behaviors are found for the normal and abnormal phases of the heartbeat [11]. For the problem of studying the human brain by electroencephalogram (EEG) I have a collaborator who is at a medical school and has access to the EEG data. We have made extensive analysis of the time series using detrended (uctuation analysis and found an index that can distinguish stroke subjects from normal ones [12]. Details of that work are presented in a separate talk in this conference and are described elsewhere in these proceedings. In conclusion, we remark that to have a common approach to a wide spectrum of nonlinear systems is not only interesting in its own right, but also useful in discovering the connections between di7erent problems. Such connections open the way to new possibilities of understanding certain problems from very di7erent perspectives and thereby arriving at conclusions that could not otherwise be made. The work described here was done in collaboration with many colleagues, two of whom that I am particularly grateful to are Z. Cao and Q.H. Zhang. This work was supported, in part, by the US Department of Energy under Grant no. DE-FG03-96ER40972. References [1] A. Bialas, R. Peschanski, Nuclr. Phys. B 273 (1986) 703; A. Bialas, R. Peschanski, Nuclr. Phys. B 308 (1988) 857. [2] Z. Cao, R.C. Hwa, Phys. Rev. Lett. 75 (1995) 1268; Z. Cao, R.C. Hwa, Phys. Rev. D 53 (1996) 6608; Z. Cao, R.C. Hwa, Phys. Rev. D 54 (1996) 6674. [3] Z. Cao, R.C. Hwa, Phys. Rev. E 56 (1997) 326. [4] R.C. Hwa, Quark-Gluon Plasma 2, World Scienti>c, Singapore, 1995; R.C. Hwa, X.N. Wang, Quark-Gluon Plasma 3, World Scienti>c, Singapore, 2003. [5] R.C. Hwa, Q.H. Zhang, Phys. Rev. C 62 (2000) 014003 and 054902. R.C. Hwa, Q.H. Zhang, Phys. Rev. C 64 (2001) 054904; R.C. Hwa, Q.H. Zhang, Phys. Rev. C 66 (2002) 014904. [6] R.C. Hwa, M.T. Nazirov, Phys. Rev. Lett. 69 (1992) 741. [7] R. Graham, H. Haken, Z. Phys. 237 (1970) 31; V. DiGiorgio, M.O. Scully, Phys. Rev. A 2 (1970) 1170. [8] M.R. Young, Y. Qu, S. Singh, R.C. Hwa, Opt. Comm. 105 (1994) 325.
6
R.C. Hwa / Physica A 338 (2004) 1 – 6
[9] J.J. Niemela, L. Skrbek, K.R. Sreenivasan, R.J. Donnelly, J. Fluid Mech. 449 (2001) 169; K.R. Sreenivasan, A. Bershadskii, J.J. Niemela, Phys. Rev. E 65 (2002) 056306. [10] R.C. Hwa, et al., OITS 738, in preparation. [11] R.C. Hwa, Nonlinear phenomena in Complex Systems 3 (2000) 93. [12] R.C. Hwa, T.C. Ferree, Phys. Rev. E 66 (2002) 021901; T.C. Ferree, R.C. Hwa, J. Clin. Neurophys., to be published.
