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International Journal of Bifurcation and Chaos, Vol. 18, No. 9 (2008) 28632869 c World Scientific Publishing Company

THERMODYNAMICS OF DISPERSING BILLIARDS WITH TIME-DEPENDENT BOUNDARIES
ALEXANDER LOSKUTOV , OLGA CHICHIGINA and ALEXEI RYABOV Physics Faculty, Moscow State University, 119992 Moscow, Russia loskutov@chaos.phys.msu.ru Received April 23, 2007; Revised Octob er 26, 2007
By means of a thermodynamic approach we analyze billiards in the form of the Lorentz gas with the open horizon. For perio dic and sto chastic oscillations of the scatterers, the average velo city of the particle ensemble as a function of time is analytically obtained. It is shown that the consequence of such oscillations is Fermi acceleration which is larger for perio dic oscillations. The described results do not depend on the size of scatterers and their position. Only the property of the horizon openness is necessary. It is found that the developed thermo dynamic approach is in a very go o d agreement with the results of the direct numerical simulations at which the corresponding billiard map is used. Keywords : Billiards; Lorentz gas; thermo dynamics; Fermi acceleration; temperature.

1. Introduction
The classical billiard symb olizes a dynamical system where the p oint particle (billiard ball) reflects elastically from the b oundaries of some closed region and p erforms free straight motion b etween reflections. Thus, the billiard particle moves along geodesic lines with a constant velocity. In the present pap er we consider billiards in Euclidean plane. In this case, the angle of incidence of the particle is always equal to the angle of reflection. A natural extension of billiard systems is to consider models for which the b oundary is not fixed, but it oscillates with time. From the physical p oint of view such a generalized billiard consisting of many particles describ es the rarefied gas in a container, the walls of which vibrate with a small amplitude. For billiards with the p erturb ed b oundaries their dynamical prop erties are very essential. If it exhibits chaotic dynamics then p erturbations of the b oundary may lead to the infinite growth of the particle velocity. This problem is

related to the unb ounded increase of energy in p eriodically forced Hamiltonian systems and is known as the Fermi acceleration [Fermi, 1949; Ulam, 1961]. For the first time, to explain the origin of highenergy cosmic particles Fermi [1949] prop osed the mechanism of the particle acceleration by means of collisions with moving massive scatterers. Later various models (the FermiUlam model and others, see, e.g. [Brahic, 1971; Lichtenb erg & Lieb erman, 1990; Lichtenb erg et al., 1980; Pustyl'nikov, 1987, 1994, 1995; Pustyl'nikov et al., 1995; Ulam, 1961; Zaslavsky, 1970]) have b een develop ed which explained this phenomenon to a certain extent. Billiards with p erturb ed b oundaries can b e considered as a certain generalization of such models. For example, in the pap ers [Koiller et al., 1995; Koiller et al., 1996] elliptic billiards have b een numerically considered. The authors have come to the conclusion that, as in the FermiUlam model, the growth of the particle velocity is b ounded by invariant curves. Applying the dynamical approach

