Abstract
Interrelations between dynamical and statistical laws in physics, on the
one hand, and between the classical and quantum mechanics, on the other hand,
are discussed with emphasis on the new
phenomenon of dynamical chaos.
The principal results of the studies into chaos in classical mechanics are
presented in some detail, including the strong local instability and robustness
of the motion, continuity of both the phase space as well as the motion
spectrum, and time reversibility but nonrecurrency of statistical evolution,
within the general picture of chaos as a specific case of dynamical behavior.
Analysis of the apparently very deep and challenging contradictions of this
picture with the quantum principles is given. The quantum view of dynamical
chaos, as an attempt to resolve these contradictions guided by the
correspondence principle and based upon the characteristic time scales of
quantum evolution, is explained. The picture of the quantum chaos as a new
generic dynamical phenomenon is outlined together with a few other examples
of such a chaos including linear (classical) waves and digital computer.
I conclude with discussion of the two fundamental physical problems:
the quantum measurement (–collapse), and the causality
principle which both appear to be related to the phenomenon of dynamical
chaos.
LINEAR AND NONLINEAR DYNAMICAL CHAOS^{1}^{1}1Lectures on the Intern. Summer School ”Nonlinear Dynamics and Chaos”, Ljubljana, Slovenia, 1994
[2mm] Boris Chirikov
Budker Institute of Nuclear Physics
630090 Novosibirsk, Russia
chirikov @ inp.nsk.su
[5mm]
1 General introduction: statistical properties of dynamical systems
The main purpose of my lectures is an overview of recent studies into a new phenomenon (or rather a whole new field of phenomena) known as the dynamical chaos both in classical and especially in quantum mechanics. The concept of dynamical chaos resolves (or, at least, helps to do so) the two fundamental problems in physics and, hence, in all the natural sciences:

are the dynamical and statistical laws of a different nature or one of them, and which one, follows from the other;

are the classical and quantum mechanics of a different nature or the latter is the most universal and general theory currently available to describe the whole empirical evidence including the classical mechanics as the limiting case.
The important part of my philosophy in discussing the problem of chaos is the separation of the human from the natural following Einstein’s approach to the science: building up a model of the real world. Clearly, the human is also a part of the world, and moreover the most important one for us as human beings but not as physicists. The whole phenomenon of life is extremely specific, and one should not transfer its peculiarities into other fields of natural sciences.
A rather popular nowadays human–oriented philosophy in physics is the information–based representation of natural laws, particularly, by substituting the information for entropy (with opposite sign). In the most general way such a philosophy was recently presented by Kadomtsev [1]. Such an approach is possible, and it might be done in a selfconsistent way but one should be very careful to avoid many confusions. In my opinion, the information is an adequate conception only for the special systems which actually use and process the information like various automata both natural (living systems) as well as man–made ones. In this case the information becomes a physical notion rather than a human view of natural phenomena. The same is also true in the theory of measurement which is again a very specific physical process, the basic one in our studies of the Nature but still not a typical one for the Nature itself. This is crucially important in quantum mechanics as will be discussed in some detail below (Lectures 4 and 5).
One of the major implications from the studies in dynamical chaos is the conception of statistical laws as an intrinsic part of dynamics without any additional statistical hypotheses (for the current state of the theory see, e.g., Ref.[2] and recent Collection of papers [3] as well as the Introduction to this Collection [4]). This basic idea can be traced back to Poincare [5] and Hadamard [6], and even to Maxwell [7], the principal condition for dynamical chaos being strong local instability of motion. In this picture the statistical laws are considered as secondary with respect to more fundamental and general primary dynamical laws.
Surprisingly, the opposite is also true!
Namely, under certain conditions the dynamical laws were found to be completely contained in the statistical ones. Nowadays this is called ’synergetics’ [8] but the principal idea goes back to Jeans [9] who discovered the instability of gravitating gas (a typical example of statistical system) which is the basic mechanism for the formation of galaxies and stars in the modern cosmology, and eventually the Solar system, a classical example of dynamical system. In this case the resulting dynamical laws proved to be secondary with respect to the primary statistical laws which include the former.
Thus, the whole picture can be represented as a chain of dynamical–statistical inclusions:
Both ends of this chain, if any, remain unclear. So far the most fundamental (elementary) laws of physics seem to be dynamical (see, however, discussion of quantum measurement in Lectures 4 and 5). This is why I begin the chain (1.1) with some primary dynamical laws.
The strict inclusion on each step of the chain has a very important consequence allowing for the so–called numerical experiments, or computer simulation, of a broad range of natural processes. As a matter of fact the former (not laboratory experiments) are now the main source of new information in the studies of the secondary laws for both dynamical chaos and synergetics. This might be called the third way of cognition, in addition to laboratory experiments and theoretical analysis.
In what follows I restrict myself to the discussion of just a single ring of the chain as marked in Eq.(1.1). Here I will consider the dynamical chaos separately in classical and quantum mechanics. In the former case the chaos explains the origin and mechanism of random processes in the Nature (within the classical approximation). Moreover, that deterministic randomness may occur (and is typical as a matter of fact) even in minimal number of freedoms (for Hamiltonian systems), thus enormously expanding the domain for application of the powerful methods of statistical analysis. The latter provides a rather simple (see, however, Lecture 3) description of the essential features for the otherwise highly intricate dynamical motion.
In quantum mechanics the whole situation is much more tricky, and still remains rather controversial. Here we encounter an intricate tangle of various apparent contradictions between the correspondence principle, classical chaotic behavior, and the very foundations of quantum physics. This will be the main topic of my discussions in Lecture 4.
One way to untangle this tangle is the new general conception  pseudochaos, of which the quantum chaos is the most important example. Another interesting example is the digital computer, also very important in view of broad application of numerical experiments in the studies of dynamical systems. On the other hand, the pseudochaos in computer will hopefully help to understand quantum pseudochaos and to accept it as a sort of chaos rather than of a regular motion as many researchers, even in this field, still do believe.
The new and surprising phenomenon of dynamical chaos, especially in quantum mechanics, holds out new hopes for eventually solving some old, long–standing, fundamental problems in physics. In Lecture 5 I will briefly discuss two of them:

causality principle (time ordering of cause and effect), and

–collapse in the quantum measurement.
The conception of dynamical chaos I am going to present here, which is not common as yet, was the result of the long–term Siberian–Italian (SI) collaboration including Giulio Casati and Italo Guarneri (Como), and Felix Izrailev and Dima Shepelyansky (Novosibirsk) with whom I share the responsibility for our joint scientific results and their conceptual interpretation.
2 Chaos in classical mechanics: dynamical complexity
The classical dynamical chaos, as a part of classical mechanics, was historically the first to have been studied simply because in the time of Boltzmann, Maxwell, Poincare and other founders of statistical mechanics the quantum mechanics did not exist. No doubt, the general mathematical theory of dynamical systems, including the ergodic theory as its modern part describing various statistical properties of the motion, has arisen from (and is still conceptually based on) the classical mechanics [10]. Yet, upon construction, it is not necessarily restricted to the latter and can be applied to a much broader class of dynamical phenomena, for example, in the quantum mechanics (Lecture 4).
2.1 Dynamical systems
In classical mechanics dynamical system means an object whose motion in some dynamical space is completely determined by a given interaction and the initial conditions. Hence, a synonym deterministic system. The motion of such a system can be described in two seemingly different ways which, however, prove to be essentially equivalent.
The first one are the motion equations of the form
which always have a unique solution
Here is a finite–dimensional vector in the dynamical space and the initial conditions (). A possible explicit time–dependence in r.h.s. of Eq.(2.1) is assumed to be regular, e.g., periodic one or, at least, that with discrete spectrum.