www.elsevier.com/locate/physa
A universal approach to the study of nonlinear systems Rudolph C. Hwa∗ Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, OR 97403-5203, USA
Abstract A large variety of nonlinear systems have been treated by a common approach that emphasizes the (uctuation of spatial patterns. By using the same method of analysis it is possible to discuss the chaotic behaviors of quark jets and logistic map in the same language. Critical behaviors of quark-hadron phase transition in heavy-ion collisions and of photon production at the threshold of lasing can also be described by a common scaling behavior. The universal approach also makes possible an insight into the recently discovered phenomenon of wind reversal in cryogenic turbulence as a manifestation of self-organized criticality. c 2004 Elsevier B.V. All rights reserved. PACS: 02.50.−r; 87.15.Ya Keywords: Nonlinear systems; Fluctuations
There are many varieties of nonlinear systems and many methods of investigation that are appropriate for the di7erent types of problems. Let me classify the nonlinear systems into four categories: small, large, simple and complex. Small does not mean simple, and large does not mean complex. Some examples can illustrate what I mean. By small I have in mind quarks, hadrons and nuclei. Large systems can include magnetic material at phase transition, lasers and cryogenic turbulence. Simple systems can be the logistic map and Lorenz attractor that form the introductory topics to chaos. Finally, complex systems are exempli>ed by the human heart, human brain and evolution. All these sample problems have been studied by various methods. What I wish to discuss in this talk is a universal approach to the treatment of all of them. The basic idea ∗
Fax: +1-541-346-5217. E-mail address: [email protected] (R.C. Hwa).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.02.018
2
R.C. Hwa / Physica A 338 (2004) 1 – 6
is to examine the (uctuations of spatial patterns even for problems where temporal evolution seems more relevant than spatial distribution. Classical nonlinear dynamics that is deterministic has many familiar features, none of which are shared by small quantum systems like quarks and gluons. The dynamics of quarks and gluons is nonlinear, since the color >elds are non-Abelian and self-coupled. It is natural to ask whether the QCD dynamics is chaotic. But the conventional criteria for chaoticity, such as the positivity of the Lyapunov exponent, are useless here, since the notion of distance between nearby trajectories is meaningless for elementary quanta. If a criterion for chaos is devised for the partons (a collective term for quarks and gluons), such a criterion must also be applicable to the logistic map so that the terminology of chaoticity has a common meaning in both problems. Let us then be speci>c in our consideration of the similarities and di7erences between parton jet and the logistic map. A high-momentum quark can initiate a parton shower by successive gluon bremsstrahlung and qqE pair production. Being quantum in nature the evolution of the process cannot be traced in space or time, and the >nal products di7er from event to event even in the total number of degrees of freedom. What can be measured experimentally is the momenta of all the particles produced in a jet (actually only the charged particle). In the momentum space those particles specify a (spatial) pattern. The (uctuation of those patterns, given the same initial condition, is what we can quantify in terms of a calculable measure. The question is whether the same measure can be applied to the logistic map, which is a simple 1D problem whose time evolution is realized by repeated iteration. Since the iteration generates a series {x0 ; x1 ; : : : ; x n ; : : : ; } in space, we can extract a subseries after the exponential separation from a nearby trajectory is over. Such a subseries of a manageable length, like 30, is a spatial pattern that will di7er from the spatial pattern of another subseries initiated from a di7erent starting point, however close to x0 . Thus in both problems we have a collection of spatial patterns that (uctuate from one to another. What we need is to >nd a good measure to quantify the behaviors of those (uctuations. A pattern can always be described at any reasonable resolution, but the highest resolution is not always most desirable because too much information makes it diFcult to compare one pattern with another. In fact, if there is no speci>c scale in a problem, it is the scaling behavior that is more economical to extract. Toward that end we divide a space into many bins of equal size and count the number n of points of the pattern in a bin. De>ne the normalized factorial moment of order q Fq =
n(n − 1) · · · (n − q + 1) ; nq
(1)
