Thermodynamics of a time dependent and dissipative oval billiard: a heat transfer and billiard approach
Edson D. Leonel, Marcus Vinicius Camillo Galia, Luiz Antonio Barreiro, Diego F. M. Oliveira
TThermodynamics of a time dependent and dissipative oval billiard: a heat transferand billiard approach
Edson D. Leonel
Departamento de F´ısica, UNESP - Univ. Estadual Paulista - Av. 24A,1515 - Bela Vista - 13506-900 - Rio Claro - SP - BrazilAbdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
Marcus Vin´ıcius Camillo Galia and Luiz Antonio Barreiro
Departamento de F´ısica, UNESP - Univ. Estadual Paulista - Av. 24A,1515 - Bela Vista - 13506-900 - Rio Claro - SP - Brazil
Diego F. M. Oliveira
Center for Complex Networks and Systems Research,School of Informatics and Computing, Indiana University, Bloomington - Indiana, USA.Department of Chemical and Biological Engineering,Northwestern University, Evanston, Illinois 60208, USA.Northwestern Institute on Complex Systems (NICO),Northwestern University, Evanston, Illinois 60208, USA. (Dated: September 20, 2018)We study some statistical properties for the behavior of the average squared velocity – hencethe temperature – for an ensemble of classical particles moving in a billiard whose boundary istime dependent. We assume the collisions of the particles with the boundary of the billiard areinelastic leading the average squared velocity to reach a steady state dynamics for large enoughtime. The description of the stationary state is made by using two different approaches: (i) heattransfer motivated by the Fourier law and; (ii) billiard dynamics using either numerical simulationsand theoretical description.
PACS numbers: 05.45.Ac, 05.45.Pq
I. INTRODUCTION
The initial mark in the investigation of billiards the-ory is related to Birkhoff [1] in beginning of last century.Since then this research area has developed significantly.Birkoff considered the investigation of the motion of afree point-like particle – representing a billiard ball – ina bounded manifold. However the modern investigationsof billiards are indeed related to the results of Sinai [2, 3]and Bunimovich [4, 5] who made rigorous demonstrationson the topic.A billiard is a dynamical system composed of a parti-cle, or an ensemble of non interacting particles, movingconfined to a domain with a piecewise-smooth boundary[6] where they collide. The specular reflections occur un-der the condition the boundary is sufficiently smooth. Insuch case the tangent component of the velocity of theparticle measured with respect to the border where colli-sion happened is unchanged while the normal componentreverses sign. There are many results nowadays consid-ering either static [7–15] and time dependent boundaries[16–19]. A phenomenon well known in time dependentboundary is the Fermi acceleration [20] as well as theso called Loskutov-Ryabov-Akinshin (LRA) conjecture[21, 22]. Fermi acceleration [20] is a phenomenon wherean ensemble of particles acquires unlimited energy fromcollisions with an infinitely heavy and moving wall. Theconjecture itself claims that if chaos is present in the dy- namics of a particle in a static version of the billiard,then this is a sufficient – but not necessary – conditionto observe Fermi acceleration when a time perturbationto the boundary is introduced. Many different billiardsexhibit Fermi acceleration under time perturbation to theboundary including the Lorentz gas [23, 24], oval billiard[25], stadium [26] and other shapes [27]. The ellipticalbilliard is an exception and the LRA conjecture does notapply to it. For the static boundary, the elliptical bil-liard is integrable [6] and the phase space is composedof rotating and librating orbits. A curve which separatesthese two different regimes is called as separatrix. Lenzet al. [28, 29] shown that when the boundary of thebilliard is allowed to be time dependent, the separatixcurve is replaced by a stochastic layer allowing cross vis-itations from regions of libration and rotation. For thestatic case both energy, E , and angular momenta aboutthe two foci, F , are constants of motion [30], leadingthe system to be integrable. However, when time per-turbation is introduced, the observable F (see Ref. [28])experiences strong and fast variations from the crossingof orbits coming from rotation and those leaving fromlibration. The successive crossings produce the stochas-ticity required in the LRA conjecture, hence leading thetime dependent elliptical billiard to exhibit Fermi accel-eration. This result is considered a counter example ofthe LRA conjecture. Latter on, investigations on differ-ent models have proved the Fermi acceleration is not a a r X i v : . [ n li n . C D ] D ec robust phenomenon since a very small amount of dissi-pation is enough to suppress the phenomenon [31]. Con-sideration of inelastic collisions in the elliptical billiard[32] has proved the successive crossings of orbits com-ing from rotation region and entering libration domain –and vice versa – are interrupted suppressing the Fermiacceleration.The motion of the time dependent boundary can berelated to a more physical situation. Due to the thermalfluctuations the position of each atom that compose theboundary is allowed to move locally. Such oscillation ofthe atoms, and hence of the boundary, can be broughtto the context of billiard which allows connections of theobservables obtained from the velocity of the particle –hence the kinetic energy – to the thermodynamics, par-ticularly the temperature and entropy.In this paper we investigate some dynamical propertiesfor an ensemble of particles confined in an oval billiardwhose boundary is moving in time. Our main goal isto understand and describe the dynamics of the averagesquared velocity for a gas of noninteracting particles. Wewill do this by using two different procedures. Becausethe boundary of the billiard is moving, as soon as theparticles collide, there is a change of energy of the par-ticle. Therefore, the first procedure considered involvesthe heat transfer Fourier equation. We write and solvethe Fourier equation considering the geometrical proper-ties of the boundary. The resulting equation is that thetemperature of the gas settles down for sufficiently longtime as the temperature of the boundary, hence the av-erage squared velocity, reaches the thermal equilibrium.The second one involves the formalism commonly usedin billiard problems. We write down the equations of themapping that describe the dynamics of the problem andextract some average properties for the squared velocityof the particles. The properties are obtained either bystraightforward numerical simulations as well as analyt-ically. The results obtained on the analytical approachare remarkable well fitted by the numerical simulations.The first approach however uses the time as the dynam-ical variable while the second one uses the number ofcollisions of the particles with the boundary. The twodynamical variables are not trivially connected amongthemselves. Therefore, by the use of an empirical func-tion, we find a straight relation between these two pa-rameters.This paper is organized as follows. In Sec. II we dis-cuss the properties of the average squared velocity bythe use of heat transfer Fourier equation. We use somegeometrical properties of the boundary to fit into the re-quired parameters of the equation. Section III is devotedto construct the billiards approach of the problem. Wethem obtain the equations that describe the dynamics ofthe model and discuss the several types of characteriza-tion including steady state, dynamical regime, numericalsimulations, critical exponents and the behavior of theprobability distribution function for the velocity of theparticles. The connection of the two parts is made in Sec. IV where a relation between the time and numberof collisions is obtained. Conclusions and discussions aremade in Sec. V. II. HEAT TRANSFER APPROACH
We discuss in this section the approach involving heattransfer. To start with we assume that there is a set ofidentical particles moving inside a closed boundary. Thedensity of the particles is considered sufficiently smallsuch that the particles are noninteracting. Figure 1(a)shows an illustration of the system. We assume theboundary of the billiard is moving in time, therefore, thisis the mechanism responsible for the exchange of energywith the particles: collisions! The boundary is at a tem-
FIG. 1: (a) Sketch of the billiard boundary and an ensemble ofnoninteracting particles. (b) Illustration of the heat transferregion. The arrows direction point the heat flux. perature T b that is considered fixed and does not changewith the dynamics of the particles. Hence the boundaryworks as a thermal bath and two obvious conclusions canbe extracted. If the temperature of the gas of particles isless than T b , then the boundary gives energy to the parti-cles raising up the temperature of the gas. On the otherhand, if the temperature of the particles is larger than T b , the heat bath absorbs energy from the particles anddissipate it along with the chain of nearby atoms of theboundary – hence reducing the temperature of the gas.There is a region near the border of the billiard wherethe particles can exchange energy which we denote as acollision region.The Hamiltonian that describes the dynamics of eachparticle is given by H = p m + V ( q x , q y , t ) , (1)where p = p x + p y corresponds to the momentum of theparticle and V is the potential energy which is written as V ( q x , q y , t ) = (cid:26) q x , q y , t ) < R ( t ) ∞ if ( q x , q y , t ) = R ( t ) , (2)where