The most important feature of dynamical systems is the absence of any random parameters or any noise in the motion equations. Particularly, for this reason I will consider a special class of dynamical systems, the so–called Hamiltonian (nondissipative) systems, which are most fundamental in physics.
Dissipative systems, being very important in many applications, are neither fundamental (because the dissipation is introduced via a crude approximation of the very complicated interaction with some ’heat bath’) nor purely dynamical in view of principally inevitable random noise in the heat bath (fluctuation–dissipation theorem). In a more accurate and natural way the dissipative systems can be described in the frames of the secondary dynamics ( inclusion in Eq.(1.1)) when both dissipation and fluctuations are present from the beginning in the primary statistical laws.
A purely dynamical system is necessarily the closed one which is the main object in the fundamental physics. Thus, any coupling to the environment is completely neglected. I will come back to this important question below.
In Hamiltonian mechanics the dynamical space, called phase space, is even–dimensional one composed of pairs of canonically conjugated ’coordinates’ and ’momenta’, each pair corresponding to one freedom of motion.
In the problem of dynamical chaos the initial conditions play a special role: they completely determine a particular trajectory, for a given interaction, or a particular realization of dynamical process which may happen to be a very specific, nontypical, one. To get rid of such singularities another description is useful, namely, the Liouville partial differential equation for the phase space density, or distribution function :
with the solution
Here is linear differential operator, and the initial density. For any smooth this description provides the generic behavior of dynamical system via a continuum of trajectories. In special case the density describes a single trajectory like the motion equations (2.1). Notice that even in this limiting case Eq.(2.3) is linear with respect to the dynamical variable .
In any case the phase space itself is assumed to be continuous which is the most important feature of the classical picture of motion and the main obstacle in the understanding of quantum chaos.
2.2 Dynamical chaos
Dynamical chaos can be characterized in terms of both the individual trajectories and the trajectory ensembles, or phase density. Almost all trajectories of a chaotic system are in a sense most complicated (unpredictable from observation of any preceding motion). Exceptional, e.g., periodic trajectories form a set of zero invariant measure, yet it might be everywhere dense.
An appropriate notion in the theory of chaos is symbolic trajectory first introduced by Hadamard [6]. The theory of symbolic dynamics was developed further in Refs.[11 – 13]. Symbolic trajectory is a projection of the true (exact) trajectory on a discrete partition of the phase space at discrete instants of time , e.g., such that fixed. In other words, to obtain a symbolic trajectory we first turn from the motion differential equations (2.1) to the difference equations over a certain time interval :
These are usually called mapping or map: . Then, while running (theoretically) exact trajectory we record each to a finite accuracy: . For a finite partition each can be chosen integer. Hence, the whole infinite symbolic trajectory
can be represented by a single number which is generally irrational, and which is some function of the exact initial conditions. The symbolic trajectory may be also called coarse–grained trajectory. I remind that the latter is a projection of (not substitution for) the exact trajectory to represent in compact form the global dynamical behavior without unimportant microdetails.
A remarkable property of chaotic dynamics is in that the set of its symbolic trajectories is complete that is it actually contains all possible sequences (2.6). Apparently, this is related to continuity of function (2.6). On the opposite, for a regular motion this function is everywhere discontinuous.
In a similar way the coarse–grained phase density is introduced, in addition to exact, or fine–grained density, which is also a projection of the latter on some partition of the phase space.
The coarse–grained density represents the global dynamical behavior, particularly, the most important process of statistical relaxation, for chaotic motion, to some steady state (statistical equilibrium) independent of initial if the steady state is stable. Otherwise, synergetics comes into play giving rise to a secondary dynamics (Lecture 1). As the relaxation is aperiodic process the spectrum of chaotic motion is continuous which is another obstacle for the theory of quantum chaos.
The relaxation is one of the characteristic properties of statistical behavior. Another one are fluctuations. Chaotic motion is a generator of noise which is purely intrinsic by definition of the dynamical system. Such a noise is a particular manifestation of the complicated dynamics as represented by the symbolic trajectories or by the difference
The relaxation , apparently asymmetric with respect to time reversal , had given rise to a long–standing misconception of the notorious time arrow. Even now some very complicated mathematical constructions are still being erected (see, e.g., Refs.[14]) in attempts to extract somehow statistical irreversibility from the reversible mechanics. This is especially surprising as such ’irreversibility’ is based on the separation of the phase density into two parts similar to Eq.(2.7). In fact, the time direction is fixed by the additional statistical condition imposed on initial which is equivalent also to the ’causality condition’ (see Lecture 5).
In the theory of dynamical chaos there is no such problem. The answer turns out to be conceptual rather than physical: one should separate two similar but different notions, reversibility and recurrency. The exact density is always time–reversible but nonrecurrent for chaotic motion that is it will never come back to the initial in both directions of time . In other words, the relaxation, also present in , is time–symmetric. The projection of , coarse–grained , which is both nonrecurrent and irreversible, emphasizes nonrecurrency of the exact solution. The apparent violation of the statistical relaxation upon time reversal, as described by the exact , represents in fact the growth of a big fluctuation which eventually will be followed by the same relaxation in the opposite direction of time. This apparently surprising symmetry of the statistical behavior was discovered long ago by Kolmogorov [15]. Another manifestation of that symmetry is the well–known principle of detailed balancing (for discussion see, e.g., Ref.[24]).
One can say that instead of imaginary time arrow there exists the process arrow pointing always to the steady state. The following simple example would help, perhaps, to overcome this conceptual difficulty. Consider the hyperbolic one–dimensional (1D) motion:
which is obviously time–reversible, yet remains unstable in both directions of time (). Besides its immediate appealing this example is closely related to the mechanism of chaos which is the motion instability. Another example of time–reversible chaos will be given in Lecture 3.
2.3 Instability and chaos: dynamical complexity
Local instability of motion responsible for a very complicated dynamical behavior is described by the linearized equations:
Here is a reference trajectory satisfying Eq.(2.1), and the deviation of a close trajectory . At average, the solution of Eq.(2.9) has a form
where is called Lyapunov’s exponent. The motion is (exponentially) unstable if . In the Hamiltonian system of freedoms there are Lyapunov’s exponents satisfying the condition . The partial sum of all positive exponents
is called (dynamical) metric entropy. Notice that it has the dimensions of frequency and characterises the instability rate.
The motion instability is only a necessary but not sufficient condition for chaos. Another important condition is boundedness of the motion, or its oscillatory (in a broad sense) character. The chaos is produced by the combination of these two conditions (also called stretching and folding). Let us again consider an elementary example of 1D map
where operation restricts (folds) to the interval (0,1). This is not a Hamiltonian system but it can be interpreted as a ’half’ of that, namely, as the dynamics of the oscillation phase. This motion is unstable with because the linearized equation is the same except taking the fractional part (). The explicit solution for both reads
The first (linearized) motion is unbounded like Hamiltonian hyperbolic motion (2.8) and perfectly regular. The second one is not only unstable but also chaotic just because of the additional operation which makes the motion bounded, and which mixes up the points within a finite interval.
The combination of two above conditions for chaos – exponential instability and boundedness – requires the motion equations to be nonlinear. In the latter example (2.12) nonlinearity is provided by the operation . However, Liouville’s Eq.(2.3) for the phase density is always linear. Hence, the local stability of that is the variation for a small deviation is described by the same Liouville’s Eq.(2.3). The motion exponential instability () results then in the contraction of the domain occupied by the initial phase density. If the simultaneous stretching in another direction is bounded (owing to nonlinearity of the motion, not Liouville’s, equation) the exponentially long domain of conserving volume fills up the whole phase space region allowed by the exact motion integrals, e.g., the whole energy surface of a conservative system. Eventually, coarse–grained density approaches a homogeneous steady state while the exact density keeps fluctuating with a characteristic wave length exponentially decreasing in time. In other words, we may say that in Liouville’s description the phase space density evolution is exponentially unstable in the wave number (vector) of rather than in itself. Notice that for a Hamiltonian system vector includes momenta as well.