where the averages are performed over all the bins of a given size . It has been shown that Fq has the virtue of >ltering out the statistical (uctuations provided that n is not too small [1]. For every pattern in any dimensional space we can determine Fq () at a resolution speci>ed by bins of size . The behavior of how the patterns (uctuate from event to event is then described by the distribution of Fq . Here, an event can correspond to a parton jet or a logistic series starting from a speci>c initial point. A large collection of events in the case of the logistic map can be just the trajectories that all start from a small neighborhood of x0 . In the case of parton jets they may
R.C. Hwa / Physica A 338 (2004) 1 – 6
3
all have precisely the same initial energy, since quantum (uctuations are suFcient to result in a wide range of possible >nal states. How to go from a distribution of Fq for a collection of patterns to a useful measure of the (uctuations of Fq involves some technicalities that are not necessary for us to delve into here. Interested readers can >nd them in Refs. [2,3]. They can be summarized by a few key equations, which will not be explained. De>ne q = Fq =Fq ;
q = q ln q ;
(2)
where · · · here denotes the average over events. Since these quantities depend on the bin size , one examines the dependence of q (). The entropy index is de>ned by q = −9 q =9 ln :
(3)
If a linear dependence of q on ln cannot be found, a more general form of q can be considered [3]. The index q is our measure of the (uctuation property of the spatial patterns. For the logistic map the index 2 has been calculated and is found to agree well with the Lyapunov exponents for values of the control parameter r where ¿ 0. For
¡ 0 the value of 2 is zero, since 2 cannot be negative. The same type of result is obtained by the Lorenz attractor. Thus we have proven that 2 ¿ 0 is also a criterion for chaotic behavior [3]. With this result we can now apply the method to the parton jets whose chaoticity can then be similarly determined. We >nd that q for quark jets are always larger than those for gluon jets at any coupling strength S [2]. If we vary S by hand in an attempt to >nd the threshold for the onset of chaos, we discover that q increase with decreasing S . The implication is that there is no threshold and that parton jets are always chaotic with quark jets being more so than gluon jets. With that result we come to the conclusion that the concept of chaoticity is not very useful. One can only say that large q means large (uctuations in the >nal states from event to event. If indeed q quanti>es spatial (uctuations, then surely the measure should be applied to the study of the Ising model in 2D, for which the (uctuation of spatial patterns is the hallmark of the critical phenomenon. That application has been carried out as an example of how the method can be useful in the study of large systems exhibiting critical behavior [3]. We have found that 2 has a sharp peak as a function of temperature, situated at Tc . More recently, the same approach has been applied to the study of heavy-ion collisions at relativistically high energies. Those are large systems by the standard of individual partons or even nucleons. The goal of heavy-ion collisions is to create quark–gluon plasma, for which many signatures have been proposed [4]. We have emphasized the phase transition from the plasma phase to the hadron phase near the end of the evolution of the many-parton system. The observables proposed are variations of the entropy index; they are focused on the voids and gaps in the momentum space of the particles detected. Limitation of space here prevents us from any discussion of the subject. Interested readers are referred to Ref. [5]. The studies of the (uctuations of spatial patterns described so far above have been done by computer simulations. It is also possible to investigate analytically certain similar problems in phase transition and determine the corresponding scaling exponents.
4
R.C. Hwa / Physica A 338 (2004) 1 – 6
One problem, in particular, is the critical behavior of systems that can be described by the Ginzburg–Landau theory. If, as in quark-hadron phase transition, the temperature is not a quantity that can be directly measured, we use the (uctuation of spatial pattern as a manifestation of the critical behavior that can be calculated. We consider the normalized factorial moments Fq as de>ned in Eq. (1) but with the averages performed over the distribution 1 DPn [] exp − d d r F[] ; Pn = (4) Z where Z is the integral above without Pn [], and is the order parameter in terms of which the free energy has the Ginzburg–Landau form F[] = a|(r)|2 + b|(r)|4
(5)
without derivative term. It turns out that Fq can be calculated analytically, and that it satis>es the scaling law [6]
Fq ˙ F2 q ;
(6)
where q = (q − 1) ;
= 1:304 :
(7)