R is the radius of the boundary written in polar co-ordinates, which assumes the following form in this work R ( θ, η, t ) = 1+ ηf ( t ) cos( pθ ), where p is any integer num-ber. A non integer number leads to an open boundary towhere the particles can escape. η is a parameter whichcontrols the circle perturbation. If η = 0, the boundaryis a circle, that is integrable [6], in billiards terminology,due to the conservation of energy and angular momen-tum, while η (cid:54) = 0 leads the phase space to be mixedwhen f ( t ) is a constant [30]. The function f ( t ) leads tothe time perturbation of the boundary and we considerin this work two different types of perturbation: (i) pe-riodic oscillations and; (ii) random oscillations. For case(i) the function is written as f ( t ) = 1 + ε cos( ωt ) where ε is the amplitude of oscillation and ω is the angular fre-quency, which we set it as fixed ω = 1. For the randomcase (ii), the function f ( t ) assumes the same expressionas in the case (i) however, at the instant of the impact,we assume there is a random phase Z ( t ), given randomnumbers Z ∈ [0 , π ], such that the velocity of the mov-ing boundary is given by (cid:126)V b ( t ) = ddt [ (cid:126)R b ( t ) + Z ( t )]. Thischoice is made in such a way to avoid the possibility ofhaving the chance of the particles moving outside of theboundary, hence a non physical situation. At the sametime, this is an easy way to introduce randomness in themodel. In this section we discuss the thermodynamicalproperties based on the heat transfer equation – Fourierlaw – and the geometrical parameters of the boundarywill be used in the approach. In next section we describethe dynamics by using the billiards formalism hence writ-ing the dynamical equations of the particle and averagingthe velocity as a function of the number of collisions aswell as along an ensemble of particles.The equation that governs the heat transference [33] iswritten as ∂Q∂t = − κ(cid:96) ∂T∂x , (3)where κ corresponds to the heat conductivity coefficient, (cid:96) is length along the boundary to where the heat canflow and is obtained from the geometrical properties ofthe boundary, ∂Q∂t denotes the flux of heat from a re-gion where there is a temperature difference ∆ T and ∂T∂x corresponds to the temperature gradient. We present ashort discussion of the Fourier equation in Appendix 1and an interpretation of the heat conductivity coefficient κ for the one-dimensional case. The minus ( − ) is re-lated to the fact the heat flow from the region of higherto the lower temperature, hence opposite to the tem-perature gradient [33]. Figure 1(b) illustrates schemati-cally the collision zone and the region to where heat canflow. The effective length (cid:96) to where heat can flow is obtained from (cid:96) = (cid:82) π R ( θ, η, ε, p, t ) dθ = (cid:82) π [1 + η [1 + ε cos( t )] cos( pθ )] dθ = 2 π .The two steps we consider to solve Eq. (3) is to obtainthe corresponding expressions for: (i) ∂Q∂t and; (ii) ∂T∂x in such a way that its solution can be obtained. Weknow that the density of particles is considered suffi-ciently small so that each particle does not interact withany other. Therefore, the energy of each particle is dueto the energy associated to the state of its motion, hencekinetic energy. From the energy equipartition theoremwe have that 12 mV ( t ) = KT ( t ) , (4)where K is the Boltzmann constant and V ( t ) corre-sponds to the squared average velocity averaged over theensemble of particles. The knowledge of V ( t ) directlygives the temperature T ( t ).We know from the thermodynamics [33] that anamount of heat transferred in a process depends on thetemperature [37] dQ = cdT , where dQ is an infinitesimalamount of heat transferred at the price of an infinitesi-mal variation dT in the temperature. The parameter c corresponds to the heat capacity of the gas of particles.For an ideal gas c = KN p where N p is the total numberof particles in the gas [34]. With these we have the lefthand side of Eq. (3) is written as ∂Q∂t = cm K ∂∂t V ( t ). Thenext step is to obtain the expression of the right side ofEq. (3). Since the temperature gradient can only happenalong the collision zone, we can consider an approxima-tion that ∂T∂x ∼ = ∆ T ∆ x = T − T b ∆ x , (5)where ∆ x is measured along the collision zone. To obtain∆ x we note that the radius of the boundary can assumetwo extrema: R max = 1 + η (1 + ε ) cos( pθ ) and R min =1 + η (1 − ε ) cos( pθ ), where R max and R min correspond tothe maximum and minimum values of the radius whentime varies. The collision zone then is a region given by∆ R = R max − R min = 2 ηε cos( pθ ). We see that ∆ R isnot constant being dependent directly on θ and has theproperty that ∆ R = 0. Therefore, an approximation for∆ x is obtained from ∆ x = (cid:113) (∆ R ) where(∆ R ) = 12 π (cid:90) π η ε cos ( pθ ) dθ. (6)A straightforward calculation gives ∆ x = √ ηε . Hencethe expression of ∆ T ∆ x = T − T b √ ηε . Incorporating these ap-proximations in the heat transfer equation we end upwith cm K ∂∂t V = − κ(cid:96) √ ηε (cid:104) m K V − T b (cid:105) . (7)Equation (7) is a first order differential equation andthat when solved properly leads to the following result V ( t ) = 2 Km T b + (cid:20) V − Km T b (cid:21) e − πκ √ η(cid:15)c t . (8)From the energy equipartition theorem, the temperatureis written as T ( t ) = T b + [ T − T b ] e − πκ √ η(cid:15)c t . (9)Let us now discuss some possibilities to study fromexperimental approach. Suppose a gas of particles is in-jected in the billiard with a low initial velocity such that T (cid:29) T b . From Eq. (9) and considering only the domi-nant term we have T ( t ) ∼ = T b + T e − πκ √ η(cid:15)c t , (10)therefore, confirming an exponential decay for short t anda convergence to the stationary state at T ( t ) = T b when t → ∞ . Other type of behavior is observed when an en-semble of particles is injected in the billiard with very lowenergy such that T (cid:28) T b . Expanding the exponentialin Taylor series and keeping only the dominant term weend up with T ( t ) = T b πκ √ η(cid:15)c t. (11)This result confirms the temperature grows at short timelinearly in time hence leading the average velocity V ( t ) = (cid:112) V to grow with square root of time, hence V ( t ) = (cid:115) T b πKκ √ mη(cid:15)c √ t. (12) III. BILLIARDS APPROACH
We now discuss how to construct the equations of themapping that describe the dynamics of the particle insideof the billiard. The mapping gives the angular position ofthe particle θ , the angle that the trajectory of the particleforms with a tangent line at the position of the collision α , the absolute velocity of the particle | (cid:126)V | and finally theinstant of the collision with the boundary t at the impact n th with the further impact ( n + 1) th . Figure 2 shows atypical illustration of a billiard and the angles used todescribe the dynamics of the model.The position of the particle at a given state( θ n , α n , | (cid:126)V n | , t n ), written as a function of time is X ( t ) = X ( θ n , t n ) + | (cid:126)V n | cos( α n + φ n )( t − t n ) , (13) Y ( t ) = Y ( θ n , t n ) + | (cid:126)V n | sin( α n + φ n )( t − t n ) , (14)where the time t ≥ t n with X ( θ n , t n ) = R ( θ n , t n ) cos( θ n )and Y ( θ n , t n ) = R ( θ n , t n ) sin( θ n ). As soon as the θ isknown, the angle φ , which corresponds to the angle be-tween the tangent line and the horizontal at X ( θ ) , Y ( θ )is φ = arctan[ Y (cid:48) ( θ, t ) /X (cid:48) ( θ, t )] where Y (cid:48) ( θ, t ) = dY /dθ and X (cid:48) ( θ, t ) = dX/dθ .Considering the particle travels with a constant speedbetween collisions, the distance traveled by the particle FIG. 2: Illustration of four snapshots of the boundary at thefour collisions. measured with respect to the origin of the coordinatesystem is given by R p ( t ) = (cid:112) X ( t ) + Y ( t ). The an-gular position θ n +1 is obtained by solving the equation R p ( θ n +1 , t n +1 ) = R ( θ n +1 , t n +1 ). The time at collision n + 1 is given by t n +1 = t n + √ ∆ X + ∆ Y | (cid:126)V n | , (15)where ∆ X = X p ( θ n +1 , t n +1 ) − X ( θ n , t n ) and ∆ Y = Y p ( θ n +1 , t n +1 ) − Y ( θ n , t n ).We notice that the referential frame of the boundary isnon inertial. We assume also the collisions of the particlewith the boundary are inelastic, hence there is a frac-tional loss of energy upon collision, which we consideronly with respect to the normal component of the veloc-ity. Then at the instant of collision the reflection lawsare (cid:126)V (cid:48) n +1 · (cid:126)T n +1 = (cid:126)V (cid:48) n · (cid:126)T n +1 , (16) (cid:126)V (cid:48) n +1 · (cid:126)N n +1 = − γ (cid:126)V (cid:48) n · (cid:126)N n +1 , (17)where the unit tangent and normal vectors are (cid:126)T n +1 = cos( φ n +1 )ˆ i + sin( φ n +1 )ˆ j, (18) (cid:126)N n +1 = − sin( φ n +1 )ˆ i + cos( φ n +1 )ˆ j, (19)Here γ ∈ [0 ,
1] is the restitution coefficient. If γ = 1we have completely elastic collisions while γ < (cid:126)V (cid:48) corresponds the velocity of theparticle measured in the non-inertial reference frame. Wecan then obtain the tangential and normal componentsof the velocity after collision n + 1 as (cid:126)V n +1 · (cid:126)T n +1 = (cid:126)V n · (cid:126)T n +1 , (20) (cid:126)V n +1 · (cid:126)N n +1 = − γ (cid:126)V n · (cid:126)N n +1 ++ (1 + γ ) (cid:126)V b ( t n +1 + Z ( n )) · (cid:126)N n +1 , (21)where (cid:126)V b ( t n +1 + Z ( n )) denotes the velocity of the bound-ary that is given by (cid:126)V b ( t n +1 ) = dR ( t ) dt (cid:12)(cid:12)(cid:12) t n +1 [cos( θ n +1 ) (cid:98) i + sin( θ n +1 ) (cid:98) j ] , (22)and Z ( n ) ∈ [0 , π ] is a random number introduced in theargument of the velocity of the moving wall to simulatestochasticity into the model.Finally, the velocity of the particle after the collision( n + 1) is given by | (cid:126)V n +1 | = (cid:113) ( (cid:126)V n +1 · (cid:126)T n +1 ) + ( (cid:126)V n +1 · (cid:126)N n +1 ) , (23)when the angle α n +1 is written as α n +1 = arctan (cid:34) (cid:126)V n +1 · (cid:126)N n +1 (cid:126)V n +1 · (cid:126)T n +1 (cid:35) . (24)With the equations above we can now discuss some ofthe statistical properties for the average velocity of theparticle. A. Stationary state
To investigate the average velocity of an ensemble ofparticles we make the following assumption. We con-sider the probability distribution for the velocity in thetwo-dimensional phase space α vs. θ is uniform. In thestochastic model, the one which gives random numbers Z in the argument of the velocity of the moving wall ateach collision, this is observed. If we take the expressionof | (cid:126)V n +1 | and average the squared velocity for the ranges θ ∈ [0 , π ], α ∈ [0 , π ] and t ∈ [0 , π ] we end up with V n +1 = V n γ V n γ ) η ε . (25)In the steady state regime the mean-squared velocityis obtained considering V n +1 = V n = V , and after iso-lating V we obtain V = (1 + γ ) η ε − γ ) . (26)If we define the root mean square velocity as V = (cid:112) V ,we have V = ηε (cid:112) (1 + γ )(1 − γ ) − / . (27)We notice from Eq. (27) that the exponent headingthe term (1 − γ ) is − / ηε ) is 1. We discuss these exponents latteron. B. Dynamical regime
An easy way to study the dynamical regime is trans-form the difference equation given in Eq. (25) into adifferential equation where the solutions can be easier totrack. We assume for a large ensemble that V n +1 − V n = V n +1 − V n ( n + 1) − n ∼ = dV dn , (28) which leads to dV dn = V γ −
1) + (1 + γ ) η ε . (29)A straightforward integration considering the initial con-dition V at n = 0 gives V ( n ) = V e ( γ − n + (1 + γ )4(1 − γ ) η ε (cid:20) − e ( γ − n (cid:21) . (30)The dynamics of V ( n ) = (cid:113) V ( n ) is described by V ( n ) = (cid:115) V e ( γ − n + (1 + γ )4(1 − γ ) η ε (cid:20) − e ( γ − n (cid:21) . (31)Two important limits are obvious from Eq. (31). Thefirst one considered is when V (cid:29) (1+ γ ) / (1 − γ ) − / ηε hence leading to an exponential decay of the velocity V ( n ) ∼ = V e ( γ − n ∼ = V e ( γ − n . (32)The second one is observed when the initial velocity issufficiently small, say V ∼ = 0, the dominant expressionfor V ( n ) is V ( n ) = (1 + γ ) / − γ ) − / ηε (cid:20) − e ( γ − n (cid:21) / . (33)A Taylor expansion in Eq. (33) gives that V ( n ) ∼ ηε √ n. (34) C. Numerical simulations
Let us now discuss the behavior of the squared aver-age velocity via numerical simulations. The range of γ we are interested in is γ → γ = 1 (conservative case)the average velocity must grow unbounded. However for0 < γ < − γ ). The simulations were madein such a way that each initial condition has a fixed initialvelocity, V = 10 − , ηε ∈ [0 . , .
02] and randomly cho-sen α ∈ [0 , π ], θ ∈ [0 , π ], t ∈ [0 , π ]. Moreover, aftereach time step, a random number [ Z ( n )] is drawn in theequation of the velocity of the moving wall introducingstochasticity into the model. For computing the averagevelocity numerically, two different procedures were ap-plied: (i) we evaluate the average velocity over the orbitfor a single initial condition and; (ii) average the velocityover an ensemble of initial conditions. Hence, the averagevelocity is written as < V > ( n ) = 1 M M (cid:88) i =1 n + 1 n (cid:88) j =0 V i,j , (35) FIG. 3: (a) Plot of < V > vs. n for different values of γ andtwo combinations of ηε . (b) Overlap of the curves shown in(a) onto a single and universal plot after the following scalingtransformations: n → n/ [(1 − γ ) z ( ηε ) z ] and < V > →
2. Similar values were obtained for allcurves we simulated for the range of γ ∈ [0 . , . ηε and vary γ , a power law fitting for
2, asshown in Fig. 4 (a). A fitting to the plot of n x vs. (1 − γ )gives z = − . ∼ = −