We may look at the above example (2.12) from a different viewpoint. Let us express initial in the binary code as the sequence of two symbols, 0 and 1, and let us make the partition of unit interval also in two equal halves marked by the same symbols. Then, the symbolic trajectory will simply repeat that is Eq.(2.6) takes the form
It implies that, as time goes on, the global motion will eventually depend on ever diminishing details of the initial conditions. In other words, when we formally fix exact we ’supply’ the system with infinite complexity which is coming up due to the strong motion instability. Still another interpretation is in that the exact is the source of intrinsic noise amplified by the instability. For this noise to be stationary the string of digits has to be infinite which is only possible in continuous phase space.
A nontrivial part of this picture of chaos is in that the instability must be exponential while a power–law instability is insufficient for chaos. For example, linear instability () is a generic property of perfectly regular motion of the completely integrable system whose motion equations are nonlinear and, hence, whose oscillation frequencies depend on initial conditions [16, 17]. The character of motion for a faster power–law instability () is unknown.
On the other hand, the exponential instability () is not invariant with respect to the change of time variable [4] (in this respect the only invariant statistical property is ergodicity [10]). A possible resolution of this difficulty is in that the proper characteristic of motion instability, important for dynamical chaos, should be taken with respect to the oscillation phases whose dynamics determines the nature of motion. It implies that the proper time variable must go proportionally to the phases so that the oscillations become stationary [4]. A simple example is harmonic oscillation with frequency recorded at the instances of time . Then, oscillation phase obeys map (2.12) which is chaotic. Clearly, the origin of chaos here is not in the dynamical system but in the recording procedure (random ). Now, if is a parameter (linear oscillator), then the oscillation is exponentially unstable (in new time ) but only with respect to the change of parameter , not of the initial (). In a slightly ’camouflaged’ way essentially the same effect was considered in Ref.[56] with far–reaching conclusions for the quantum chaos (Lecture 4).
Rigorous results concerning the relation between instability and chaos are concentrated in the Alekseev  Brudno theorem [13] (see also Refs.[4, 18]) which states that the complexity per unit time of almost any symbolic trajectory is asymptotically equal to the metric entropy:
Here is the so–called algorithmic complexity, or in more familiar terms, the information associated with a trajectory segment of length .
The transition time from the dynamical to statistical behavior according to Eq.(2.15) depends on the partition of the phase space, namely, on the size of a cell which is inversely proportional to the biggest integer in symbolic trajectory (2.6). The transition is controlled by the randomness parameter [19]:
where is the dynamical time scale. As both we have a somewhat confusing situation, typical in the theory of dynamical chaos, when two limits do not commute:
For the left order ( first) parameter , and we have temporary determinism (), while for the right order , and we arrive at the asymptotic randomness ().
Instead of the above double limit we may consider the conditional limit
which is also a useful method in the theory of chaotic processes. Particularly, for strong dynamical correlations persist in a symbolic trajectory which allows for the prediction of trajectory from a finite–accuracy observation. This is no longer the case for when only statistical description is possible. Nevertheless, the motion equations can still be used to completely derive all the statistical properties without any ad hoc hypotheses. Here the exact trajectory does exist as well but becomes the Kantian thing–in–itself which can be only observed but neither predicted nor reproduced in any other way.
The mathematical origin of this peculiar property goes back to the famous Gödel theorem [20] which states (in modern formulation) that most theorems in a given mathematical system are unprovable, and which forms the basis of contemporary mathematical logic as well as of the algorithmic theory of dynamical systems (see Ref.[21] for detailed explanation and interesting applications of this relatively less known mathematical achievement). A particular corollary, directly related to symbolic trajectories (2.6), is that almost all real numbers are uncomputable by any finite algorithm. Besides rational numbers some irrationals like or are also known to be computable. Hence, their total complexity, e.g., is finite, and the complexity per digit is zero (cf. Eq.(2.15)).
The main object of my discussion here, as well as of the whole physics, is a closed system which requires neglecting the external perturbations. However, in case of strong motion instability this is no longer possible, at least, dynamically. What is the impact of a weak perturbation on the statistical properties of a chaotic system? The rigorous answer was given by the robustness theorem due to Anosov [22]: not only statistical properties remain unchanged but, moreover, the trajectories get only slightly deformed providing (and due to) the same strong motion instability. The explanation of this striking peculiarity is in that the trajectories are simply transposed and, moreover, the less the stronger is instability.
In conclusion let me make a very general remark, far beyond the particular problem of chaotic dynamics (see also Ref.[89]). According to the Alekseev  Brudno theorem (2.15) the source of stationary (new) information is always chaotic. Assuming farther that any creative activity, science including, supposed to be such a source we come to an interesting conclusion that any such activity has to be (partly!) chaotic. This is the creative side of the chaos.
3 Chaos in classical mechanics: statistical complexity
The theory of dynamical chaos does not need any statistical hypotheses, nor does it allow for arbitrary ones. Everything is to be deduced from the dynamical equations. Sometimes the statistical properties turn out to be quite simple and familiar [2,23]. This is usually the case if the chaotic motion is also ergodic (on the energy surface). However, quite often, and even typically for a few–freedom chaos, the phase space is divided, and the chaotic component of the motion has a very complicated structure which results in a high complexity not only of individual trajectories (Lecture 2) but also of the statistical picture of the motion. Before to proceed further let us consider a few simple examples.
3.1 Simple physical examples of dynamical chaos
In these Lectures I restrict myself to finite–dimensional systems where the peculiarities of dynamical chaos are most clear (see Lecture 4 for some brief remarks on infinite systems). Consider now a few examples of chaos in minimal dimensionality. In a conservative system of one freedom () chaos is impossible. Such a system is completely integrable since there is one motion integral, the energy, per one freedom. The motion is periodic that is perfectly regular. The solution (2.2) of motion equations (2.1) is explicitly expressed in the standard way as the integral of the Hamiltonian. Chaos requires at least two freedoms (for conservative systems). For a regular (quasiperiodic) motion two independent (commuting) and isolating (single–valued) integrals would be necessary which is not always the case. At this point I would like to mention a rather widespread confusion that any motion equations possess integrals, the initial conditions (see Eq.(2.2)). This is certainly true but those integrals are nonisolating, in fact they might be infinitely many–valued. In the latter case, the trajectory is not restricted to an invariant surface of lower dimensions, and may be even ergodic that is occupy the whole energy surface. Let me mention also that there are some minor differences between several possible definitions of the (complete) integrability. One is based on the motion integrals in some particular dynamical space. Another one (more narrow) corresponds to a stronger condition of the existing of the motion integrals in action–angle variables^{2}^{2}2I denote actions by having in mind the subsequent quantization in Lecture 4 when they become integers if . . In this case the invariant surface of the completely integrable system is an –dimensional torus. Below I will asume the latter definition of integrability.
In case of time–dependent Hamiltonian the chaos is possible even in one freedom. This is because such a system is equivalent to the conservative 2–freedom one in the extended phase space with a new Hamiltonian [2]
where , and (for periodic time–dependence of frequency ). In this case one speaks sometimes on one–and–a–half freedoms () as the time dependence is fixed.
3.1.1 Charged particle confinement in adiabatic magnetic traps
This is Budker’s problem [24, 25], very important in the studies of controlled nuclear fusion. A simple model of two freedoms (axisymmetric magnetic field) is described by the Hamiltonian:
Here magnetic field describes the motion along magnetic line, and does so across the line (a projection of Larmor’s rotation).
In Ref.[24] a slightly different model with ’potential energy’ (which is actually the transverse part of particle’s kinetic energy) was considered in detail. Here I chose model (3.2) to apply the results below to a completely different physical system.