This result is general enough so that it should be applicable to any critical system that has multiplicity n as an observable. The veri>cation of the above scaling law for heavy-ion collisions has been sti(ed by statistical (uctuations at SPS energies, and has not been carried out at RHIC. However, there is a system that can readily be applied to, and that is the photocount at the threshold of lasing in single-mode lasers. It has been known for a long time that a laser at the threshold has the dynamics of a second-order phase transition [7]. An experiment was designed to measure the (uctuation of the number of photons produced at threshold, and Eqs. (6) and (7) have been veri>ed to a high degree of accuracy [8]. Thus in a large system such as lasers the method of studying the (uctuation of spatial patterns has been found to be fruitful. Another large system that we have investigated is cryogenic turbulence. An experiment using liquid helium in a large cylinder with a larger temperature di7erence between the top and the bottom plates has shown that at high Rayleigh number there exists a rotation of the whole (uid, called mean wind [9]. The wind direction, however, can change abruptly at irregular intervals that seem random. Suspecting that it is the manifestation of a critical behavior, we analyzed the distribution of the intervals between wind reversals by use of a moment analysis developed in Ref. [5] and then compared the result with that obtained by applying the same method to the Ising system. Amazingly, we found that the results agree remarkably well [10]. Thus the wind phenomenon can be understood in the picture where the wind is the ordered motion of the whole system, while the plumes that are emitted randomly are the disordered motion. The latter can disrupt the former and cause the wind to reverse direction at intervals of all lengths as if the system is at a critical point. What makes this description plausible is the use of the moment analysis that can be applied to both the turbulence and the Ising problems, which are two systems usually studied by totally
R.C. Hwa / Physica A 338 (2004) 1 – 6
5
di7erent methods. Since the results of our analysis on the two systems are similar, and since there is no control parameter in the turbulence problem that has been tuned, we come to the conclusion that the phenomenon of wind reversal is a manifestation of self-organized criticality. Finally, for complex systems such as the human heart or the human brain we have also made progress in the use of our method, suitably adapted for individual problems. We have studied the electrocardiogram (ECG) of a person entering into cardiac >brillation. As is usual for a physicist working in the medical >eld, it is the acquisition of data that is more diFcult than the analysis. I was fortunate to have a piece of data to examine, and the result is not surprising, since one can even see visually the change of the nature of heartbeat pulses after the onset of >brillation. The point is then to >nd a good measure to quantify the change. Using wavelet decomposition an entropy-like measure is de>ned, and di7erent scaling behaviors are found for the normal and abnormal phases of the heartbeat [11]. For the problem of studying the human brain by electroencephalogram (EEG) I have a collaborator who is at a medical school and has access to the EEG data. We have made extensive analysis of the time series using detrended (uctuation analysis and found an index that can distinguish stroke subjects from normal ones [12]. Details of that work are presented in a separate talk in this conference and are described elsewhere in these proceedings. In conclusion, we remark that to have a common approach to a wide spectrum of nonlinear systems is not only interesting in its own right, but also useful in discovering the connections between di7erent problems. Such connections open the way to new possibilities of understanding certain problems from very di7erent perspectives and thereby arriving at conclusions that could not otherwise be made. The work described here was done in collaboration with many colleagues, two of whom that I am particularly grateful to are Z. Cao and Q.H. Zhang. This work was supported, in part, by the US Department of Energy under Grant no. DE-FG03-96ER40972. References [1] A. Bialas, R. Peschanski, Nuclr. Phys. B 273 (1986) 703; A. Bialas, R. Peschanski, Nuclr. Phys. B 308 (1988) 857. [2] Z. Cao, R.C. Hwa, Phys. Rev. Lett. 75 (1995) 1268; Z. Cao, R.C. Hwa, Phys. Rev. D 53 (1996) 6608; Z. Cao, R.C. Hwa, Phys. Rev. D 54 (1996) 6674. [3] Z. Cao, R.C. Hwa, Phys. Rev. E 56 (1997) 326. [4] R.C. Hwa, Quark-Gluon Plasma 2, World Scienti>c, Singapore, 1995; R.C. Hwa, X.N. Wang, Quark-Gluon Plasma 3, World Scienti>c, Singapore, 2003. [5] R.C. Hwa, Q.H. Zhang, Phys. Rev. C 62 (2000) 014003 and 054902. R.C. Hwa, Q.H. Zhang, Phys. Rev. C 64 (2001) 054904; R.C. Hwa, Q.H. Zhang, Phys. Rev. C 66 (2002) 014904. [6] R.C. Hwa, M.T. Nazirov, Phys. Rev. Lett. 69 (1992) 741. [7] R. Graham, H. Haken, Z. Phys. 237 (1970) 31; V. DiGiorgio, M.O. Scully, Phys. Rev. A 2 (1970) 1170. [8] M.R. Young, Y. Qu, S. Singh, R.C. Hwa, Opt. Comm. 105 (1994) 325.
6
R.C. Hwa / Physica A 338 (2004) 1 – 6
[9] J.J. Niemela, L. Skrbek, K.R. Sreenivasan, R.J. Donnelly, J. Fluid Mech. 449 (2001) 169; K.R. Sreenivasan, A. Bershadskii, J.J. Niemela, Phys. Rev. E 65 (2002) 056306. [10] R.C. Hwa, et al., OITS 738, in preparation. [11] R.C. Hwa, Nonlinear phenomena in Complex Systems 3 (2000) 93. [12] R.C. Hwa, T.C. Ferree, Phys. Rev. E 66 (2002) 021901; T.C. Ferree, R.C. Hwa, J. Clin. Neurophys., to be published.