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(i.e. using the corresp onding billiard dynamical systems) the Fermi acceleration has b een analyzed on the example of the Lorentz gas and stadium-like billiards [Loskutov et al., 1999, 2000; Loskutov & Ryab ov, 2002]. Stadium-like billiards are defined as a closed domain with the b oundary consisting of two focusing curves connected by the two parallel lines. It was found that in billiards with chaotic prop erties there is a linear growth of the particle velocity with time. However, for a nearly rectangle stadium with p eriodically oscillated b oundary a new interesting phenomena is observed. Dep ending on the initial values, the particle ensemble can b e accelerated, or its velocity will not grow, i.e. there is lack of the Fermi acceleration. This means that the b oundary p erturbation leads to the separation of the particle ensemble in velocities. In these pap ers the following conjecture has b een advanced: chaotic dynamics of a billiard with a fixed b oundary is a sufficient condition for the Fermi acceleration in the system when a b oundary p erturbation is introduced. In pap ers [de Carvalho et al., 2006; Karlis et al., 2006] (see also references therein) the validity of this conjecture has b een confirmed on the examples of some billiards of sp ecific forms. Recently, more deep insight into the problem of the Fermi acceleration has b een presented [Kamphorst et al., 2007]. As it is known, the p eriodic Lorentz gas (with the fixed b oundary) with the op en horizon is a billiard which has strong chaotic prop erties (ergodicity, mixing, decay of correlations, etc.). This allows us to use a qualitative different method and apply a thermodynamic analysis to the system (see, e.g. [Kozlov, 2000, 2004; Chernov et al., 1993; Bonetto et al., 2002; Moran et al., 1987]). Using such an approach one comes to the conclusion that the obtained results should not depend on the bil liard geometry,i.e.onthe size,curvature and disp ositions of scatterers. The unique condition is the limitedness of the mean free path of the billiard particle, i.e. the Lorentz gas should p ossesses the b ounded or the op en horizon. It should b e noted that for the detailed analysis of such a complicated system as billiards it is necessary to use several approaches: dynamical, thermodynamical and computer simulations. The first method gives the exact description of some properties of the corresp onding dynamical systems. The thermodynamical way, on the basis of a quite simple semi-phenomenological model, allows us to get the most general ideas ab out the system.

In the present pap er, on the example of the Lorentz gas with the op en horizon and timedep endent p eriodically and stochastically oscillating scatterers (i.e. the billiard b oundary), we develop the thermodynamic approach to the description of such a system. We show that for b oth cases there is a Fermi acceleration. Thus, we supp ort the ab ove describ ed conjecture advanced in pap ers [Loskutov et al., 1999, 2000] ab out the Fermi acceleration in billiards with the p erturb ed b oundaries.

2. The Lorentz Gas
A p eriodic Lorentz gas is a system containing a set of heavy discs (scatterers) with radius R emb edded at sites of an infinite lattice with p eriod a. Particles move freely among these discs. The billiard table in such a configuration is the whole plane except for the scatterers. Because particles do not interact with each other, then it is sufficient to consider only one particle. In Fig. 1 one can see a variant of the Lorentz gas in the p eriodic triangle lattice [Bunimovich & Sinai, 1981; Hakmi et al., 1995; Machta & Zwanzig, 1983]. Dep ending on the radius R of the scatterers, this model p ossesses qualitatively different prop erties. If R a/2 then the system has an infinite hori i zon. If R a/ 3, then the particle paths b ounded by only one cell. When a/2 < R < a/ 3 the free path of the particle is b ounded, but it may move

Fig. 1. Configuration of Lorentz gas model. The scatterers (circles of radius R) are located at sites of a lattice with period a.

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freely in the billiard table. In this case the Lorentz gas has the op en horizon. For the infinite horizon, owing to the probability of long free paths, the statistical prop erties of a billiard are qualitatively changed. In particular, the mean free path does not converge, and there is an algebraic correlation decay with time [Baldwin, 1991; Bunimovich, 1985; Bunimovich & Sinai, 1981; Chernov, 1997; Garrido, 1997]. At the same time, for the Lorentz gas with the b ounded horizon and the op en horizon an exp onential decay of correlations takes place, and the mean free path can b e obtained as follows [Chernov, 1997]: , (1) = P where is the accessible billiard region and P is the scatterer p erimeter. Supp ose that the radii of the scatterers are p erturb ed by a certain law, i.e. their b oundaries are changed in such a way that R(t) = R + r (t), where max |r (t)| R. We will analyze two different cases: stochastic and p eriodic (and in-phase) b oundary oscillations. In the first case we assume that the phase of oscillations (and, as a consequence, the velocity of scatterer b oundaries) at the collision time is a random value. The latter case corresp onds to the situation when all the b oundaries are p erturb ed by one and the same law and in the same phase. Physically this may b e considered as an introduction of an external alternating field.

