Physical versus mathematical billiards: From regular dynamics to chaos and back Free

Billiards are dynamical systems generated by an uniform motion of a point particle within a domain (billiard table) with a piecewise smooth boundary. Upon reaching the boundary, the particle gets elastically reflected. It is a commonly held opinion that such mathematical billiards adequately describe dynamics of real physical particles within the same billiard table. By a physical particle, we mean here a hard sphere of radius $r$⁠, which gets elastically reflected off the boundary of a billiard table. Clearly, one can follow the evolution of a (spherical) physical particle by considering motion of its center. Therefore, dynamics of a physical billiard is equivalent to dynamics of mathematical billiard within a smaller billiard table, which is obtained by shrinking billiard table by moving points of its boundary on the distance $r$ toward the interior of the table.

One can immediately see that for all standard examples of billiards, transition from mathematical to physical billiards does not change dynamics (besides making all free passes shorter). Indeed, a physical billiard within a triangle becomes mathematical billiard within a smaller similar triangle. Analogously, a physical billiard within a circle becomes a mathematical billiard within a smaller circle, Sinai billiard remains a Sinai billiard with larger scatterers, and squash billiards result in smaller squashes. [Recall that the boundary of squash billiards (sometimes called tilted stadia) consists of two circular arcs connected by two tangent to them straight segments.] A squash becomes stadium if the arcs are semicircles. Hence, the general opinion is that transition to physical billiards is not interesting because it brings nothing new.

In this paper, we show that this general opinion is wrong. It turned out that in fact virtually any possible changes of dynamics may occur in transitions to physical billiards. Namely, such a transition may generate appearance of KAM-tori when mathematical billiard was completely chaotic. Also, a nonchaotic mathematical billiards may generate chaotic physical billiards. Moreover, both such transitions may occur for any positive $r>0$⁠, i.e., demonstrating a “soft” transition as well as via “hard” transitions when a value of the radius of the moving physical particle must exceed some strictly positive critical value. We present concrete examples of all four such types of transitions. All the examples we consider are two dimensional because the goal of this paper is just a proof of concept. Therefore, we try to built the most visual and simple examples.

This paper opens up a new direction for mathematical and physical studies. For instance, real pipes, containers, and channels always have nonideally smooth boundaries, which rather can be viewed as piecewise smooth random surfaces. Our examples show that generally (with positive probability) such impurities may slow down a flow by generating elliptic periodic orbits in some ranges of radii of propagating particles. The families of KAM-tori “surrounding” these orbits correspond to a kind of local vortices where particles propagating in a pipe can be trapped. Such vortices may appear and disappear with the change of sizes of propagating particles. This observation should be taken into account, especially for nanochannels⁷ where the width of a channel is small and comparable to the size of particles.

Also, physical billiards seem more natural for quantum mechanics because of Heisenberg’s uncertainty principle. Hence, (mathematical) point particles do not seem to be relevant for the studies of quantum chaos. Instead, one should consider the evolution of wave packets, i.e., of finite-size “particles.” Therefore, a quantum billiard generally may be more chaotic than the corresponding classical billiard.¹² 

Consider a free motion of a hard sphere with radius $r$ within a domain (billiard table) $Q$ in $d$-dimensional Euclidean space. We always assume that the boundary $∂Q$ of $Q$ consists of a finite number of codimension one smooth manifolds of class $C2$⁠, which are called regular components of the boundary. Interior points of regular components are called regular points of the boundary, and points of intersection of regular components will be called corner points.

At any interior point $q$ of a regular component of the boundary $∂Q$⁠, there exists a unique inner (i.e., directing toward the interior of $Q$⁠) unit normal vector $n(q)$⁠. Upon reaching the boundary of a billiard table, the particle gets elastically reflected.

One gets a standard (mathematical) billiard if a moving particle is a point, i.e., its “radius” $r=0$⁠. We call a billiard physical if a moving particle is a (smooth) hard sphere with positive radius. Physical (as well as mathematical) billiards are Hamiltonian systems. Therefore, these dynamical systems have a natural invariant measure, which is a volume in the phase space. We will consider a billiard map which arises if one follows billiard trajectories only at the moments immediately after reflections. Such billiard map generates a measure preserving dynamical system (see, e.g., Ref. 6). For our purposes, in this paper, neither an exact expression for a billiard map nor coordinates in a corresponding phase space are needed. Therefore, we will stick with just visual purely geometric consideration. Also, for the sake of clarity, we will deal in what follows only with two-dimensional billiards. A regular component of the boundary $∂Q$ is called dispersing (focusing) if it is convex inside (outside) of a billiard table (Fig. 1). The choice of internal vectors $n(q)$ ensures that the curvature of dispersing (focusing) components is positive (negative) at all their points. A regular boundary component is called neutral if its curvature is identically zero. It is easy to see that the moving particle with a positive radius $r$ may never hit some of the points of the boundary of the billiard table. For instance, in case when dimension of the billiard table is two, it happens if two regular components of the boundary intersect under the angle less than $π$⁠. Indeed, in such a case, a disk (particle) with positive radius cannot get into this corner and hit points at the boundary of the billiard table, which are close to a point of intersection of these regular components.