Assume the adiabaticity parameter
where is the full particle velocity (), stands for the frequency of transverse oscillation, and is a characteristic time for crossing the magnetic field minimum at . Under this condition both actions, and , which are also adiabatic invariants, are approximately conserved. This would imply bounded –oscillations that is the confinement of a particle in magnetic trap. However, the adiabatic invariant is only an approximate motion integral, and Budker’s problem is the evaluation of a long–term variation of that, if any, which would result in a leakage of particles out of the trap.
The unperturbed (adiabatic) Hamiltonian for model (3.2) is defined by with , and reads:
Consider the case of large –oscillation, with amplitude . Then, the frequency
Hence, last expression for in Eq.(3.4), and
where the brackets denote averaging over –oscillation.
Now, the central part of the problem – evaluation of variation from the equation:
where we drop sub . This equation can be derived either via canonical transformation of the original Hamiltonian (3.2) to action–angle variables or directly from the exact motion equations.
Under adiabatic condition (3.3) variation is exponentially small in parameter . So, this part of the problem is essentially nonperturbative that is it cannot be solved using conventional perturbation techniques of the expanding in an asymptotic power series in small parameter . Instead, a new perturbation parameter should be introduced absorbing nonadiabatic exponential. To this end we integrate Eq.(3.7) over half–period of –oscillation substituting in r.h.s the unperturbed solution.
The integration is performed in the complex plane of phase
around the cut from the branch point at and to infinity. Here
, and is the phase value at . For the integral can be reduced to –function, and we obtain for the amplitude in Eq.(3.8)
This is the required perturbation parameter.
Now we can derive a map describing the particle motion over many –oscillations. Beside Eq.(3.8) we need another one for phase . From Eqs.(3.5) and (3.6) we have
where a new variable is introduced. Now, from Eqs.(3.8) and (3.11) we arrive at the map –oscillation: over half–period of
In the second equation the new value of momentum () is substituted which determines the change in phase up to the next crossing the plane where the first equation operates. As is the exact motion integral the map (3.12) is canonical, particularly preserving the phase plane area .
The map describes the global dynamics of the model and is relatively simple for further analysis both in numerical experiments as well as by means of asymptotic perturbation series in the new small parameter . It can be still simplified by linearizing the second Eq.(3.12) around a resonance at such that with any integer . Upon dropping the latter term the map is reduced to the so–called standard map which (in standard notations) reads[23]:
where , and new parameter
The term ’standard’ emphasizes a universal character of the map to which many (but, of course, not all) various physical models can be reduced as we shall see right below. Both maps, (3.12) and (3.13), can be formally considered as describing one–freedom system driven by the periodic external perturbation in the form of short –pulses. Hence, a nickname ’kicked rotator’ for model (3.13). Yet, contrary to a common belief, the map can describe also a conservative system as is the case in our example. Then, it is called the Poincare map [2].
Unlike global map (3.12) the standard map describes the dynamics locally in momentum, e.g., for Eq.(3.12). This dynamics is determined by a single parameter
The latter inequality determines the region of chaotic motion in parameter for the standard map, and that in phase space for model (3.2) [23]. In the latter case the chaos condition becomes:
where is the so–called pitch–angle at . The second inequality (3.16) determines chaotic cone in particle’s velocity space. Thus, the motion in this model has always a chaotic component which, however, is never ergodic on the energy surface . The chaotic component is bounded by the chaos border at .
All particles within the chaos cone will be eventually lost diffusing to smaller which correspond to large amplitudes . The diffusion rate in per map’s iteration is obtained from Eq.(3.13):
Particle’s life time within the cone can be roughly estimated in the number of –oscillations as
It is fairly long for big . Besides, most particles are in stable region (3.16) and are confined there forever. So, Budker’s adiabatic magnetic trap turns out to be a very good confinement device indeed (at least for a single particle!).
A peculiar feature of model (3.2) is ’open’ (infinite) energy surfaces ( if ). Moreover, ergodic (microcanonical) measure of energy surface is also infinite. It is defined by the integral
where is a particular value of energy, and stands for the element of the full phase space. Using and integrating over phases and energy we obtain
which diverges as .
Notice that ergodic measure is proportional to measure for both maps, global (3.12) and local (3.13). This ensures a correct description of the global dynamics by the map. If we would change dynamical variables, e.g. , it were no longer the case, and only local map could be used. For this reason the special (’preferable’) variables ( in our example) are called ergodic variables [24].
Generally, the description in discrete time (map, or difference equations) and in continuous time (differential motion equations) is not completely identical just because of a different time variable. An interesting and instructive example is Lyapunov’s exponent (Lecture 2) in the model under consideration. For the standard map it depends (as anything else) on the single parameter [23]:
However, for the global map (3.12) local depends on momentum (3.15) and must be averaged over the whole chaotic component:
Thus, Lyapunov’s exponent for the map (per iteration) diverges as does the measure of the chaotic component (3.20).
The result drastically changes in continuous time. Now, we must divide local (3.21) by the half–period of –oscillation . We have
where is some finite constant. Thus, Lyapunov’s exponent per unit continuous time is zero ! This qualitatively different result seems to imply violation of the main condition for chaos (Lecture 2). Resolution of the apparent contradiction is in that for any finite time Lyapunov’s exponent remains finite, and the motion is still chaotic but, apparently, with some unusual statistical properties.
3.1.2 Internal dynamics of the Yang  Mills (gauge) fields in classical approximation
Surprisingly, this Matinyan’s problem[26] for a completely different physical system can be also represented by Hamiltonian (3.2) with symmetrized ’potential energy’:
The dynamics is always chaotic with divided phase space similar to model (3.2) [27]. Model (3.24) describes the so–called massive gauge field that is one with the quanta of nonzero mass (in the classical limit!).
The massless field corresponds to the ’potential energy’
and looks ergodic in numerical experiments. This model can be analysed as a limiting case of Budker’s model (3.2) with additional parameter in the potential energy
To this end we change variables: , which brings the new Hamiltonian into the form of old one (3.2), and we can use the results above. Particularly, adiabaticity parameter (3.3) decreases with for a given energy . Hence, the chaos cone (3.16) rapidly expands covering eventually the whole velocity space in agreement with numerical experiments [26, 27].
For any finite the energy surfaces are also open and infinite in measure, and Lyapunov’s exponent takes the opposite limits in discrete and in continuous time. This is not the case for massive field (3.24). Here the energy surfaces are closed and finite while Lyapunov’s exponent does not qualitatively depend on the time variable albeit it may have different values in both cases which is not important for the nature of the motion.
3.1.3 Perturbed Kepler motion
This is a particular case of the famous 3–body problem. Now we understand why it has not been solved since Newton: the chaos is generally present in such a system. One particular example is the motion of comet Halley perturbed by Jupiter which was found to be chaotic with estimated life time in the Solar system of the order of 10 Myrs [28], and with very complicated divided phase space.
The simplest model is described by a global map similar to (3.12):
where is comet’s total energy, and stands for Jupiter’s phase (angle) on its round orbit of unit radius (and unit velocity) at the moment when comet is in perihelion where the perturbation effect is the strongest. Function , where is Jupiter’s orbital frequency, is the Kepler law. In our units but we will keep it in the expressions for the next example. Perturbation parameter is essentially determined by the ratio of Jupiter and the Sun masses (). Actually, it is somewhat larger because of close encounters between Jupiter and the comet depending on the relative position of their orbits. For simplicity we assume in Eq.(3.27) like in Eq.(3.12) albeit the actual dependence is somewhat different owing to the same close encounters. A relatively weak perturbation by Saturn was also found to be important for global dynamics of comet Halley.
