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For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity, we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.

The idea to model apparently disordered spectra, like those of heavy nuclei, using random matrices was sug- gested in the mid-50’s by Wigner, and then formalized in the early 60’s in the work of Dyson and Mehta [

The study of classical strongly chaotic systems (Anosov systems) has revealed that central limit theorems (CLT) hold [

Do classical fully chaotic systems also exhibit universality properties? This question was addressed by Argaman et al. [

An answer was proposed by Laprise et al. [

This article extends the results of reference [

The answers we found can be summarized as follows: For the 2D Sinai billiard and the 2D Bunimovich stadium billiard the level spacing distribution

In classical systems, chaos information is encoded in trajectories. According to the Alekseev-Brudno theorem [

This motivates us to look at the length of trajectories and its fluctuation properties. Let

denote the action over

denote the length of the trajectory

where i and j are respectively the indices of the final and initial boundary points of the trajectory. Both matrices are viable for statistical analysis of classical chaos. In the case of billiard systems, we consider trajectories where the billiard particle moves with constant velocity u and constant kinetic energy E. Then the action and the length matrix are essentially equivalent,

When solving for two boundary points

For

range

the number of trajectories and retaining only the trajectories corresponding to the starting angles

If one considers integrable quantum systems and analyzes them in terms of the NNS distribution of energy levels and spectral rigidity, then in most cases one finds a Poissonian distribution

spectral rigidity

were found in the case with weak harmonic coupling, yielding

Laprise et al. [

As example of an integrable classical billiard we consider the 2D rectangular billiard, shown in

The shape is determined by the parameters a, b, which were chosen to be

For a given pair of boundary points, we found that the behaviour of the number of trajectories versus the number of rebounds is linear (not shown). The error behaviour of trajectories as a function of the number of rebounds has been obtained by taking into account

The symmetry of the rectangular billiard is mirror symmetry under reflection about the x- and y-axes (with origin at centre of rectangle). The symmetry shows up in the shape of trajectories. For example, a trajectory (1) going from starting point

group of discrete translations,” which imply strong correlations among length matrix elements and among eigenvalues of the length matrix. We expect that this will manifest itself in the statistical behaviour in the level spacing distribution

computed the correlation coefficient to obtain

This is possibly evidence for universal behaviour in the integrable case. Comparing the behaviour of the rectangular billiard with the circular billiard [

For general closed 2D billiards, the mean free path length

where

The rule of dynamics is free motion in the interior region and elastic specular collision at the central disc and the exterior square wall. We have classified trajectories using the scheme presented in Section 2. A global

characteristic feature of chaos is encoded in the number of classical trajectories. For the Sinai billiard we found that the number

Such exponential behaviour in chaotic billiards is clearly distinct from the behaviour in integrable billiards, where the number of trajectories increases linearly with the number of rebounds (see rectangular billiard). We

fixed a value for

to all possible trajectory indices

In order to make sure that the chaotic behaviour is not due to numerical noise, we estimated the numerical error

For

We found a relative error of about

The histogram of the length matrix elements itself is shown in

Mathematical note. The BGS-conjecture does not state that the matrix elements of a quantum Hamiltonian must be distributed like a GOE ensemble. The conjecture rather only says that the statistical fluctuations of the eigenvalue spacings obtained from the quantum Hamiltonian are the same as those from a GOE ensemble, giving a Wignerian distribution. In other words, it is possible that the matrix elements of the quantum Hamil- tonian be distributed quite differently from a Gaussian and that its level spacing distribution be nevertheless Wignerian.

Such a situation, where the distribution of matrix elements is not GOE, but the level fluctuation statistics is GOE, occurs in nuclear physics. An example is the distribution of the Hamiltonian matrix elements obtained from nuclear shell model calculations [

Let us consider the 2D Bunimovich stadium billiard with semi-axes a and b (

The rule of dynamics is free motion in the interior region and elastic specular collision at the exterior square wall. For a given pair of boundary points, the number of trajectories

Such behaviour is qualitatively similar to that found in the Sinai billiard. We also measured the numerical

error following the method used in the Sinai billiard. Likewise, we found a regime of exponential behaviour followed by a regime of saturation (not shown). On average the exponential increase is given by

We have classified trajectories by the the number of rebounds

generate an ensemble of length matrices

For each trajectory index

In order to see if the observed Wigner distribution in the level spacing distribution depends on the statistical method of averaging over several trajectories (i.e. length matrices

The numerical experiments with chaotic billiards investigated above led us to the following observations: 1) For a given pair of boundary points

The leading Gaussian behaviour in the distribtion

discs. There are two types of 2D periodic Lorentz gases (or Sinai billiards on a torus): One has a finite horizon, where free paths between collisions are bounded (the scatterers are sufficiently dense to block every direction of motion. The other type has an infinite horizon, where the particle can move indefinitely without collision with

any disc. The Sinai billiard investigated above (see

Actually, for the periodic Lorentz gas with finite horizon it can be shown rigorously that the distribution of length of trajectories becomes a Gaussian distribution in the limit of many bounces. This holds when the initial points of trajectories are distributed randomly on the billiard boundary. Then Chernov and Markarian [

for all

is the time of the n-th collision.