To compare our thermodynamical analysis with the known dynamical models [Loskutov et al., 2000], we consider the Lorentz gas in a triangular lattice (Fig. 1). However, the prop osed approach is much wider, and it can b e applied to the description of the other typ es of billiards (see b elow). It is well known that in the Lorentz gas with the op en horizon the particle motion is stochastic. Therefore, the temp erature of these particles can b e associated with their average kinetic energy. The Hamiltonian of such a system corresp onding to a single particle with the mass m is H= p2 p2 y x + + U (qx ,qy ), 2m 2m

where U is the energy of interaction with scatterers / which fill area L = R2 such that U (qx ,qy L) = 0, U (qx ,qy L) = . For the two-dimensional motion, using the known result on the energy distribution into degrees of freedom, we get the expression for the particle energy: p2 2m = mv 2
2

= kT .

(2)

By the same manner, the temp erature of the scatterers can b e determined by means of the kinetic energy of their motion, i.e. Mu 2
2

=

Mu 2 0 = kTR . 4

(3)

3. Stochastic Perturbations
To construct a thermodynamic model corresp onding to a billiard with oscillating walls, it is logical to consider a numb er of identical particles with the mass m. In this case scatterers should b e presented as heavy conglomerates consisting of the same rigidly connected particles. The b oundary of such scatterers consists of N particles. Then collision of the moving particle with a scatterer may b e describ ed as the collision with one of the particles forming this scatterer. Also, this collision has a sense of the interaction of two thermodynamic subsystems (i.e. the moving particles and the fixed particles in scatterers) having their own temp eratures. Then the interaction area S will corresp ond to a collision region and may b e defined as the scatterer p erimeter divided by N . It is obvious that the parameters m, S and N should not app ear in the final result.

At u0 = 0 this temp erature turns out to b e infinite, TR = , b ecause we consider the model with scatterers of the infinite mass, M = mN . The conclusion ab out the infinite mass directly follows from the main assumption related to the Lorentz gas with the fixed scatterers: The angle of incidence is equal to the angle of reflection, i.e. the absence of recoil. Thus, the Fermi acceleration can b e considered as a result of the heat exchange b etween the particle as a thermodynamic system and thermostat of the infinite temp erature.

3.1. The heat conductivity equation
The heat flow from scatterers may b e defined by the temp erature gradient. Therefore, the heat conductivity equation can b e written as follows: T Q = -S , t x (4)

where is a heat conductivity coefficient, S is a heat transfer area (the temp erature gradient should b e

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taken along the normal to this area). For the twodimensional Lorentz gas this area S is defined as a collision area of the free particle and the scatterer b oundary. On the other hand, the heat flow can b e expressed by the change of the particle energy: T c mv Q =c = t t k t 2
2

two-dimensional diffusion coefficient along the normal to the scatterer b oundary, c is the heat capacity. This diffusion coefficient can b e written in the form D=
2 vx v2 = , 2 3

(9)

,

(5)

where = / v is thefree pathtime. Thefreepath may b e presented as (see expression (1)): = . 2R (10)

where c is the heat capacity of the particle gas. In this case this parameter is equal to the Boltzmann constant k b ecause the total input heat is transformed into the energy of the two-dimensional particle motion, i.e. into the temp erature (see expression (2)). For such a process in the ideal gas the heat capacity is well known and defined by the expression c = (i/2)kNp , where Np = 1 is the number of moving particles, i = 2 is the numb er of degrees of freedom. Let us find the explicit form of the right-hand side of Eq. (4). On the one hand, the effective temp erature gradient b etween the particle and scatterers turns out to b e infinite, |T / x| , b ecause TR = . However, on the other hand, as a consequence of p oint collisions the interaction area tends to zero, S 0. Thus, there is an indeterminate form 0 · which we can evaluate as follows. It is obvious that N . Then the interaction area corresp onds to the particle size and can b e written as 2R . (6) S= N The temp erature gradient in Eq. (4) one can approximately write as T / x T/x, where x is the distance where there is a temp erature difference, and T is the difference in the temp erature b etween the scatterer and the particle gas. In turn, x we may approximately write as the free path: x x = vx / v = /4. For T , taking into account the infinite temp erature of scatterers, we get: -T = TR - T TR = mNu 2 /4k . Thus, eval0 uating an indeterminate form on the right-hand side of (4), finally we get: 2Ru2 m T 0 . (7) x k The heat conductivity coefficient can b e introduced in a usual manner: -S = cD n, (8)