Because of negligible comet mass the perturbation from Jupiter is fixed, which corresponds to a time–dependent Hamiltonian, and phase is simply proportional to time. In such a case the ergodic variable was shown to be the energy (, canonically conjugated to map’s phase ) rather than comet’s action in continuous time [29].
The stability parameter of the local (standard) map for model (3.27) reads:
Chaotic component corresponds to higher energies and goes up to when the comet will leave out (or was captured into) the Solar system. In the latter case the whole motion (capture – diffusion – ejection) is a sort of delayed (on the diffusion stage) scattering of comet by the Solar system.
From Eq.(3.28) the chaos border for comet Halley is roughly at or, in frequency, . The actual comet frequency is now which is close to chaos border where the structure of phase space is very complicated, with many stable domains of various size.
Detailed studies [28] have shown that current is only 5% apart from the border of a big stable region. Additional perturbations, including ones of unknown nature, both in future as well as in the past could change the character of comet’s motion from chaotic to regular and vice versa. Neglecting this possibility, the comet life time in the Solar system can be roughly estimated from inhomogeneous diffusion equation (see Eq.(3.27)):
which gives
This is essentially less than the result from computer simulation of map (3.27): . The difference is explained by an anomalously slow diffusion near the chaos border.
Another example of the perturbed Kepler dynamics is a new, diffusive, mechanism for ionization of the Rydberg (highly excited) Hydrogen atom in the external monochromatic electric field. It had been discovered in laboratory experiments [30], and was explained by the dynamical chaos in classical approximation [31]. In this system a given field plays a role of the third body. The simplest model of the diffusive photoelectric effect has 1.5 freedoms, and is described by exactly the same global map (3.27) [32], now with but, of course, with different perturbation parameter
where is field strength, and we use now atomic units: . Of course, this is essentially quantum problem but for a large quantum number (electron’s action variable) the classical approximation proved to be fairly good [31]. We will come back to the quantum effects in this system in Lecture 4. I remind that in variable energy and Kepler frequency .
Stability parameter
is expressed here in dimensionless variables and which are reduced to the values of the corresponding atomic quantities at energy level (action) . For given field strength and frequency parameter increases with . Hence, in the chaotic component, the electron is diffusing up to eventual ionization. If the critical field that is much less than the atomic field. In the interval the field frequency may be considerably lower than that required for the conventional (one–photon) ionization while the ionization rate is much higher provided chaos condition (3.32).
3.1.4 Billiards and cavities
In a (non–dissipative) billiard of, at least, two dimensions the ball motion is chaotic for almost any shape of the boundary except special cases like circle, ellipse, rectangle and some others (see, e.g., Refs.[2, 10]). However, the ergodicity (on the energy surface) is only known for singular boundaries (of a singly–connected region). If the boundary is smooth enough the structure of motion becomes a very complicated admixture of chaotic and regular domains of various size. In the latter case the description via global and local maps of the kind considered above is very useful (see, e.g., Refs.[2, 33] and below).
Another view of a billiard model is the wave cavity in the limit of geometric optics. This provides a helpful bridge between classical and quantum chaos.
Generally, the mechanism of exponential instability in billiards is related to the particle scattering from a convex (towards the particle) boundary [34]. A simple example is the doubly–connected region with a convex internal boundary. The more important example is the collision of several convex balls within any boundary which is a classical model for the gas of molecules. The first simple estimate for the Lyapunov exponent in such a model was made already by Poincare [5]:
where is the mean distance between the balls, and the ball velocity and radius, respectively. The maximal instability rate is reached at .
Surprisingly, a concave boundary may also cause the instability if its curvature is large enough [35]. This is explained by the so–called ’overfocusing’: first, close trajectories converge upon reflection from the boundary but later, after passing the focus, they eventually diverge. A well studied example is the ’stadium’, the planar billiard with the boundary composed of two semicircles connected by two straight lines. For any nonzero length of the latter the ball motion is not only chaotic but also ergodic.
Here we consider two examples of chaotic billiards with a moving boundary. In this case the chaos is possible already in one freedom that is for the ball motion along a straight line. One example is Ulam’s model (see Ref.[2]) for the mechanism of cosmic rays acceleration proposed by Fermi [36]. The Fermi model was a ’gas’ of huge magnetic clouds in cosmic space and the protons. In the steady state the mean energy of both must be equal which would imply an enormous acceleration of protons. Ulam checked this idea in numerical experiments with a very simple one–freedom model: a particle between two parallel walls, apart, one of which is oscillating with a given velocity . Surprisingly, the computation showed no significant acceleration beyond wall’s velocity . This was explained in Ref.[37] using the chaos theory just developed at that time.
Under condition the particle motion is described again by the global map (3.12) in variables (particle’s velocity), (at collision time), and with . Hence,
and the chaotic component is bounded from above, indeed:
Acceleration turns out to be the bigger the smaller the amplitude of the wall oscillation! It was a surprising result which would be difficult to imagine without a theory. Of course, in the original Fermi model there was no such restriction since the cloud motion was assumed to be random which has been later confirmed by the chaos theory for the model of gas mentioned above.
Dynamical variables are not ergodic. Still, the local map can be used to evaluate the conditions for chaos. If we would change the velocity to energy the map were no longer canonical, and a more complicated map had to be constructed. It was also assumed that the wall has infinite mass, so that its motion is fixed. We may lift this condition to study the ergodicity of the whole system [38]. Assume that the wall with a finite mass (particle’s mass) is a linear oscillator of frequency . From the energy conservation and condition (3.34) we can derive the chaos border on energy surface in the form:
where may be called the ergodicity parameter. Chaotic component corresponds to and increases with . Yet, the motion is never completely ergodic. The measure of chaotic component can be evaluated using the arguments applied above to Eq.(3.19). Since now the wall frequency is fixed (in continuous time). Chaos is restricted to where is the border value. Hence, the relative measure of chaotic component
where the latter expression corresponds to big . For given parameters of the model the essential ergodicity is achieved in the low energy limit only.
Another version of Ulam’s model was studied in Ref.[39] (see also Ref.[2]). The new model is the ’open’ billiard with only a single oscillating wall in the homogeneous field which brings the particle back to the wall. The only difference in the global map is the phase shift between collisions: where is particle’s acceleration in the field. Then,
is independent of , and the chaotic acceleration becomes unbounded.
3.1.5 Reversible chaos in magnetic field
Magnetic lines can be formally considered as the ’trajectories’ of some dynamical system, the distance along a line playing a role of ’time’. Owing to Maxwell’s equation div the line dynamics is Hamiltonian. Consider a toroidal magnetic field which is used in magnetic traps, like stellarator or tokamak, for plasma confinement [40].
Three–dimensional magnetic lines have freedoms corresponding in the latter example to a one–freedom oscillation (line’s rotation in the plane transverse to the torus closed axis) driven by the external perturbation due to the variation of magnetic field in (along the axis). The transverse surface plays here a role of the ’phase plane’ for the line oscillation which is generally nonlinear that is with the frequency depending on the initial conditions (a distance from the axis).
Under certain conditions the lines become chaotic [41] which is called the ’braided’ magnetic field. Particularly, the lines are ’diffusing’:
where is the dynamical scale (2.16), and the Lyapunov exponent for magnetic lines (per unit length). Notice that here is not restricted by the torus circumference. Instead, , and line’s diffusion is only bounded by a chaos border at large , e.g., near the current wires producing the magnetic field.
In sufficiently strong electron’s Larmor radius is negligibly small, and the electron follows a magnetic line: where is the longitudinal velocity. Hence, the electron is also diffusing:
and it will be evetually lost.