We now formally describe the process of transition from a mathematical billiard to the corresponding physical one. Consider a physical particle moving in the same billiard table as a point particle. To represent dynamics of a (hard) homogeneous spherical particle of radius $r$⁠, it is enough to follow the motion of its center. It is easy to see that the center of particle moves in the smaller table, which one gets by moving any point $q$ of the boundary by $r$ to the interior of the billiard table along the internal normal vector $n(q)$ (see Fig. 2). Dynamics of the center of a physical particle is equivalent to a mathematical billiard in this smaller billiard table, which we will call a reduced billiard table. To overcome some technicalities, we assume throughout this paper that billiard tables satisfy the following “no internal corners” (NIC) condition.

For example, the billiard table $Q$ in Fig. 1 does not satisfy condition NIC because some segments connecting points on two adjacent focusing components do not belong $Q$⁠. On the other hand, the billiard table in Fig. 2 satisfies condition NIC. Observe also that this condition does not allow billiard tables to be nonconvex polygons. This restriction (condition NIC) will be lifted in a consequent paper, but here we want to avoid unnecessary technicalities.

A purpose in this paper is just a proof of concept. Therefore, our examples will be simple and visual ones. Moreover, in order to avoid long technical proofs typical for the mathematical theory of billiards, all statements about changes in dynamics that occur in transitions from mathematical to physical billiards would follow from already known results on mathematical billiards.

In this section, we consider the appearance and disappearance of KAM-islands. These examples will demonstrate that transitions from mathematical to physical billiards may result in changes of the phase portrait in the corresponding dynamical system. Following our general strategy, in this paper, only the simplest situation will be discussed when KAM-islands are generated by elliptic periodic orbits of period two. Indeed, it is the simplest case because in billiards there are no fixed points. Necessary and sufficient conditions for linear stability of period two points in billiards are well known.¹⁴ They read as

where $k1$ and $k2$ are curvatures of the boundary at the endpoints of a period two orbit and $L$ is the length of the segment connecting these points.

Observe that period two orbit ending on two dispersing or on two neutral components is always unstable. The case when only one end of period two orbit belongs to a neutral component can be reduced to consideration of period two orbit bouncing between two focusing or two dispersing components by a standard in geometric optics trick of reflecting a billiard table with respect to a neutral component of the boundary. Therefore, only the cases when both components are focusing or one is focusing and the other one is dispersing are of interest.

Take first a period two orbits bouncing between two focusing components. Let $q1$ and $q2$ are the ends of this periodic orbit. Denote by $R1$(⁠$R2$⁠) a radius of curvature of the first (second) focusing component at the point $q1$(⁠$q2$⁠). Without loss of generality, we assume that $R1>R2$⁠. Then conditions of linear stability (1) and (2) become $L<R1+R2$ and $(L−R1)(L−R2)>0$⁠, respectively. Suppose that this periodic orbit is linearly stable and $L<R1+R2$⁠, $L>R1$⁠, and $L>R2$⁠.

Let a physical particle with radius $r$ moves in a corresponding billiard table. Then, in the reduced billiard table, this period two orbit remains. However, its length becomes $L−2r$⁠, while both radii of curvature at its end points will become $R1−r$ and $R2−r$⁠, respectively. Therefore, this period two point becomes unstable when $L<R1−r$ but $L>R2−r$⁠. In other words, period two orbit loses linear stability when radius $r$ of a physical particle equals $L−R1$⁠.

However, if radius of the particle continue to increase after passing the value (⁠$R2−L$⁠), then another change in dynamics occurs. Indeed, it follows from (2) that period two orbit acquires a linear stability if the radius of moving particle exceeds $L−R2$ (but of course $r$ must remain to be less than $L/2−r$⁠, otherwise, this period two orbit will just disappear).

Therefore, by increasing the radius of moving particle, it is possible to achieve several changes in dynamics of the corresponding billiard. Indeed, in our example, a periodic orbit is stable if radius of moving particle is relatively small, then it becomes unstable when this radius is within a range of larger values, and finally, this orbit again acquires stability when radius of the particle becomes large enough. In fact, a radius $r$ of the physical particle can be viewed as a kind of a bifurcation parameter in this family of physical billiards.

It is well known¹¹ that linearly stable periodic orbit not always generates KAM-island. To make this happen, some quantity called the first Birkhoff coefficient¹¹ must not vanish. To make the things easier again, we assume that both focusing components that we consider are arcs of some circles. In this case, the ellipticity condition is reduced to two inequalities.^5,9 Namely, $4(L−R1)(L−R2)$ must not be equal to $R1R2$ or to $2R1R2$⁠. These conditions are very easy to satisfy by choosing appropriate values of parameters.

Generally, it is well known that ellipticity is an open property, in the sense that sufficiently smooth perturbations of the focusing components near the ends of elliptic periodic orbit in two-dimensional billiards will again have an elliptic periodic orbit (close to the perturbed one).