1. Finally if we keep constant(1 − γ ) a fitting to the behavior of < V sat > vs. ηε gives α = 1 . ∼ = 1 while a plot of n x vs. ηε yields z = − . ∼ = 0. When the two scaling laws (37)are used to check the exponents, the results obtained areremarkably in well agreement with the simulations. D. Averaging the velocity along n As given by Eq. (30), the squared velocity was ob-tained considering an average over an ensemble of par-ticles. However, the simulations were made using either ensemble average as well as average on time. Therefore,we have to find a corresponding expression of the squaredvelocity when it is also averaged over the number of col-lisions n . The average squared velocity is written as < V ( n ) > = 1 n + 1 n (cid:88) i =0 V ( i ) . (38)The summation over the exponential terms convergessince their arguments are negative. The convergence ofthe exponential terms is n (cid:88) i =0 e ( γ − ) i = (cid:34) − e ( γ − )( n +1) − e γ − (cid:35) , (39)hence the root mean squared velocity is written as V rms ( n ) = (cid:113) < V ( n ) > , therefore V rms ( n ) = (cid:118)(cid:117)(cid:117)(cid:116) (1 + γ ) η ε − γ ) + 1( n + 1) (cid:20) V − (1 + γ ) η ε − γ ) (cid:21) (cid:34) − e ( n +1) ( γ − − e ( γ − (cid:35) . (40)A plot of Eq. (40) is represented as a continuous line inFig. 3(a).Two important limits for Eq. (40) are:1. n = 0, that leads to V rms (0) = V ;2. Considering the limit of n → ∞ , we have V rms ( n → ∞ ) = (cid:115) (1 + γ ) η ε − γ ) . (41)With Eq. (40) we can discuss the behavior of V rms forshort n . In the limit of γ ≈ n + 1)in the denominator of Eq. (40), the expansion of theexponential of the numerator must go until second orderwhile the denominator can go only to the first. Groupingthe terms properly we obtain the expression of V rms ( n )when V ∼ = 0 as V rms ( n ) ∼ = (1 + γ ) ηε (cid:112) ( n + 1) . (42)When n (cid:29) (cid:112) ( n + 1) ∼ = √ n then we have V rms ( n ) ∼ = (1+ γ ) ηε √ n . E. Critical exponents
The five relevant critical exponents that describe thescaling properties of the average velocity curves are β , α i and z i with i = 1 ,
2. The exponents α and α areobtained for the regime of n → ∞ . From Eq. (41) we ob-tain α = − / α = 1. The exponent β comes fromEq. (42). When n (cid:29) β = 1 /
2. Finally, thecrossover iteration number n x can be estimated when Eq.(42) intersects Eq. (41). A straightforward calculationgives n x = 4(1 + γ ) (1 − γ ) − . (43)Then we conclude that z = − z = 0. F. Velocity distribution
Let us discuss here how is the shape of the velocity dis-tribution for the dynamics in the dissipative case. It isimportant to notice that the lowest velocity for a movingparticle is limited to the lowest velocity of the movingboundary, hence V l = − η(cid:15) . Upward velocities are un-bounded but unlimited energy growth is not observeddue to the dissipation. The lower limit for the veloc-ity plays a major rule on the distribution of the veloc-ity and to illustrate this we discuss the following case.Suppose an ensemble of initial conditions with differentangular variables, α , θ , but with the same initial veloc-ity is given. The initial velocity is chosen in such a waythat it is located in a region above of lower velocity limitand, at the same time, below than the saturation. Thedynamics evolves as follows. For short number of colli-sion with the boundary, part of the ensemble of particlesraises the velocity while the other part reduces velocity.This distribution is Gaussian, as shown in Fig. 5 in blue(dark gray) color for and initial velocity of V = 0 . (cid:15)η = 0 . γ = 0 . P ( V ) n=10n=100 P ( V ) n=50000 FIG. 5: Plot of the normalized probability distribution forthe velocity for an ensemble of 10 particles in the dissipativeand stochastic oval billiard. Blue (dark gray) was obtainedafter 10 collisions while red bars (light gray) was obtainedafter 100 collisions. The in-box was obtained after 50 , V = 0 . (cid:15)η = 0 .
02 and γ = 0 .
999 for p = 2. and p = 2, although other values would lead to similarresults. Moreover, a total of 2 . × different initialconditions were considered in the ensemble. As soon asthe dynamics evolves, the Gaussian distribution flats it-self in both sides until the left hand side curves touchesthe lower limit of the velocity. See the red (light gray)bars obtained for the distribution after 100 collisions withthe boundary. At this point, the distribution experiencesa break of symmetry and hence if the initial velocity ofthe distribution is lower that the saturation, the aver-age velocity starts to grow until approaches the satura-tion. From this point of symmetry break and beyond thispoint, the distribution is not Gaussian anymore and ithas similar shape as shown in the in-box of Fig. 5. Suchdistribution was obtained after 50 ,
000 collisions of theensemble of particles with the boundary. Although thedistribution is not Gaussian anymore due to the breakof symmetry at V = V l , the distribution has clearly apeak and decays monotonically for large enough valuesof velocity warranting convergence for the average ve-locity as well other momenta of the distribution. It isworth to mention that such a break in the symmetry inthe probability distribution was previously observed fora one dimensional Fermi Ulam model [36]. There, theauthors show that the velocity/energy distribution canbe described perfectly by a folded normal distribution. IV. CONNECTION BETWEEN THE TWOAPPROACHES