Now, consider the impact of electron’s collisions with other particles in plasma [42]. For a small Larmor radius the main collision effect would be the electron velocity reversal () which is equivalent to the time reversal for magnetic lines (). Neglecting again a finite Larmor radius, the electron will follow back the same line. If the time reversals were periodic so would be the electron motion as well, and the diffusion were completely stopped (in this approximation). However, the collisional time reversals is a random process. Hence, the ’time’ spread
where is the mean scattering length, would itself grow only diffusively. This implies an anomalously slow electron diffusion (cf. Eq.(3.39)):
Various perturbations destroy the exact reversibility of the electron motion. Let us consider the impact of a finite Larmor radius . Then, the deviation would grow exponentially up to
at the next collision. The latter inequality is the condition for exponential, rather than diffusive, divergence of trajectories. This is to be compared with the collisionfree diffusion (3.40) for :
The minimum is reached at if to satisfy inequality in Eq.(3.43).
The strong diffusion suppression is a striking manifestation of the time reversibility in dynamical chaos. Notice that a finite residual electron diffusion is the result of a partial reversal of its velocity ( only).
3.2 Critical phenomena in dynamics
The examples considered above suggest that a few–freedom chaotic dynamical system has typically the divided phase space with many chaos borders. Each of those is characterized by the so–called critical structure [43] which is a hierarchy of chaotic and regular domains on ever decreasing spatial and frequency scales. This makes statistical description a very difficult problem. Particularly, any averaging has to be done over the chaotic component of the motion whose measure is no longer simple Hamiltonian (3.19) as for ergodic motion. Nevertheless, the critical structure can be universally described in terms of renormalization group which proved to be so efficient in other branches of theoretical physics. In turn, such a renormgroup may be considered as an abstract dynamical system which describes the variation of the whole motion structure, for the original dynamical system, in dependence of its spatial and temporal scale. Logarithm of the latter plays a role of ’time’ (renormtime) in that renormdynamics. At the chaos border the latter is determined by the motion frequencies. The simplest renormdynamics is a periodic variation of the structure or, for a renorm–map, the invariance of the structure with respect to the scale [44]. Surprisingly, this scale invariance includes the chaotic trajectories as well. The opposite limit – renormchaos – is also possible, and was found in several models (see Ref.[43]). Remarkably, for a two–dimensional map, which also may describe the two–freedom conservative system, an extremely complicated renormdynamics can be reduced to a most simple one–dimensional map
where is the so–called rotation number that is the ratio of the two motion frequencies [43]. This map was introduced by Gauss in the number theory and has been well studied by now [10]. Particularly, the Lyapunov exponent (per iteration) , and almost any intial generates a random trajectory which corresponds to random fluctuations of the motion structure from one scale to the next. Exceptional rationals give rise to a periodic oscillation of the structure and, hence, to scale invariance in steps.
Even though the critical structure occupies a very narrow strip along the chaos border it may qualitatively change the statistical properties of the whole chaotic component. This is because a chaotic trajectory unavoidably enters from time to time the critical region and ’sticks’ there for a time the longer the closer it comes to the chaos border. The sticking results in a slow power–law, rather than exponential, correlation decay for large time:
Moreover, exponent , and for the two–dimensional map was found numerically to be approximately in agreement with a simple theoretical analysis [43]. In higher dimension the dependence
was conjectured based on the same physical theory. Here is the number of linearly independent (incommensurate) frequencies both internal (unperturbed) and driving.
Slow decaying correlation (3.46) implies a singular power spectrum which is the Fourier transform of :
As the spectrum approaches that of ’mysterious’ noise (see, e.g., Ref.[45]). In the minimal dimension () the singular spectrum is .
The diffusion determined by correlation (3.46) turns out to be anomalously fast [46] as the standard diffusion rate
diverges for . In such a case the dispersion (second moment of the distribution function, e.g., in the example below) is given by a double integral of correlation or by the differential equation
which can be applied to a more general problem [47]. A particular example is the standard map (3.13) (in variables ) for special values of parameter with any integer [23]. In this case there are two fixed points satisfying , and momentum growing proportionally to (discrete) time. The fixed points are stable but relative area of both stable domains around is rather small: and rapidly decreases with . The biggest one is only. Within a stable region the particle is accelerating independent of initial conditions. This is how the so–called microtron works, the first cyclic accelerator for relativistic electrons proposed by Veksler [48].
More interesting is the behavior of a chaotic trajectory. From time to time it approaches the chaos border of a tiny stable domain and sticks there for a while being accelerated much more rapidly than in the rest (!) of the chaotic component. Since there are two stable domains with opposite acceleration the resulting motion would be also diffusive but anomalously fast: for the average increase in momentum becomes with [46]. A more accurate calculation leads to the relation [25]:
Here is map’s discrete time, and is taken from numerical experiments [49] where such enhanced diffusion was observed for the first time. Actually, the normal diffusion rate was measured and found to be times (!) larger than expected . Remarkably, the rate of anomalous diffusion (3.51) does not depend on stable area . Yet, the crossover time from normal to anomalous diffusion does so.
In higher dimension for , and . This is the fastest homogeneous diffusion possible. The motion would be close to the straight acceleration but in both directions of variation!
4 Quantum pseudochaos
The mathematical theory of dynamical chaos – ergodic theory – is selfconsistent. However, this is not the case for the physical theory unless we accept the philosophy of the two separate mechanics, classical and quantum. Even though such a view cannot be excluded at the moment it has a profound difficulty concerning the border between the two. Nor is it necessary according to recent intensive studies of quantum dynamics. Then, we have to understand the mechanics of dynamical chaos from the quantum point of view. Our guiding star will be the correspondence principle which requires the complete quantum theory for any classical phenomenon, in the quasiclassical limit, assuming that the whole classical mechanics is but a special part (the limiting case) of currently most general and fundamental physical theory, the quantum mechanics. Now it would be more correct to speak about the quantum field theory but here I restrict myself to finite–dimensional systems only.
4.1 The correspondence principle
In attempts to build up the quantum theory of dynamical chaos we immediately encounter a number of apparently very deep contradictions between the well established properties of classical dynamical chaos and the most fundamental principles of quantum mechanics.
To begin with, the quantum mechanics is commonly understood as a fundamentally statistical theory which seems to imply always some quantum chaos, independent of the behavior in the classical limit. This is certainly true but in some restricted sense only. A novel development here is the isolation of this fundamental quantum randomness as solely the characteristic of a very specific quantum process, the measurement, and even as the particular part of that  the so–called –collapse which, indeed, has so far no dynamical description.
No doubt, the quantum measurement is absolutely necessary for the study of microworld by us, the macroscopic human beings. Yet, the measurement is, in a sense, foreign to the proper microworld which might (and should) be described separately from the former. Explicitly[4] or, more often, implicitly such a philosophy has become by now common in the studies of chaos but not yet beyond this field of research (see, e.g., Ref.[50]).
This approach allows us to single out the dynamical part of quantum mechanics as represented by a specific dynamical variable in the Hilbert space satisfying some deterministic equation of motion, e.g., the Schrödinger equation. The more difficult and vague statistical part is left behind for a better time. Thus, we temporarily bypass (not resolve!) the first serious difficulty in the theory of quantum chaos. The separation of the first part of quantum dynamics, which is very natural from mathematical viewpoint, had been first introduced and emphasized by Schrödinger who, however, certainly underestimated the importance of the second part in physics.
However, another principal difficulty arises. As is well known, the energy (and frequency) spectrum of any quantum motion bounded in phase space is always discrete. And this is not the property of a particular equation but rather a consequence of the fundamental quantum principle  the discreteness of phase space itself, or in a more formal language, the noncommutative geometry of quantum phase space. Indeed, according to another fundamental quantum principle – the uncertainty principle – a single quantum state cannot occupy the phase space volume (in what follows I set ). Hence, the motion bounded in a domain of volume is represented by eigenstates, the property even stronger than the general discrete spectrum (almost periodic motion).