In the next example, we will consider another type of appearance of a linearly stable period two orbit from a linearly unstable one. Consider now a period two billiard orbit, which moves between a focusing and a dispersing component. We will keep the same notations as above besides the assumption that $R1>R2$⁠. Now, $R1$ (⁠$R2$⁠) is a radius of curvature of the focusing (dispersing) component of the boundary at the corresponding end the orbit. In this case, conditions of stability (1) and (2) become $L>R1−R2$ and $(L−R1)(L+R2)<0$⁠. Therefore, $L$ must be smaller than $R1$⁠. Suppose that in the mathematical billiard, this orbit is linearly unstable because $L>R1$⁠. Then, this orbit becomes linearly stable when radius $r$ of the particle exceeds $L−R1$⁠, and generically, a new KAM-island will appear.

However, as we will see in Sec. IV, variations of the radius of physical particle may lead to essential transformations of shapes of billiard tables when some of their boundary components just disappear. Such transformations generally cannot be interpreted as bifurcations.

In Sec. III, we showed that phase portraits of mathematical billiards may change in the transition from mathematical to physical billiards. These changes resulted in the appearance and disappearance of KAM-islands, i.e., they were, in a sense, local ones. In this section, we give examples of global transitions from nonchaotic billiards to the fully chaotic ones and vice versa. By fully chaotic dynamical systems, we mean the ones which have almost everywhere nonvanishing Lyapunov exponents. It is well known that such systems have a finite or countable number of ergodic components of positive measure, which make (up to a set of measure zero) the entire phase space. Moreover, these dynamical systems have positive Kolmogorov-Sinai entropy, are mixing on each ergodic component, and have as well other strong chaotic properties (see, e.g., Refs. 13 and 10).

For the sake of clarity, we will present only simple and visual examples of such transitions. Moreover, proofs of the corresponding claims will directly follow from already known (rigorously proved) results on mathematical billiards.

Consider first a billiard table $Q$ depicted in Fig. 3. Its boundary consists of six regular components. Two of them are focusing and formed by equal arcs AB and CD of some circles of the same radius $R$⁠. The neutral components AE and CF are straight segments tangent to both these arcs at the points A and C. The last smooth component EF of the boundary $∂Q$ is dispersing. We also assume that the angles between the segments AE and AC and between the segments AC and CF are both equal to $π/3$ and that the segments AC and BD are parallel. Besides, the arcs AB and CD are placed so that their centers $G1$ and $G2$ are at a distance strictly less than $2R$ from each other and the segment $G1G2$ contains the centers of both circles to which the arcs AB and CD belong. Then, the period two orbit $G1G2$ is linearly stable. Conditions of nonlinear stability of such orbit are given in Sec. III. We assume that these conditions are satisfied and thus our period two orbit is the center of a KAM-island. It is also assumed that dispersing component EF is placed at such distance from the segment BD, which is larger than twice the distance between parallel segments AC and BD. Let a hard disk with radius $r$ generates a physical billiard within $Q$⁠.

We now consider opposite transition from (completely) chaotic mathematical billiard to nonchaotic physical billiard. Moreover, the resulting physical billiard will be strongly nonchaotic. More precisely, this physical billiard does not have any subset with chaotic dynamics, and its Kolmogorov-Sinai entropy equals zero. Take a billiard table having a shape of a regular triangle with one vertex smoothened by an arc of a circle with radius $R$ (Fig. 4). It is well known (see, e.g., Refs. 1 and 2) that such a mathematical billiard is strongly chaotic, i.e., completely hyperbolic, ergodic, etc. It is easy to see that physical billiard in this table becomes equivalent to a mathematical billiard in a regular triangle when radius of the particle becomes equal $R$⁠. Therefore, we have

Finally, we give an example of another transition when physical billiards within billiard tables of chaotic mathematical billiards acquire a linearly stable periodic orbit (i.e., generically a KAM-island) and thus become nonchaotic. Consider a billiard table $Q$ depicted in Fig. 5. The boundary $∂Q$ of this billiard table contains eight regular components. Two regular components are arcs of the circles with centers $O1$ and $O2$⁠. One of these components is focusing and the other is dispersing. Four components adjacent to these two are neutral ones tangent to focusing or dispersing component at the endpoints. We also assume that the circle with the center $O1$ is tangent to the circle with the center $O2$ (Fig. 5). The last two regular components of the boundary are dispersing ones. It follows from the paper on track billiards³ that this (mathematical) billiard is chaotic.

We demonstrated that in transitions from mathematical to physical billiards various changes in dynamics occur. Moreover, several such transitions may occur when radius of the particle changes. Hence, there exist several different ranges of sizes of the particle. Dynamics of the corresponding billiard change when radius of the particle crosses the boundaries between these ranges. These results raise many questions for the future mathematical and physical research, including theoretical, numerical, and experimental studies. Indeed, various billiards with simple shapes considered here could be build in physical labs as it has been done for Sinai billiards, stadia, and mushrooms. Particularly, propagation of particles in channels may have different properties depending on the size of particles. Indeed, a birth and destruction of KAM-islands results in the appearance and disappearance of “vortices” in a flow, respectively. As it was demonstrated in this paper, even several changes in dynamics may occur when a size of the moving particle varies.

This work was partially supported by the NSF Grant No. DMS-1600568.