This result seems to support our numerical findings of a (near) Gaussian distribution of length of trajectories. However, the scenario where the above mathematical result holds and the scenario of our numerical study differ in two respects, namely in the distribution of boundary points and in the horizon of billiard. For the purpose of statistical analysis in terms of RMT we are interested in the distribution

In the case of the stadium billiard, Bálint and Gouëzel [

Let us suppose that

Now we want to address the following questions:

1) Concerning universality observed in classical chaotic billiards, what are the underlying physical principles? We will give a heuristic description-not a rigorous derivation-of the physical principles leading to the pheno- menon of universality. Let us consider chaotic billiards in the regime of macroscopic times, i.e., when the billiard particle does a large number of bounces. Consider a trajectory carrying out

2) What is the physical significance of such universality? If one considers classically chaotic billiards in the regime of macroscopic times, where

3) Are there connections between universality and physical quantities which are easily observed in real systems? In particular, is such universality related to transport properties of the classical system? The answer is yes, and we will show in Section 6 how a transport coefficient can be obtained from the length matrix

Moreover, the behaviour of the chaotic billiard systems when approaching the regime of universality is characterized by laws specific for the particular billiard system. As example consider the decay laws of correla- tion functions. Consider a billiard trajectory and consider as observable f the segment of trajectory from n to

or polynomial fall off behaviour

where A and B are constants,

Here we suggest for chaotic billiard systems that the approach towards universality (i.e., increasing the number of rebounds) contains further physical information characteristic for the system. In particular, we expect that the distribution

for

4) Are those universality properties related to thermodynamical observables? If we consider the chaotic billiards consisting of a single particle moving in a rigid environment of scatterers, then it does not make sense to talk about thermodynamics. Thus for systems considered in this work the answer is no. However, if one considers billiards of many particles, then thermodynamics (as a function of temperature) will influence the dynamics. We shall defer the study of such effects on universality to future work.

5) Is such universality related to spectral statistics of the corresponding quantum system, i.e., what is the relation between universality in classical chaotic systems and universality in chaotic quantum systems, as defined via the Bohigas-Giannoni-Schmit conjecture [

which relates the quantum Hamltonian H to the classical Euclidean action

Above we have shown for the Bunimovich stadium billiard and for the Sinai billiard that they display universality properties via the statistical behaviour of the matrix of length of trajectories. Here we will show that such universal behaviour is related to relevant physical quantities. In particular we will extract transport properties from the length matrix (note the analogy to computation of transport properties in quantum chaos in semi-classical regime via Gutzwiller formula). As examples we consider the stadium billiard.

In systems for which the CLT is verified, the diffusive character is manifested by a linear relation between the time of travel and the variance of position. The diffusion coefficient d (in 2 dimensions) is given by the Einstein relation

where

where travel time

Based on this approach, we define a diffusion coefficient with respect to the variable

for the Bunmovich stadium billiard, which is a chaotic system with concave repeller/scatterer. We have chosen to analyze such system, and compute its transport properties because it reveals a very interesting non-standard diffusion behaviour. In a numerical modelling study of the stadium billiard, Borgognoni et al. [

We carried out numerical simulations using trajectories of the length matrix

We define the diffusion coefficient by

We have done statistical tests of

we obtained for the linear fit

This paper is about classical chaos occuring widely in nature, for example in astrophysics, meteorology and dynamics of the atmosphere, fluid and ocean dynamics, climate change, chemical reactions, biology, physiology, neuroscience, or medicine. We have suggested to extend random matrix theory, used in chaotic quantum systems, to classically chaotic and integrable systems. We have studied fully chaotic as well as integrable billiards and used a statistical description based on the length of trajectories to discriminate chaotic versus inte- grable behaviour.

Results:

1) In chaotic billiards (stadium and Sinai billiards), the NNS distribution

2) The distribution of length matrix elements

3) For the integrable rectangular billiard we find a correlation coefficient

4) In contrast, for integrable quantum systems the NNS distribution generally shows Poissonian behaviour with correlation coefficient

Future directions:

1) We plan to do numerical studies to investigate if universality also holds in chaotic potential systems.

2) We hope that our findings may contribute to obtain a unified description of both, quantum and classical chaos, and help understanding why quantum chaos is typically weaker than classical chaos, e.g., via an effective quantum action [

3) The global statistical approach to classical chaos proposed here may help to give insight into the problem of ergodicity breaking in Hamiltonian systems (e.g., dense packing of discs in the Lorentz gas model [

We are thankful to Prof. L. J. Dubé for insightful discussions on chaotic dynamics and to O. Blondeau-Fournier for discussions and his assistance in performing simulations and analyses presented in Section 3.1 and 4.1. H. Kröger is grateful to Prof. Chernov for discussions on central limit theorems in chaotic billiards. This work has been supported by NSERC Canada.

Jean-FrançoisLaprise,AhmadHosseinizadeh,HelmutKröger, (2015) Universality in Statistical Measures of Trajectories in Classical Billiard Systems. Applied Mathematics,06,1407-1425. doi: 10.4236/am.2015.68132