Thus, the heat conductivity of the system particlescatterers is determined as follows: = 1 kn v 2 . 3 (11)

By using Eqs. (7) and (11), the right-hand side of Eq. (4) is presented in the form: u2 m v 2 Q =0 2 . t 3 (12)

Now, combining the right-hand sides of Eqs. (5) and (12), we get: d v dt
2

=

2u2 2 0 v. 32

Taking into account that for the ideal gas v 2 = v 2 , where is a constant, we find the following expression for the change of the particle velocity in the Lorentz gas with the stochastically p erturb ed scatterer b oundaries: u2 d v = 0. dt 3 (13)

Thus, in the investigated system the Fermi acceleration is observed, and for the large enough particle velocity its average value grows as a linear function of time. It should b e noted that the obtained result (13) is valid for the scattering billiard with an arbitrary shap e of the b oundary for which the free path is b ounded.

4. Periodic Boundary Oscillations
For p eriodic oscillations of the b oundary it is necessary to use another approach. Let the b oundary b e p erturb ed in a regular way such that its velocity is u = u0 cos t. In this case one can say that we do the work A on gas. Strictly sp eaking, the process is nonequilibrium, and the gas of billiard particles does not put

where n = 1/ is a particle concentration, is the area of the free space in the cell, D is

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pressure up on scatterers. However, we may consider a small enough area nearby the scatterer and admit that here the particle concentration n is constant. Then the pressure is determined as [Chichigina et al., 2003]: p(t) = mn(v + u0 cos t)2 /2. As a result of the scatterer motion the accessible billiard area (see (1)) is changed. This change is defined by the scatterer velocity and the scatterer p erimeter: dV = 2Ru0 cos tdt. At that, the elementary work may b e written as follows: dA = pdV = Ru0 mn(v + u0 cos t)2 cos tdt. Assuming that the particle concentration in this chosen area is fixed and equal to n = (2R)-1 , we get: dA = u0 m/2(v + u0 cos t)2 cos tdt. This work exp ends on the increase of the internal energy. Now, taking the average over the p eriod of the scatterer oscillation and over the particle velocities, we obtain the differential equation which describ es the change of the mean kinetic energy of the particle: d mv dt 2
2

=

dAT mu2 v 0 = . dt 2

Owing to v 2 = v 2 , where 1, the Fermi acceleration for the case of p eriodic oscillations has the following form: u2 d 0 v= . dt 2 (14)

Numerical realizations of particle tra jectories were different from each other in the initial directions of the particle velocity which were chosen in a random way. Two qualitative different cases were considered: stochastic oscillations of the scatterer b oundaries with equidistributed phases and p eriodic oscillations. In b oth cases, the particle dynamics was determined by the map describ ed in [Loskutov et al., 2000]. In the stochastic case the oscillation velocity of the b oundary was defined as follows: un = u0 cos n , where n is a uniformly distributed random value over the interval [0, 2 ]. For p eriodic oscillations un = u0 cos tn , where tn is the instant of the nth collision with the b oundary. For each case, 5000 realizations of the tra jectory of the billiard particle have b een constructed. Dynamics of the particle ensemble was simulated up to 3 · 105 time units, and some tra jectories (of the particles with high velocities) include up to 107 iterations. The averaged dep endencies of the particle velocity for the stochastic vst and p eriodic vreg b oundary p erturbations are shown in Fig. 2. In this figure full lines corresp ond to the regular case, and dotted lines corresp ond to the stochastic case. Straight lines are analytical results (13) and (14). As follows from this figure, the acceleration has a linear character that is in good agreement with the obtained analytical expressions. The results are the same for other parameters of the Lorentz gas with the op en horizon.

Thus, p eriodic oscillations of the scatterers also lead to the Fermi acceleration. However, in the comparison with expression (13) the acceleration is higher for the stochastic b oundary oscillations.

5. Numerical Simulations
In this part we present numerical simulations and compare them with the analytical results obtained ab ove. All the calculations have b een made by the dynamical approach, i.e. via the corresp onding maps describing the particle motion in the Lorentz gas (Fig. 1) with the b ounded horizon [Hakmi et al., 1995; Loskutov et al., 2000]. Consider the Lorentz gas model with the following parameters: the amplitude of the b oundary oscillation of scatterers u0 = 0.01, the scatterer radii R = 0.56, the cell size a = 1, the frequency of b oundary oscillations = 1, the initial velocity v0 = 1. Thus, for the given billiard geometry the analytical value of the free path is = 0.17.

Fig. 2. Average particle velocities as functions of time in the Lorentz gas. The following parameters were used: u0 = 0.01,a = 1,R = 0.56 and v0 = 1. The average curve was obtained on the basis of 5000 realizations of different directions of the initial velocity v0 , which were selected in a random way.

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6. Conclusion
The Lorentz gas is a widely recognized model of the ideal rarefied gas. For the Lorentz gas with the op en horizon a numb er of its statistical prop erties is proved: mixing, exp onential decay of correlations, the existence of the diffusion coefficient [Bunimovich & Sinai, 1981]. Thus, for the analysis of the particle motion in this system one can use the thermodynamical approach. In a certain sense, such an approach is a universal one b ecause in this case the geometry (i.e. the disp osition of the scatterers, their forms, sizes and curvature) does not play an essential role. A quite natural generalization of the classical Lorentz gas is the system with oscillating scatterers. In the present pap er, on the basis of the thermodynamical analysis we have shown that at the scatterer p erturbations of the Lorentz gas with the op en horizon the average velocity of the particle ensemble (for b oth, stochastic and p eriodic oscillations) grows linearly with time (see (13), (14)), i.e. the Fermi acceleration phenomenon is observed. This confirms the conjecture advanced by the authors of the pap ers [Loskutov et al., 1999, 2000] that chaotic dynamics of a billiard with a fixed b oundary is a sufficient condition for the Fermi acceleration in the system when a b oundary p erturbation is introduced. It is also found and explained that the acceleration is higher in the case of p eriodic oscillations than for the stochastic p erturbations. The model prop osed in the present pap er and based on the assumption that the scatterers consist of an infinite numb er of particles of one and the same kind, is not uniquely p ossible. But, nevertheless, the results obtained on its basis are in a very good agreement with the conclusions which follow directly from the dynamical theory of the lattice Lorentz gas (see [Bunimovich & Sinai, 1981; Loskutov et al., 1999, 2000]). Also, using this model one can obtain the value of the Fermi acceleration for the arbitrary distributed scatterers, scatterers of the arbitrary form and the motion, and for three-dimensional Lorentz gas. The unique claim here is the condition of the op en horizon. In addition, on the basis of the prop osed model one can easily explain the difference in the results for the regular and stochastic motions of the scatterers. Namely, the Fermi acceleration is a dynamical effect that app eared in the time-dep endent Lorentz

gas. It is known that the transformation of the heat energy into a mechanical form is restricted by the second law of thermodynamics. In the case of stochastic oscillations we have exactly this typ e of the transformation. At the same time, the axiomatic notion of the work (i.e. a mechanical form of the energy transfer) used for the case of p eriodic scatterer oscillations, implies a straight energy transfer from scatterers to moving particles. Finally, we should emphasize that the principal result of our pap er is not the analytical expressions for the Fermi acceleration. They are necessary for the verification of the model premise. The main conclusion is that our approach allows to get results for a quite wide class of the parameters of the Lorentz gas.

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