The results discussed in Section II involving the heattransfer equation were obtained as a function of the time t while in Section III the results were discussed using thenumber of collisions n . It is important to mention thatthe time t and the number of collisions n are variables nottrivially connected with each other. It happens becausea particle moving with high speed can experience manymore collisions with the boundary when compared witha particle with low energy at the same interval of time.In this section we discuss a way to make a connection ofthe two variables therefore linking the results discussedin Secs. II and III.Given the particle travels with constant velocity be-tween collisions, the length of time between two collisionsis ∆ t = d/ | (cid:126)V | where d is the distance traveled by the par-ticle and | (cid:126)V | is its absolute velocity. Therefore, the totaltime spent at n collisions is written as τ = n (cid:88) i =0 d i | (cid:126)V i | . (44)The summation in Eq. (44) seems to be not easy to bemade. As an attempt to have an explicit expression in-volving the relevant parameters of the system consideredwe will do the summation in two stages evaluating thenthe numerator separately then the denominator.From the numerator we can estimate the mean freepath, which we represent as d = 1( n + 1) n (cid:88) i =0 d i , (45)where d i is defined as the distance from two collisions as d i = (cid:112) [ x ( θ i +1 ) − x ( θ i )] + [ y ( θ i +1 ) − y ( θ i )] , (46)where x ( θ ) = R ( θ ) cos( θ ) and y ( θ ) = R ( θ ) sin( θ ). Thedynamics of each particle is chaotic, therefore when wedo an average over θ ∈ [0 , π ] we obtain d = (cid:115) η (cid:20) ε (cid:21) . (47)The second part we have to consider is (cid:80) ni =0 1 V i . Todo that we consider the variation of the velocity from thecollision i to ( i + 1) is small so that the summation canbe approximated by n (cid:88) i =0 V i ∼ = (cid:90) n V ( n (cid:48) ) dn (cid:48) . (48)The expression for τ obtained for the explicit form of V as shown in Eq. (31) is not an easy equation to dealwith and the result is reported in the Appendix 2 forthe interested reader. Instead of dealing with the wholeequation we consider an easier approach. As discussed inRef. [35], the behavior of the average squared velocity,when settled in scaling variables can be described by afunction of the type f ( x ) = (cid:20) x x (cid:21) β , (49)where β is the accelerating exponent. In our case β =1 /
2. Therefore, the scaled variables considered are: f → V √ (1 − γ ) ηε and x → n (1 − γ ). Incorporating these twoequations as the behavior of V ( n ) we end up with thefollowing equation to be solved τ = d (cid:112) (1 − γ ) ηε (cid:90) dn (cid:113) n (1 − γ )1+ n (1 − γ ) , (50)After doing the integration (see Appendix 3 for the resultof the integral) and keeping only the leading term in n we obtain τ ∼ = d (cid:112) (1 − γ ) ηε n. (51) V. CONCLUSIONS
We have studied some dynamical and statistical prop-erties of gas of non interacting particles in a time de-pendent and dissipative oval billiard. We have investi-gated the behavior of the average velocity of the particlesas a function of time and the number of collisions withthe moving boundary by using two different approaches,namely, involving (i) heat transfer and (ii) billiards. Wehave obtained an empirical expression for the averagesquared velocity by using the equilibrium condition atthe steady state regime. Such an expression allowed us tomake a connection with the thermodynamic, more pre-cisely by using the Fourier law for heat transfer. Theresulting equation have shown that the temperature ofthe gas reaches the thermal equilibrium for sufficientlylong time. Our results also have demonstrated that theaverage squared velocity grows as a power law and aftera crossover it tends to a constant plateau. Furthermore,the stronger is the dissipation the faster is the transitionfrom growth to saturation. Finally, by using an empiricalfunction to describe the behavior of the average squaredvelocity, we have shown that time and number of colli-sions are linearly correlated.
ACKNOWLEDGMENTS
EDL thanks FAPESP (2012/23688-5) and CNPq(303707/2015-1) for financial support. MVCG acknowl- edges CNPq for financial support. DFMO thanks toJames S. McDonnell Foundation.
Appendix 1
This appendix is devoted to a short discussion on theheat flow equation [33]. The heat can indeed quantifyan amount of energy which is transferred due to a tem-perature gradient. The amount of heat flowing along thetemperature gradient depends on the thermal conductiv-ity κ . The heat flows from a region of high to low temper-ature, therefore this flow is contrary to the temperaturegradient. In a generic 3-D system, the heat flux vector (cid:126)J is written as (cid:126)J = − κA(cid:126) ∇ T , where A corresponds to asection of area perpendicular to where the flow of heatis flowing while (cid:126) ∇ T gives the gradient of temperature.The signal ( − ) is introduced to represent a flow contraryto the temperature gradient, i.e. from higher to lowertemperature. The vector (cid:126)J indeed represents a certainamount of energy which is flowing through an area A ata given interval of time due to a gradient of temperature.In the system we are considering in this paper, the flowof heat is not crossing a perpendicular area, but rather itcrosses the border of the billiard. Hence, in the case 2-Das discussed, the heat transfer equation is written as J = ∂Q∂t = − κ(cid:96) ∂T∂x , (52)where J represents the amount of heat which is trans-ferred around the border (cid:96) of the billiard at a given in-stant of time due to a temperature gradient representedas ∂T∂x . In our case, the thermal conductivity coefficient κ denotes the constant of proportionality between theamount of energy flowing in the border of the billiard (cid:96) per unit of time due to a temperature gradient. Appendix 2
When we consider V ( n ) as given by Eq. (31) to obtainthe expression for τ the direct integral is τ = d (cid:90) n dn (cid:48) (cid:115) V e ( γ − n (cid:48) + (1+ γ )4(1 − γ ) η ε (cid:20) − e ( γ − n (cid:48) (cid:21) . (53)A straight integration yields0 τ = 8 (cid:113) η (cid:0) ε (cid:1) ηε (cid:112) (1 + γ )(1 − γ ) × arctanh (cid:114) (1+ γ ) η ε − γ ) + (cid:16) V − (1+ γ ) η ε − γ ) (cid:17) e ( γ − nηε (cid:113) (1+ γ )(1 − γ ) − arctanh V ηε (cid:113) (1+ γ )(1 − γ ) . (54)With some algebra one can isolate n as a function of τ from the equation above. Appendix 3
The solution of the integral τ = d (cid:112) (1 − γ ) ηε (cid:90) dn (cid:113) n (1 − γ )1+ n (1 − γ ) , (55) is given by τ = d (cid:112) (1 − γ ) ηε
12 2 (cid:112) ( n − n γ + n ) (cid:112) (1 − γ ) (cid:112) (1 − γ ) (cid:112) − n ( − − n + nγ ) (cid:113) − n (1 − γ ) − − n + nγ n + d (cid:112) (1 − γ ) ηε
12 ln (cid:20) − (cid:18) − − n +2 nγ − √ ( n − n γ + n ) √ − γ √ (1 − γ ) (cid:19)(cid:21)(cid:112) (1 − γ ) (cid:112) − n ( − − n + nγ ) (cid:113) − n (1 − γ ) − − n + nγ n . (56)After grouping the terms and considering only theleading term we have τ = d (cid:112) (1 − γ ) ηε (cid:34) n (cid:115) n (1 − γ ) (cid:35) . (57)Expanding the square root and keeping only the first or-der we have τ = d (cid:112) (1 − γ ) ηε (cid:20) n + 12(1 − γ ) (cid:21) , (58) therefore τ ∼ = d (cid:112) (1 − γ ) ηε n. (59) [1] G. Birkhoff, Dynamical Systems, American Mathemati-cal Society, Providence, RI, USA, 1927.[2] Ya. G. Sinai, Russian Mathematical Surveys, , 137(1970).[3] Ya G. Sinai, Not. Am. Math. Soc.
412 (2004).[4] L. A. Bunimovich, Functional Analysis and Its Applica-tions, , 73 (1974).[5] L. A. Bunimovich and Ya. G. Sinai, Communications inMathematical Physics, , 479 (1981).[6] N. Chernov and R. Markarian, Chaotic Billiards . Amer-ican Mathematical Society, Vol. 127, 2006.[7] L. A. Bunimovich, Commun. Math. Phys., , 295(1979).[8] L. A. Bunimovich, Chaos , 187 (1991).[9] L. A. Bunimovich and L. V. Vela-Arevalo, Chaos, ,026103 (2012). [10] S. Tabachnikov, Geometry and Billiards (Providence, RI:American Mathematical Society) (2005).[11] C. P. Dettmann and O. Georgiou, Physica D 238, 2395-2403 (2009).[12] C. P. Dettmann and O. Georgiou, J. Phys. A.: Math.Theor. 44, 195102, (2011).[13] M. Robnik, Nonlinear Phenomena and Chaos (Bristol:Hilger) 303 (1986).[14] D. P. Sanders, Phys. Rev. E , 016220 (2005).[15] A. B¨acker, R. Ketzmerick, S. L¨ock, M. Robnik, G. Vid-mar, R. H¨ohmann, U. Kuhl, H.-J. St¨ockmann, Phys. Rev.Lett. , 174103 (2008).[16] D. F. M. Oliveira, M. Robnik, Int. J. Bif. Chaos, ,1250207 (2012).[17] A. B. Ryabov and A. Loskutov. J. Phys. A: Math. Theor., , 125104 (2010). [18] V. Gelfreich, D. Turaev, J. Phys. A , 212003 (2008).[19] V. Gelfreich, D. Turaev, Comm. Math. Phys. , 769(2008).[20] E. Fermi, Phys. Rev, , 1169 (1949).[21] A. Loskutov, A. B. Ryabov and L. G. Akinshin, J. Phys.A, , 7973 (2000).[22] A. Loskutov, A. B. Ryabov and L. G. Akinshin, J. Exp.and Theor. Physics, , 966 (1999).[23] A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V.Constantoudis, P. Schmelcher, Phys. Rev. E, 76 016214(2007).[24] D. F. M. Oliveira, E. D. Leonel, Chaos , 026123 (2012).[25] E. D. Leonel, D. F. M. Oliveira and A. Loskutov, Chaos, , 033142 (2009).[26] C. P. Dettmann and O. Georgiou, Phys. Rev. E, ,036212 (2011).[27] R. E. de Carvalho, F. C. de Souza and E. D. Leonel, J.Phys. A: Math. Gen., , 3561 (2006).[28] F. Lenz, F. K. Diakonos and P. Schmelcher, Phys. Rev. Lett., , 014103 (2008).[29] F. Lenz, C. Petri, F. R. N. Koch, F. K. Diakonos and P.Schmelcher, New J. Phys., , 083035 (2009).[30] M. V. Berry, Eur. J. Phys. , 91 (1981).[31] D. F. M. Oliveira and E. D. Leonel, Physica A , 1009(2010)[32] E. D. Leonel and L. A. Bunimovich, Phys. Rev. Lett.,104