According to the existing ergodic theory such a motion is considered to be regular which is something opposite to the known chaotic motion with continuous spectrum and exponential instability, again independent of the classical behavior. This seems to never imply any chaos or, to be more precise, any classical–like chaos as defined in the ergodic theory. Meanwhile, the correspondence principle requires conditional chaos related to the nature of motion in the classical limit.
4.2 Pseudochaos
Now the principal question to be answered reads: where is the expected quantum chaos in the ergodic theory? Our answer to this question[51] (not commonly accepted as yet) was concluded from a simple observation (principally well known but never comprehended enough) that the sharp border between the discrete and continuous spectrum is physically meaningful in the limit only, the condition actually assumed in the ergodic theory. Hence, to understand the quantum chaos the existing ergodic theory needs some modification by introducing a new ’dimension’, the time. In other words, a new and central problem in the ergodic theory becomes the finite–time statistical properties of a dynamical system, both quantum as well as classical.
Within a finite time the discrete spectrum is dynamically equivalent to the continuous one, thus providing much stronger statistical properties of the motion than it was (and still is) expected in the ergodic theory in case of discrete spectrum. In short, the motion with discrete spectrum may exhibit all the statistical properties of the classical chaos but only on some finite time scales.
A simple example of (classical) pseudochaos is a symbolic trajectory (Lecture 2) of some period composed of random elements whatever the origin of the randomness. In any event, most finite sequences are random, indeed, according to the algorithmic theory of dynamical systems [21]. In this example there is a single time scale () for all statistical properties while, generally, there are several different scales related to a particular property (see below).
The conception of time scale is a fundamental one in our theory of quantum chaos [51]. This is certainly a new dynamical phenomenon, related but not identical at all to the classical dynamical chaos.^{3}^{3}3There are very special, even exotic I would say, examples of the ’true’, classical–like, chaos in quantum systems (see [51, 4] and references therein). In all such cases the quantum motion is not only unbounded in some phase space variables but, moreover, the latter grow exponentially in time. We call it pseudochaos, the term pseudo intending to emphasize the difference from the asymptotic (in time) chaos in the ergodic theory. Yet, from the physical point of view, we accept here, the latter, strictly speaking, does not exist in the Nature. So, in the common philosophy of the universal quantum mechanics the pseudochaos is the only true dynamical chaos (cf. the term pseudoeuclidian geometry in special relativity). The asymptotic chaos is but a limiting pattern which is, nevertheless, very important both in the theory to compare with the real chaos and in applications as a very good approximation in macroscopic domain as is the whole classical mechanics. Ford calls it mathematical chaos as contrasted to the real physical chaos in quantum mechanics [52]. Another curious but impressive term is artificial reality [53] which is, of course, a selfcontradictory notion reflecting, particularly, confusion in the interpreting such surprising phenomena like chaos.
Until recently the conception of classical dynamical chaos was completely incomprehensible, especially for physicists. One particular point of confusion was (and still remains to some extent) the Second Law of thermodynamics, the entropy increase in a closed system. Meanwhile, the entropy defined by the exact phase density is the motion integral in any Hamiltonian system. Some physicists are still reluctant to assume thermodynamic entropy determined via the coarse–grained density in which case it may well increase under conditions of dynamical chaos. From many researchers I know that they actually observed dynamical chaos in numerical or laboratory experiments but… did their best to get rid of it as some artifact, noise or other interference! Now the situation in this field is upside down: most researchers (not me!) insist that if an apparent chaos is not like that in the classical mechanics (and in the existing ergodic theory) then it is not a chaos at all. The most controversial conception in today’s disputes is just the quantum chaos. The curiosity of the current situation is that in most studies of the ’true’ (classical) chaos the digital computer is used where only pseudochaos is possible that is one like in quantum (not classical) mechanics!
The statistical properties of the discrete–spectrum motion is not a completely new subject of research, it goes back to the time of intensive studies in the mathematical foundations of statistical mechanics before the dynamical chaos was discovered or, better to say, was understood (see, e.g., Ref.[54]). We call this early stage of the theory traditional statistical mechanics (TSM). It is equally applicable to both classical as well as quantum systems. For the problem under consideration here one of the most important rigorous results with far–reaching implications was the statistical independence of oscillations with incommensurate (linearly independent) frequencies , such that the only solution of the resonance equation
in integers is for all . This is a generic property of the real numbers that is the resonant frequencies (4.1) form a set of zero Lebesgue measure. If we define now the statistical independence of means that trajectory is ergodic in –cube . This is a consequence of ergodicity of the phase trajectory in –cube .
Statistical independence is a basic property of a set to which the probability theory is to be applied. Particularly, the sum of statistically independent quantities
which is the motion with discrete spectrum, is a typical object of this theory. However, the familiar statistical properties like Gaussian fluctuations, postulated (directly or indirectly) in TSM, are reached in the limit only[54] which is called thermodynamic limit. In TSM this limit corresponds to infinite–dimensional models [10] which provide a very good approximation for macroscopic systems, both classical and quantal.
However, what is really necessary for good statistical properties of sum (4.2) is a big number of frequencies which makes the discrete spectrum continuous (in the limit). In TSM the latter condition is satisfied by setting . The same holds true for quantum fields which are infinite–dimensional. In quantum mechanics another mechanism, independent of , works in the quasiclassical region where is some big quantum parameter, e.g. quantum number, and stands for a characteristic action of the system. Indeed, if the quantum motion (4.2) (with instead of ) is determined by many () eigenstates we can set independent of . The actual number of terms in expansion (4.2) depends, of course, on a particular state under consideration. For example, if it is just an eigenstate the sum reduces to a single term. This corresponds to the special peculiar trajectories of classical chaotic motion whose total measure is zero. Similarly, in quantum mechanics for most states if the system is classically chaotic. This important condition was found to be certainly sufficient for good quantum statistical properties (see Ref.[51] and below). Whether it is also a necessary condition remains as yet unclear.
Thus, with respect to the mechanism of the quantum chaos we essentially come back to TSM with exchange of the number of freedoms for quantum parameter . However, in quantum mechanics we are not interested, unlike TSM, in the limit which is simply the classical mechanics. Here, the central problem is the statistical properties for large but finite . This problem does not exist in TSM describing macroscopic systems. Thus, with an old mechanism the new phenomena were understood in quantum mechanics.
The direct relation between these two seemingly different mechanisms of chaos can be traced back in some specific dynamical models [4]. One interesting example is the nonlinear Schrödinger equation [88]. From a physical point of view it describes the motion of a quantum system interacting with many other freedoms whose state is expressed via the function of the system itself (the so–called mean field approximation). This approximation becomes exact in the limit which is a particular case of the thermodynamic limit. Therefore, the mechanism for chaos in this system is apparently the old one. On the other hand, the nonlinear Schrödinger equation has generally exponentially unstable solutions, hence the mechanism of chaos here seems to be the new one. Thus, for this particular model both mechanisms describe the same physical process. We would like to emphasize that the ’true’ chaos present in these apparently few–dimensional models actually refers to infinite–dimensional systems.
4.3 Characteristic time scales in quantum chaos
The existing ergodic theory is asymptotic in time, and hence contains no time scales at all. There are two reasons for this. One is technical: it is much simpler to derive the asymptotic relations than to obtain rigorous finite–time estimates. Another reason is more profound. All statements in the ergodic theory hold true up to measure zero that is excluding some peculiar nongeneric sets of zero measure. Even this minimal imperfection of the theory did not seem completely satisfactory but has been ’swallowed’ eventually and is now commonly tolerated even among mathematicians to say nothing about physicists. In a finite–time theory all these exceptions acquire a small but finite measure which would be already ’unbearable’ (for mathematicians). Yet, there is a standard mathematical trick, to be discussed below, for avoiding both these difficulties.
The most important time scale in quantum chaos is given by the general estimate
where is a system–dependent parameter. This is called the relaxation time scale referring to one of the principal properties of the chaos – statistical relaxation to some steady state (statistical equilibrium). The physical meaning of this scale is principally simple, and it is directly related to the fundamental uncertainty principle () as implemented in the second Eq.(4.3) where is the full average energy level density (also called Heisenberg time). For the discrete spectrum is not resolved, and the statistical relaxation follows the classical (limiting) behavior. This is just the ’gap’ in the ergodic theory (supplemented with the additional, time, dimension) where the pseudochaos, particularly quantum chaos, dwells. A more accurate estimate relates to a part of the level density. This is the density of the so–called operative eigenstates only that is those which are actually present in a particular quantum state , and which actually control its dynamics.
The formal trick mentioned above is to consider not finite–time relations we really need in physics but rather the special conditional limit (cf. Eq.(2.18)):
Quantity is here a new rescaled time which is, of course, nonphysical but very helpful technically. The double limit (4.4) (unlike the single one ) is not the classical mechanics which holds true, in this representation, for and with respect to the statistical relaxation only. For the behavior becomes essentially quantum (even in the limit !) and is called nowadays mesoscopic phenomena. Particularly, the quantum steady state is quite different from the classical statistical equilibrium in that the former may be localized (under certain conditions) that is nonergodic in spite of classical ergodicity.
Another important difference is in fluctuations which are also a characteristic property of chaotic behavior. In comparison with classical mechanics the quantum plays, in this respect, an intermediate role between the classical trajectory (exact or symbolic) with big relative fluctuations and the coarse–grained classical phase space density with no fluctuations at all. Unlike both the fluctuations of are which is another manifestation of statistical independence, or decoherence, of even pure quantum state (4.2) in case of quantum chaos. In other words, chaotic represents statistically a finite ensemble of systems even though formally describes a single system. Quantum fluctuations clearly demonstrate also the difference between physical time and auxiliary variable : in the double limit () the fluctuations vanish, and one needs a new trick to recover them.
The popular term mesoscopic means here an intermediate behavior between classical () and quantum (e.g., localization) one. In other words, in a mesoscopic phenomenon both classical and quantum features are combined simulteneously. Again, the correspondence principle requires transition to the completely classical behavior. This is, indeed, the case according to Shnirelman’s theorem or, better to say, to a physical generalization of the theorem [55]. Namely, the mesoscopic phenomena occur in the so–called intermediate quasiclassical asymptotics where is already very big but still less than a certain critical which determines the border of transition to a fully classical behavior. The latter region, ensured by the above theorems in accordance with the correspondence principle, is called the far quasiclassical asymptotics.
The striking well known examples of mesoscopic phenomena are superconductivity and superfluidity. A mesoscopic parameter here is the temperature which determines the behavior of microparticles, electrons and atoms, respectively. The far asymptotics corresponds here to where both essentially quantum phenomena disappear.
The relaxation time scale should not be confused with the Poincare recurrence time which is typically much longer, and which sharply increases with decreasing of the recurrence domain. Time scale characterizes big fluctuations (for both the classical trajectory, but not the phase space density, and the quantum ) of which recurrences is a particular case. Unlike this characterizes the average relaxation process.
More strong statistical properties than relaxation and fluctuations are related in the ergodic theory to the exponential instability of motion. Their importance for the statistical mechanics is not completely clear. Nevertheless, in accordance with the correspondence principle, those stronger properties are also present in quantum chaos as well but on a much shorter time scale
where is classical metric entropy (2.11). This time scale was discovered and partly explained in Refs.[57] (see also Refs.[51, 4]). We call it random time scale. Indeed, according to the Ehrenfest theorem the motion of a narrow wave packet follows the beam of classical trajectories as long as the packet remains narrow, and hence it is as random as in the classical limit. Even though the random time scale is very short, it grows indefinitely as . Thus, a temporary, finite–time quantum pseudochaos turns into the classical dynamical chaos in accordance with the correspondence principle. Again, we may consider the conditional limit:
Notice that scaled time is different from in Eq.(4.4).
Particularly, if we fix time , then in the limit we obtain the transition to the classical instability in accordance with the correspondence principle while for fixed, and we have the proper quantum evolution in time. For example, the quantum Lyapunov exponent
The quantum instability () was observed in numerical experiments, indeed [58, 4]. What does terminate the instability for ? A naive explanation that the major size of the originally most narrow quantum packet reaches the full swing of a bounded motion is obviously too simplified. This is immediately clear from the comparison with the classical packet behavior (Lecture 2). Also, the quantum packet squeezing is not principally restricted since only 2–dimensional area (per freedom) is bounded from below in quantum mechanics. Instead, numerical experiments show that the original wave packet, after a considerable stretching similar to the classical one, is rapidly destroyed. Namely, it gets split into many new small packets. A possible explanation[59] (see also Ref.[4]) is related to the discreteness of the action variable in quantum mechanics which leads to the “rupture” of a very long stretched packet into many pieces. Such a mechanism determines a new destruction time scale which, for the quantized standard map (see below), is given by the estimate:
This roughly agrees with the results of numerical experiments [58, 4]. As expected (see Eq.(4.5)).
There is another mechanism which produces deviation of the quantum packet evolution from the classical motion [59]. We call it inflation because of the increase in time of the phase space area occupied by the quantum phase space density (the Wigner function ) contrary to the classical density which is conserved (Liouville’s theorem). The inflation can be analyzed using the quantum Liouville equation for the Wigner function [60]. In case of standard map this equation reduces to:
and gives the following estimate for the inflation time scale:
The inflation time is of the order of destruction time (4.8) and of the random time scale (4.5) as well which implies, particularly, a considerable squeezing of a wave packet.
An important implication of the above picture of packet’s time evolution is the rapid and complete destruction of the so–called generalized coherent states [61] in quantum chaos.
In quasiclassical region () scale (4.3). This leads to an interesting conclusion that the quantum diffusion and relaxation are dynamically stable contrary to the classical behavior. It suggests, in turn, that the motion instability is not important during statistical relaxation. However, the foregoing correlation decay on short random time scale is crucial for the statistical properties of quantum dynamics. Dynamical stability of quantum diffusion has been proved in striking numerical experiments with time reversal [65]. In a classical chaotic system the diffusion is immediately recovered due to numerical ”errors” (not random !) amplified by the local instability. On the contrary, the quantum ”antidiffusion” proceeds untill the system passes, to a very high accuracy, the initial state, and only than the normal diffusion is restored. The stability of quantun chaos on relaxation time scale is comprehensible as the random time scale is much shorter. Yet, the accuracy of the reversal (up to (!) ) is surprising. Apparently, this is explained by a relatively large size of the quantum wave packet as compared to the unavoidable roundingoff errors unlike the classical computer trajectory which is just of that size [68]. In the standard map the size of the optimal, leastspreading, wave packet [51]. On the other hand, any quantity in the computer must well exceed the rounding–off error . Particularly, , and .
4.4 Quantum localization: the kicked rotator model
The standard map (3.13) was shown in Lecture 3 to provide the local description of motion for many more realistic classical models. So, the quantized standard map seems to be a good approach in the studies of quantum chaos as well. This can be done in two ways. The first one is to derive exact unitary operator over some time interval :
where is the Hamiltonian operator. Generally, this is a very difficult mathematical problem which we will not discussed (see Ref.[62]). Instead, we consider here the second way: the direct quantization of the classical standard map (3.13) which is, of course, only approximate solution of the whole problem. I am not aware of any thorough analysis of the accuracy and limitations of this simple method. However, the direct comparison of such a quantum map with the numerical solution of Schrödinger equation for the diffusive photoeffect in Rydberg Hydrogen atom confirms that the former is a reasonable approximation, indeed [32].
Quantization of standard map with Hamiltonian
leads to the unitary operator [63]: