On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas
Péter Bálint, Thomas Gilbert, Péter Nándori, Domokos Szász, Imre Péter Tóth
JJournal of Statistical Physics manuscript No. (will be inserted by the editor)
On the limiting Markov process of energy exchanges in ararely interacting ball-piston gas
Péter Bálint †‡ ·
Thomas Gilbert§ · Péter Nándori ¶ (cid:107) · Domokos Szász † ·
Imre Péter Tóth †‡ To David Ruelle and Yasha Sinai on the occasion of their 80th birthdays (cid:63)
Version of September 21, 2018
Abstract
We analyse the process of energy exchanges generated by the elastic colli-sions between a point-particle, confined to a two-dimensional cell with convex bound-aries, and a ‘piston’, i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered asthe elementary building block of a spatially extended high-dimensional billiard mod-eling heat transport in a class of hybrid materials exhibiting the kinetics of gases andspatial structure of solids. Using heuristic arguments and numerical analysis, we arguethat, in a regime of rare interactions, the billiard process converges to a Markov jumpprocess for the energy exchanges and obtain the expression of its generator.
Keywords
Transport processes & heat transfer · chaotic billiards · mean free path · stochastic processes Fourier’s law of heat conduction [3], according to which the heat current in a materialis proportional to the gradient of its local temperature, has over the last two centuries (cid:63)
We are deeply honored and privileged to dedicate this paper to David Ruelle and YashaSinai, great founding fathers of rigorous statistical physics. Their works were fundamental increating the subject and have profoundly changed the way people think about it. In particular,beyond its wider and deeper impact, Ruelle’s 1998 work [1], as well as his Brussels lecture [2],were key to reinvigorating the general interest toward understanding Fourier’s law of heatconduction and raising hopes for a satisfactory answer to this difficult problem. The theory ofhyperbolic billiards, initiated by Sinai, helped engineer models which offer the most promisingperspectives in this direction. It is our hope this paper testifies to their influence. † Institute of Mathematics, Budapest University of Technology and Economics, Egry József u.1, H-1111 Budapest, Hungary ‡ MTA-BME Stochastics Research group, Egry József u. 1, H-1111 Budapest, Hungary§ Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles,C. P. 231, Campus Plaine, B-1050 Brussels, Belgium ¶ Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park,MD 20742, USA (cid:107)
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012USA a r X i v : . [ m a t h . D S ] J u l P. Bálint et al. proved a powerful phenomenological tool for describing the process of energy transferin physical systems. Yet, in spite of being well understood at a macroscopic level,the derivation of this law from a microscopic point of view arguably remains one ofmathematical physics’ great challenges. Thus, in their millenium review, Bonetto etal. [4], after offering “a selective overview of the current state of our [then] knowledge(more precisely of our ignorance) regarding the derivation of Fourier’s Law”, proceededto the observation that“There is however at present no rigorous mathematical derivation of Fourier’slaw [. . . ] for any system (or model) with a deterministic, e.g. Hamiltonian,microscopic evolution.”An intriguing review of Fourier’s work and its influence with a detailed chronologyof some of the main developments can be found in reference [5]. Other useful reviewsdevoted to the non-equilibrium statistical mechanics of low-dimensional systems includereferences [6] and [7].Building upon the earlier work of Bunimovich et al. [8], Gaspard and Gilbert thenset out in 2008 [9] to consider the regime of rare interactions of a class of models,henceforth referred to as GG-models, which, from the point of view of ergodic theory[8], are intermediate between the gas of hard balls and the periodic Lorentz gas. Ingeneral, the GG-model can be thought of as a billiard chain (or Z d -network in dimension ≤ d ≤ n ) of n -dimensional cells with (semi-) dispersing walls, each containing asingle ball particle trapped inside it. Ball particles are moreover let to interact amongneighbours under the control of the geometry of the interface between their respectivecells. Hence by excluding mass transport the model focuses on energy transport solely.A standard two-step strategy for analysing the process of heat transport is to firstidentify the conditions under which a mesoscopic description can be attained fromthe microscopic one, and second, by taking the hydrodynamic limit of the mesoscopicprocess to obtain, in the diffusive scaling, the heat equation at the macroscopic scale.The completion of this step also implies gaining an analytical form for the coefficientof heat conductivity.The present work does not deal with the analysis of the second step of this strategy.In particular, we will not address the precise form of the coefficient of thermal conduc-tivity associated with our model. We note, however, that Sasada, inspired by StefanoOlla’s remarks on the results announced in reference [9], recently reported [10] that thecoefficient of heat conductivity figuring in the papers of Gaspard and Gilbert, see inparticular reference [11], corresponds to the contribution to the heat conductivity fromthe static correlations alone, while the true transport coefficient should also include acontribution from dynamic correlations. Whereas the latter contribution appears to bevery small in comparison to the former, it does not vanish. This issue, which deservesfurther clarification, will be the subject of future work. Another important remarkhere is that for handling the second step of the strategy for GG-models the necessaryspectral gap has already been obtained in reference [12]; see also reference [13]. Theanalogous lower estimate of the spectral gap for the model to be introduced below isso far an open question.We mention here that progress in this area has been paralleled by interestingprospects aiming at understanding the emergence of Fourier’s law when small noiseis added to simple deterministic models, e.g. to weakly anharmonic crystals [14, 15],or harmonic oscillators [16]. Such systems have also been considered in regimes of rareinteractions [17, 18] Another line of research aims at studying energy exchanges be- tochastic limit of a rarely interacting ball-piston gas 3 tween the cells of regular lattices through the mediation of point particles [19]; see alsoreference [20] for recent progress based on the seminal work of Kipnis et al. [21]. Amixture of those two approaches was treated in reference [22].Going back to the first step of the strategy described above, in the GG-model,the reduction from the deterministic dynamics at the microscopic scale to a stochasticprocess at the mesoscopic scale, emerges in the rare interaction limit (RIL), out of atwo-stage relaxation process. In the RIL, the rate of interactions between neighbouringball particles is arbitrarily low, which implies that a form of local equilibrium is achievedon the scale of every cell, each kept at constant energy. The convergence to localequilibrium is indeed controlled by the rate of collisions between a particle and thewalls of its cell, which can be made much larger than the rate of binary collisions, i. e.of collisions between two neighbouring particles. Since averaging takes place, energytransfers between neighbouring particles behave stochastically, with every cell of thesystem acting as a fundamental unit whose state is specified by the energy of its ballparticle. In effect, the RIL yields for the Hamiltonian kinetic equation of any finitesubsystem the generator of a Markov jump process for the energies of the ball particles.The GG-model can consist of networks of two-dimensional particles (discs) confinedto identical cells [23]. It can also consist of networks of three (or higher)-dimensionalparticles (spheres) confined to identical cells [24]. One can think of extensions of suchmodels with several particles trapped in every cell; their numbers may be identicalor vary from cell to cell and the cells of the network may no be all identical. Thedimensionality of the dynamics would then vary from cell to cell. One main goal of thepresent paper is to introduce yet another version of the GG-model, the simplest of itsclass, for which we deem the (mathematically rigorous) completion of the first step ofthe GG-strategy realistic.Let us explain why we think that such a modification is necessary. The ‘simplest’task in the original GG-programme is the treatment of a two-cell system with two in-teracting discs. This is actually isomorphic to a 4-dimensional semi-dispersing billiard.However, statistical properties of higher-dimensional (larger than two) billiards are sofar understood exclusively for finite-horizon strictly dispersing billiards [25], and eventhen only under the notorious ‘complexity hypothesis’.It was therefore suggested in reference [26] that by exploiting the RIL feature of theapproach of reference [9], a way out of this high-dimensional quagmire would be to applythe method of ‘standard pairs’ [27], a very efficient tool stemming from the developmentof Markov approximation methods, which permits the use of statistical properties oflower dimensional projections of the model—in this case of well-understood planarSinai-billiards. Be that as it may, this idea led to further technical difficulties.For this reason we introduce below a ball-piston model , which belongs to the class ofGG-models, but for which the dimension of the (simplest) isomorphic semi-dispersingbilliard shrinks from 4 to 3. In this model, a disc particle caged in a two-dimensionalcell with dispersing walls is let to interact with a ‘piston’ moving in a one-dimensionalinterval. This is isomorphic to a 3-dimensional semi-dispersing billiard. We believethat coping with its RIL is already a realistic question, which is the subject of a longerproject, involving four of us, and still in progress; as to the first related publication,see [28]. Suffice it to say here that the application of the method of standard pairs tothe ball-piston model is met with difficulties similar to those alluded to above. However,in this model their occurrence can be tracked and explicitly described and ultimately,as we hope, treated. P. Bálint et al.
In this respect, it is worth mentioning that the method of standard pairs has alreadybeen successfully applied to models of heat conduction. Thus, in reference [29] a modelof weakly interacting Anosov flows was considered for which the appropriate long timelimit is a chain of interacting diffusion processes rather than the kind of Markov jumpprocesses which are expected to arise in rarely interacting systems. More recently, inreference [30], the authors obtained an exponential limit law for the first encounterof two small discs on a planar Sinai billiard table, a result somewhat closer to ourprogram.A further aim of the present work is the calculation of several relevant character-istics of the ball-piston model, in particular:(i) The unconditional and conditional mean free times of binary collisions (conditioningon the outgoing energy partition between the two particles after a binary collision);(ii) The transition kernel of the Markov jump process expected to emerge in the RIL;(iii) The restriction of the invariant measure to the binary collision surface at fixedenergy.Since some of these calculations are new, even as to their mathematical content, weendeavour to formulate our arguments in a language which we hope will be accessibleto both mathematicians and physicists.Finally, we describe a computational test of the conjecture that the emerging meso-scopic limit of the rarely interacting ball-piston model is indeed the Markov jumpprocess we claim it is. This conjecture relies on the assumption that, in the RIL, twosuccessive binary collisions are separated by enough wall collision events that averagingtakes place. We test this by considering several outgoing laws from a binary collision andcompare the ingoing laws at the next binary collision event to the relevant equilibriumlaw by using the Kullback-Leibler divergence [31]. We provide numerical evidence thatreducing the rate of binary collisions yields a limiting distribution of energy exchangesconsistent with the expected result.The paper is organised as follows. The ball-piston model is introduced in section 2.Section 3 is devoted to the calculation of the ball-piston mean free time and collisionrate. In section 4 we introduce the notion of conditional mean free time and proceed toits calculation. The derivation of the transition kernel of the conjectured Markov jumpprocess is provided in section 5, together with a description of our numerical test ofthe validity of the conjecture, as well as a discussion of numerical results. Concludingremarks are given in section 6. Specific calculations are provided in appendix A (volumeintegrals), appendix B (wall collision frequencies) and appendix C (restriction of theinvariant measure to the binary collision surface at fixed energy).
The ball-piston gas, shown in figure 1, is a collection of alternating balls and pistons,arranged in a periodic structure, with every particle confined to its own cell. Ballsand pistons are particles of two different types. On the one hand, balls are point-particles with two degrees of freedom. They move in two-dimensional closed cells whoseboundaries are defined by impenetrable circular obstacles placed at the vertices of asquare lattice. Pistons, on the other hand, have only one degree of freedom. They areone-dimensional vertical or horizontal line-segments that are allowed to move back andforth along perpendicular intervals placed between two neighbouring ball cells. Whereas tochastic limit of a rarely interacting ball-piston gas 5
Fig. 1: Ball-piston gas in a random configuration of the positions and velocities ofthe balls and pistons. The arrows’ lengths and colours reflect the magnitude of thecorresponding particle or piston’s kinetic energy (blends of blue for low energy values,red and yellow for high energy values). Here periodic boundary conditions are applied:the piston cells of the right-most column are identical to those of the left-most column,and similarly for the upper and lower rows.pistons are unaffected by the presence of the circular obstacles in the ball cells, theydo interact elastically with balls whenever collisions occur, thereby exchanging theirhorizontal or vertical velocity components (both balls and pistons have unit masses).By choosing the lengths of the piston cells large enough that their extremities lay insidethe ball cells (one symmetrically on each side), we allow for energy exchanges betweenevery ball and piston pair, the likelihood of which depends on the piston’s penetrationlength into the ball cell. In turn, whereas mass transport of either species is prohibitedby the confining walls in every cell, energy exchanges between balls and pistons induceheat transport on the scale of the ball-piston gas.The present study focuses on a minimal version of the ball-piston gas, such asshown in figure 2a. Here a single pair of ball and piston—in this case a horizontallymoving vertical line-segment—is let to interact. This model can in fact be viewed asa three-dimensional billiard, rendered in figure 2b: it is equivalent to the free motionof a point-particle in a three-dimensional cavity, undergoing elastic reflections upon itsboundary. The corresponding collision map is a four-dimensional symplectic map.The parameters relevant to the definition of the model are displayed in figure 2.A point-particle (ball) of unit mass moves freely in the interior of a cell (the ball cell)whose boundaries are delimited by four arc-circles of common radius ρ , / < ρ < / √ , centered at the four corners of a unit cell, and performs elastic collisions withthem. A vertical line segment of height η and unit mass, which we call piston, moveshorizontally back and forth between the two edges of an interval of length λ + 2 δ (thepiston cell), centered at the middle point of the cell’s right edge, where λ = (cid:112) ρ − P. Bálint et al. θ ρ δη λ / ( /
2, 1 / )(- /
2, 1 / ) ( / - / )(- / - / ) (a) Minimal ball-piston model (b) Three dimensional rendition Fig. 2: (a) The minimal ball-piston model consists of a single ball-piston pair, hereshowed in a random configuration. The relevant parameters are defined in the text. (b)Three-dimensional rendition of the billiard boundary.measures the length of the interval between the two intersections of opposite discs(and such that tan θ = λ ). The parameter δ , < δ < ρ/ √ − λ/ , measures thelength of penetration of the piston inside the ball cell and therefore determines thepossibility of interactions between the ball and the piston. The height of the piston, η = 1 − (cid:112) ρ − ( λ/ δ ) , is such that, at its left-most position, it lies inside theboundary of the ball cell. The positions of the ball and piston must be initially chosenso that the ball is located to the left of the piston; the ball cannot move passed thepiston.We let Γ ⊂ R denote the three-dimensional ball-piston configuration space and ∂ Γ = ∂ Γ bp ∪ ∂ Γ bw ∪ ∂ Γ pw its boundary, where ∂ Γ bp is the surface of ball-pistoncollisions, ∂ Γ bw the surface of ball-wall collisions, and ∂ Γ pw the surface of piston-wallcollisions. In figure 2b, the first term refers to the slanted darker surface of triangularshape, the second to the vertical walls, and the third to the flat top and bottom walls.The phase space of the billiard flow M is the product of Γ and S , the sphere of unitradius in R , M = Γ × S .A point q = ( q , q , q ) ∈ Γ specifies the horizontal and vertical coordinates of theball, ( q , q ) , and the piston’s position, q . They are such that: ( q ± ) + ( q ± ) ≥ ρ , (1 − λ ) − δ ≤ q ≤ (1 + λ ) + δ ,q ≤ q . (1)The associated velocity vector is v = ( v , v , v ) ∈ S . The system’s total (kinetic)energy is the sum (cid:15) b + (cid:15) p = of the ball and piston energies, (cid:15) b = ( v + v ) and The upper bound on δ is imposed so as to prevent overlap between the piston and a similarhypothetical vertical piston centered on either of the top and bottom edges of the cell, suchas in the cells depicted in figure 1.tochastic limit of a rarely interacting ball-piston gas 7 (cid:15) p = v respectively. The corresponding phase-space point is denoted x = { q , v } ∈ M . The phase space of the billiard map is denoted by M and is given by the productof ∂ Γ and the set of vectors v ∈ S whose scalar product with the unit vector normalto the billiard surface at point q ∈ ∂ Γ is non-negative. We have the decomposition M = M bp ∪ M bw ∪ M pw . We write x = { q , v } ∈ M a phase point of the billiard map.The natural invariant measure of the flow is denoted by µ and is normalised sothat µ ( M ) = 1 . Likewise the invariant measure of the billiard map is denoted by ν andsuch that ν ( M ) = 1 . Let S t x denote the flow generated by the billiard dynamics on M . The first hittingtime is the function τ : M (cid:55)→ R + : x (cid:55)→ inf { t > | S t x ∈ M } . (2)For x ∈ M , τ ( x ) is the return time to the billiard surface [32], or free (flight) time[33]. Similarly, we define the ball-piston free flight time to be the time separating twosuccessive collisions between the ball and piston, τ bp : M bp (cid:55)→ R + : x (cid:55)→ inf { t > | S t x ∈ M bp } . (3)By ergodicity, the mean free time, which is defined to be the infinite n limit of thetime to the n th collision (with any of the surface elements of the three-dimensionalbilliard cavity) divided by the number of collisions n , almost surely exists and is in-dependent of the initial condition if the latter is sampled with respect to the naturalinvariant measure of the billiard flow. It is then equal to τ = E ν ( τ ) ≡ (cid:90) M τ ( x ) d ν ( x ) , (4)measured in terms of the natural invariant measure of the billiard map on M .The ball-piston mean free time, i.e. the average time separating successive collisionsof the ball-piston pair, is defined similarly to be τ bp = E ν bp ( τ bp ) ≡ (cid:90) M bp τ bp ( x ) d ν bp ( x ) , (5)where the measure ν bp is ν conditioned on M bp , ν bp = ν ( M bp ) − ν | M bp , which is thenatural invariant measure of the first return map from M bp to itself.As explained in references [33,34], the presence of the hitting time under the integralin equations (4) and (5) has the effect of lifting the integral on M to a measure on M .We thus obtain an explicit formula for the ball-piston mean free time by taking theratio between the normalising factors of the two invariant measures, that of the flowto that of the map restricted to M bp . For the billiard flow, we write d µ ( x ) = c µ d q d v ,and, for the conditional measure of the billiard map d ν bp ( x ) = c ν bp d q d v ( v · n ) , where n is the ( q -independent) unit vector normal to ∂ Γ bp , n = 1 √ − , , . (6) P. Bálint et al.
These normalising factors are, respectively, c − µ = (cid:90) Γ d q (cid:90) S d v = 4 π | Γ | , (7) c − ν bp = (cid:90) ∂ Γ bp d q (cid:90) S : v · n > d v ( v · n ) = π | ∂ Γ bp | , (8)where we have substituted | S | = 4 π and | B | = π , the volume of the unit ball in R .The ratio between equations (7) and (8) yields the ball-piston mean free time, ormean return time to the ball-piston collision surface, τ bp = c ν bp c µ = 4 | Γ || ∂ Γ bp | . (9)This formula is but a special case of the well-known formula for the mean-free time ofthree-dimensional billiards [33].The simple geometry of the minimal ball-piston model allows for an explicit com-putation of the volume and surface integrals in equation (9): | Γ | = ( λ + 2 δ ) (cid:110) − λ − ρ (cid:2) π − λ ) (cid:3)(cid:111) − (cid:110) δ ( λ + 4 δ ) + (cid:2) − (cid:112) − δ ( λ + δ ) (cid:3)(cid:2) δ ( λ + δ ) + 3 λ (cid:3) (cid:111) − ρ ( λ + 2 δ ) (cid:104) arctan λ − arctan λ + 2 δ (cid:112) − δ ( λ + δ ) (cid:105) , (10) | ∂ Γ bp | = 12 √ (cid:110) ( λ + 2 δ ) (cid:2) − (cid:112) − δ ( δ + λ ) (cid:3) − λ (cid:111) + √ ρ (cid:104) arctan λ − arctan λ + 2 δ (cid:112) − δ ( δ + λ ) (cid:105) ; (11)see appendix A for details.The small δ regime, when ball-piston collisions become arbitrarily rare, is of par-ticular interest, as discussed in section 5. On the one hand, lim δ → | Γ | is simply thearea (A.1) multiplied by λ , the length of the piston’s interval of motion (1) in the limit δ → . On the other hand, the piston’s height in this regime is η (cid:39) λδ , so that theregion the piston can penetrate inside the ball cell forms approximately a triangle ofarea λδ in the ( q , q ) plane. As discussed in appendix A, this triangle is essentiallythe projection on the ball cell of the Poincaré section of ball-piston collisions. Accord-ingly, | ∂ Γ bp | (cid:39) √ λδ . To leading order, the inverse of the mean free time is thereforeproportional to the parameter squared , lim δ → ( τ bp δ ) − = 12 √ (cid:104) − λ − ρ (cid:0) π − λ (cid:1)(cid:105) − , (12) For ρ = , however, we have λ = 0 so that, in the small δ regime, | Γ | (cid:39) δ (1 − π ) and | ∂ Γ bp | (cid:39) √ δ . The two limits ρ → and δ → are therefore not interchangeable: lim ρ → / lim δ → δ | Γ || ∂ Γ bp | = 14 √ − π ) (cid:54) = lim δ → lim ρ → / δ | Γ || ∂ Γ bp | = 34 √ − π ) . tochastic limit of a rarely interacting ball-piston gas 9 which, up to the prefactor, is the inverse of area (A.1).Since the number of collisions up to some time t almost surely increases like t/τ as t → ∞ , it is natural to call f ≡ τ − the collision frequency. However, f can alsobe identified with a probability rate. Indeed, ergodicity implies that f is also, for anytime interval, the expected number of collisions measured in that time interval dividedby its length. Thus, in particular, f can be expressed using the probability to observeat least one collision up to time t , which is µ (cid:0) { x ∈ M | S t x ∩ M (cid:54) = ∅} (cid:1) . Namely, f = lim t → t µ (cid:0) { x ∈ M | S t x ∩ M (cid:54) = ∅} (cid:1) . (13)By the same token, f bp ≡ τ − bp is the ball-piston collision frequency, and also definesa probability rate in the sense that f bp = lim t → t µ (cid:0) { x ∈ M | S t x ∩ M bp (cid:54) = ∅} (cid:1) . (14)We emphasise that equations (13) and (14) justify referring to f and f bp as collision(probability) rates, regardless of the actual distributions of the waiting times τ and τ bp .In particular, these quantities may have distributions which are not exponential, and,accordingly, the collision event process may not be Poisson.Similar considerations apply to the ball-wall and piston-wall collision events. Werefer to appendix B for a computation of the corresponding return times. Say the billiard flow is in a stationary regime and we are observing the successivecollision events that result in energy exchanges between the ball and piston. Let usassume we are only interested in a marginal set of events such that the ball-pistonenergy partition has the value { (cid:15) b , (cid:15) p } , with (cid:15) b + (cid:15) p = . Specifically, we ask: forpoints x ∈ M whose velocity vectors v = ( v , v , v ) are such that ( v + v ) = (cid:15) b and v = (cid:15) p , what is the corresponding mean free time? A formula similar to equation(5) is obtained for this quantity, which we call the conditional mean free time: τ bp ( (cid:15) p ) = E ν bp | (cid:15) p ( τ bp ) ≡ (cid:90) M bp ( (cid:15) p ) τ bp ( x ) d ν bp | (cid:15) p ( x ) . (15)where the measure ν bp | (cid:15) p is the measure ν bp conditioned on the subset M bp ( (cid:15) p ) ⊂ M bp of phase-space points x = { q , v } such that q ∈ ∂ Γ bp and v = ( v , v , v ) ∈ S with v = ±√ (cid:15) p and v > v (consistent with v · n > ).To obtain an explicit formula, it is enough to transpose the normalising factors inequations (7) and (8) to the associated subvolumes of phase-space, τ bp ( (cid:15) p ) = c ν bp | (cid:15) p c µ (cid:15) p . (16)These normalising factors are computed as follows. Note that the three-dimensionalvelocity vector v is allowed to take values on two circles of radii √ − (cid:15) p parallel to et al. the plane ( v , v ) , whose two heights correspond to the two allowed signs of the pistonvelocity v , v = √ − (cid:15) p cos α ,v = √ − (cid:15) p sin α ,v = σ √ (cid:15) p , (17)where σ = ± ; see figure 6 in appendix C. The inverse of the factor c µ (cid:15) p is thus theproduct of the volume | Γ | and twice the perimeter of the unit circle, c − µ (cid:15) p = (cid:90) Γ d q (cid:90) × S d v = 4 π | Γ | , (18)which turns out to be identical to c µ , equation (7). This quantity must indeed be inde-pendent of the parameter (cid:15) p , since only the orientations of the velocities are relevant.This is, however, not so of the factor c ν bp | (cid:15) p , which, as described in appendix C, isfound to be: c − ν bp | (cid:15) p = (cid:90) ∂ Γ bp d q (cid:90) × S : v · n > d v ( v · n ) , = 4 π | ∂ Γ bp | π (cid:34)(cid:113) − (cid:15) p + √ (cid:15) p arcsin (cid:114) (cid:15) p − (cid:15) p (cid:35) , (cid:15) p < , √ (cid:15) p , (cid:15) p ≥ . (19)To perform an actual measurement of the conditional mean free time (16), onemust sample initial conditions with respect to the density c ν bp | (cid:15) p ( v · n ) + (20)on M bp , where ( x ) + = x if x > , and otherwise; see appendix C for details.It is, however, worth noting the conditional mean free time (16) may be computedmost simply as follows. Consider an equilibrium time series of the billiard dynamicsand select the subset of ball-piston collision events with energy partitions close to thedesired one. Indeed, definition (15) can immediately be extended to arbitrary energyintervals. Considering, in particular, the interval ( (cid:15) p − ε, (cid:15) p + ε ) for small ε > , wehave the identity τ bp ( (cid:15) p ) = lim ε → τ bp ( (cid:15) p − ε, (cid:15) p + ε ) ε , (21)which guarantees the convergence, as one decreases the parameter ε , to the desiredresult of a measurement performed on a coarser set.The reason that makes the conditional mean free time (16) particularly interestingis that its inverse f bp ( (cid:15) p ) ≡ τ bp ( (cid:15) p ) − can again be viewed as a rate, f bp ( (cid:15) p ) = lim t → t µ (cid:15) p (cid:0) { x ∈ M | S t x ∩ M bp (cid:54) = ∅} (cid:1) . (22)In the rare interaction limit δ → , we expect the conditional distribution of τ bp ( (cid:15) p ) to indeed become exponential, with rate f bp ( (cid:15) p ) , consistent with the expectation thatthe energy process converges to a Markov jump process; see section 5. tochastic limit of a rarely interacting ball-piston gas 11 Note that, by definition of c ν bp | (cid:15) p , we recover the inverse of c ν bp after integrating theinverse of the former quantity over the values of the piston energy (cid:15) p , weighted by thedensity of its marginal equilibrium distribution, a Beta distribution of shape parame-ters and . This relation implies a similar one between the ball-piston conditionalcollision rate (22) and the collision rate (14), (cid:90) d (cid:15) p √ (cid:15) p f bp ( (cid:15) p ) = f bp . (23)This does not contradict the identities f bp ( (cid:15) p ) = τ bp ( (cid:15) p ) − and f bp = τ − bp .The product of the mean free time (9) by the collision rate (22) allows to define adimensionless energy-dependent collision frequency which is independent of the billiardgeometry, φ bp ( (cid:15) p ) ≡ τ bp f bp ( (cid:15) p ) = π (cid:34)(cid:113) − (cid:15) p + √ (cid:15) p arcsin (cid:114) (cid:15) p − (cid:15) p (cid:35) , < (cid:15) p ≤ , √ (cid:15) p , < (cid:15) p ≤ . (24) ��� ��� ��� ��� ��� ������������������������ ϵ ϕ � � ( ϵ ) ●●●●●●●●●●●●●●●●●●●●●●●●●● ���� ��� ���� ����� - � �� - � �� - � �� - � �� - � δ τ � � - � Fig. 3: Numerical computations of the product of the mean free time by the conditionalcollision rate, compared with the rescaled ball-piston collision frequency, φ bp ( (cid:15) p ) , givenby equation (24) (solid white curve), plotted as function of the piston energy (cid:15) p = (cid:15) (with the ball energy (cid:15) b = − (cid:15) ). The parameter δ takes on a number of differentvalues which can be read off the horizontal axis of the graph in the inset. There weshow the numerical computations of the inverse of the mean free time as function of δ in comparison to equation (9) (dotted black curve). The colours of the data in theinset match those of the data sets in the main graph. The integral of the surface element on S along the two horizontal circles at heights ±√ (cid:15) p is / √ (cid:15) p , which is the density of the Beta distribution of shape parameters and , properlynormalised.2 P. Bálint et al. A numerical computation of the conditional mean free times of the minimal ball-piston model for a range of energy values was carried out taking ρ = ( √ − / (5 √ (cid:39) . and varying δ in the interval < δ ≤ . . The comparison withequation (24) is shown in figure 3. All data sets collapse on the analytic result, withinan accuracy that is controlled by the number of initial conditions. For each parameter δ and energy value (cid:15) p , a number of initial conditions were generated with respectto the distribution of density (20). A comparison between equation (9) and numericalcomputations of the mean free time is shown in the inset of the same figure. The mean free time (9) determines the time scale of the stochastic process of energyexchanges of the ball-piston pair. Given an equilibrium ingoing energy configuration { (cid:15) b , (cid:15) p } , at collision, the density of the equilibrium measure (20) yields the probabilityto find the system in the outgoing energy configuration { (cid:101) (cid:15) b , (cid:101) (cid:15) p } .5.1 Stochastic kernelPrior to resolving the collision event, let us assume the system is in the configuration ( q , v ) , with position vector q ∈ ∂ Γ bp and velocity vector v such that the ball-pistonpair has the energy configuration { (cid:15) b , (cid:15) p } , parametrised by equation (17), and such that v · n < . After the collision, we must have the outgoing velocity vector ˜v , ˜v · n > ,with components (cid:101) v = (cid:112) (cid:101) (cid:15) b cos ˜ α = σ √ (cid:15) p , (cid:101) v = (cid:112) (cid:101) (cid:15) b sin ˜ α = √ (cid:15) b sin α , (cid:101) v = ˜ σ (cid:112) (cid:101) (cid:15) p = √ (cid:15) b cos α . (25)In particular, (cid:101) (cid:15) b = (cid:15) p + (cid:15) b sin α and (cid:101) (cid:15) p = (cid:15) b cos α .The probability per unit time of this transition is | ∂ Γ bp | π | Γ | ( √ (cid:15) b cos α − σ √ (cid:15) p ) + d α , (26)which we now wish to rewrite in terms of the outgoing piston energy (cid:101) (cid:15) p . From the thirdline in equation (25), we see that α can be written explicitly in terms of the ingoingand outgoing energies, √ (cid:15) b cos α = ˜ σ (cid:112) (cid:101) (cid:15) p . (27)The measure element thus transforms to d α = d (cid:101) (cid:15) p (cid:112) (cid:101) (cid:15) p ( (cid:15) b − (cid:101) (cid:15) p ) Θ h ( (cid:15) b − (cid:101) (cid:15) p ) , (28) The energy values include (cid:15) p = 1 / , / , . . . , / , to which are added (cid:15) p = 1 / , / , . . . , / and (cid:15) p = 1 / − / , / − / , . . . , / − / .tochastic limit of a rarely interacting ball-piston gas 13 where we have inserted the Heaviside step function, Θ h ( x ) = 1 if x ≥ , otherwise,to keep track of the condition v · n < .Now summing over σ and ˜ σ and multiplying the above expression by 2, whichreflects the fact that the collision process is independent of the sign of v , the probabilityper unit time (26) transposes to W ( (cid:15) b , (cid:15) p | (cid:101) (cid:15) b , (cid:101) (cid:15) p ) d (cid:101) (cid:15) p , (29)where the probability density, W ( (cid:15) b , (cid:15) p | (cid:101) (cid:15) b , (cid:101) (cid:15) p ) = | ∂ Γ bp | π | Γ | (cid:88) σ, ˜ σ (˜ σ √ (cid:101) (cid:15) p − σ √ (cid:15) p ) + (cid:112) (cid:101) (cid:15) p ( (cid:15) b − (cid:101) (cid:15) p ) Θ h ( (cid:15) b − (cid:101) (cid:15) p ) , = | ∂ Γ bp | π | Γ | max( √ (cid:101) (cid:15) p , √ (cid:15) p ) (cid:112) (cid:101) (cid:15) p ( (cid:15) b − (cid:101) (cid:15) p ) Θ h ( (cid:15) b − (cid:101) (cid:15) p ) , (30)can be interpreted as the rate of probability of a transfer of energy ζ = (cid:15) b − (cid:101) (cid:15) b fromthe ball at energy (cid:15) b to the piston whose energy changes from (cid:15) p to (cid:101) (cid:15) p = (cid:15) p + ζ , − (cid:15) p ≤ ζ ≤ (cid:15) b .By construction, we recover the conditional collision rate (22) after integratingequation (30) over ζ , f bp ( (cid:15) p ) = (cid:90) d ζ W ( (cid:15) b , (cid:15) p | (cid:15) b − ζ, (cid:15) p + ζ ) . (31)5.2 Convergence to a Markov processAlthough equation (30) is a property of the equilibrium system, we argue that it alsoprovides an accurate description of the energy exchange process between the balland piston away from equilibrium, provided we consider the limiting regime of rareinteractions—that is, when the penetration length of the piston into the domain of theball is arbitrarily small, δ (cid:28) . Indeed, under this assumption, the ball and pistontypically undergo many wall-collision events between every binary collision, so that arelaxation to equilibrium of the ball-piston pair at fixed energies effectively takes placebefore the next occurrence of a binary collision.As a result, in the limiting regime of rare interactions, the process of energy ex-changes is expected to converge to a Markov jump process with kernel (30). Indeed,the Markov property essentially means that if one knows not only the present energypartition, but also has information about its history, this additional information doesnot improve one’s ability to predict the future evolution of the energy partition. Cor-respondingly, convergence to a Markov process means that if δ > decreases, theninformation about the past influences the future less and less. So we need to see thatif we start the system with a given energy partition between the disk and piston, butaway from the equilibrium measure (conditioned on the energy surface), then, as δ → ,we measure nearly the same jump rates as in (30). This is expected to hold exactlybecause of the relaxation to (conditional) equilibrium during the many wall-collisionsthat typically precede the first energy exchange.According to this argument, the time-evolution of the ball-piston pair energy dis-tribution P ( { (cid:15) b , (cid:15) p } , t ) may be described by the following master equation: et al. ∂ t P ( { (cid:15) b , (cid:15) p } , t ) = (cid:90) d ζ (cid:104) W ( (cid:15) b + ζ, (cid:15) p − ζ | (cid:15) b , (cid:15) p ) P ( { (cid:15) b + ζ, (cid:15) p − ζ } , t ) − W ( (cid:15) b , (cid:15) p | (cid:15) b − ζ, (cid:15) p + ζ ) P ( { (cid:15) b , (cid:15) p } , t ) (cid:105) . (32)The method used to derive this result relies on geometric and measure-theoretic ar-guments. An alternative approach based on kinetic theory, such as used in refer-ences [11, 23, 35], yields the same results.5.3 Numerical test of the Markov propertyWe want to further substantiate the assertion that—with an appropriate rescaling oftime—the actual process of energy exchanges produced by the ball-piston billiard hasa well-defined limit when δ → , and that the limiting process is indeed Markovian anddescribed by equation (32). Based on the heuristic arguments presented in section 5.2,we should therefore check that, as δ decreases, the collision rate and post-collisionenergy distribution become arbitrarily close to the limits associated with the processgenerated by equation (32), independently of our information about the past, thatis, independently of the (non-equilibrium) initial distribution we sample our energypartition with.Moreover the notion that we have limited information about the past reflects a lackof precise knowledge of the initial conditions. Our initial distribution should thereforebe smooth. Finally, since there is, of course, no hope of checking that this is convergenceholds for every initial distribution, we pick a specific family of smooth initial measureson the energy surface, and check convergence for it.In particular we consider, for different values of δ , an initial state ( q , v ) of thebilliard map, with position q uniformly distributed on the collision surface ∂ Γ bp andvelocity v as in (17), such that v · n > , with angle α and sign σ now distributed away from the distribution of density (20), and measure the distribution of ingoing velocitiesat the first ball-piston collision event. The velocity v , now such that v · n < , may againbe parametrised as in (17), with values of α and σ different from the outgoing initialvelocity, but (cid:15) p unchanged. Since we are considering a marginal velocity distribution,apart from the two values of the sign σ , this distribution is a function of a single realvariable, α . Irrespective of the choice of piston energy (cid:15) p (with ball energy (cid:15) b = − (cid:15) p ),we expect to find a distribution of ingoing α and σ that, as δ → , becomes arbitrarilyclose to the distribution on the constant (cid:15) p circles induced by the equilibrium measure.That is because, in the absence of ball-piston collisions, the wall-collision events willtypically induce a relaxation of the billiard dynamics to the measure of density (20),which is a true invariant measure of the non-interacting ball-piston dynamics whentheir energies are fixed to the corresponding values. In other words, when δ is small,the billiard dynamics is likely to perform many wall collision events before first hittingthe ball-piston collision surface M bp . The first hitting distribution is thus expectedto converge to the distribution of density (20), which happens to be an equilibriumdistribution of the ball-piston dynamics when interactions are turned off ( δ ≡ ).To be specific, let h ( n ) (cid:15) p ( α, σ ) ∝ ( √ (cid:15) b cos α − σ √ (cid:15) p ) n + , (33) tochastic limit of a rarely interacting ball-piston gas 15 and normalise these densities so that (cid:80) σ = ± (cid:82) d α h ( n ) (cid:15) p ( α, σ ) = 1 . In particular, thedensity h (0) (cid:15) p is uniform on the set v · n > and h (1) (cid:15) p is the density (20) induced by theequilibrium distribution, albeit with a different normalisation.In figure 4, we plot the histograms of the ingoing velocity distributions obtainedby sampling initial conditions with respect to the non-equilibrium density h (0) (cid:15) p . Eachsubfigure corresponds to a different value of δ , varied horizontally, and (cid:15) p , varied verti-cally. Histograms are measured by dividing the intervals of allowed values of α into bins. As seen from the figure, the differences between the measured distributions andthe corresponding distributions induced by the equilibrium measure are never large,but are most noticeable when δ is large. To quantify the convergence of the measureddistributions to those induced by the equilibrium measure, we computed the (coarsegrained) relative entropy of the measured distribution with respect to h (1) (cid:15) p , also knownas Kullback-Leibler divergence [31]. Using the notation ˜ h ( n ) (cid:15) p to denote the ingoing dis-tribution into which the initial distribution h ( n ) (cid:15) p evolves until the first energy exchange, D kl (˜ h ( n ) (cid:15) p | h (1) (cid:15) p ) = (cid:88) σ = ± (cid:90) d α ˜ h ( n ) (cid:15) p ( α, σ ) log ˜ h ( n ) (cid:15) p ( α, σ ) h (1) (cid:15) p ( α, σ ) , (34)where the integral over α is evaluated by summing the measured averaged density overthe total number of bins. The results of measurements of this quantity using differentoutgoing velocity distributions h ( n ) (cid:15) p , n = 0 , , , , and exact values for the density h (1) (cid:15) p ( α, σ ) in the denominator of equation (34) are shown in figure 5. Whereas the decayto the equilibrium noise level of the Kullback-Leibler divergence with the parameter δ appears to be qualitatively different when the piston energies are larger than theball energies or vice versa, our measurements clearly show a systematic return to thestatistics induced by the equilibrium measure as the parameter δ → and thus providea confirmation of the observations drawn from figure 4.5.4 Moments of the kernelWe end this section by noting that the stochastic evolution (32) proves particularlyuseful to study energy exchanges in rarely interacting systems consisting of many par-ticles, such as the ball-piston gas shown in figure 1. In this context, we note that thefirst three moments of the energy transfer rate share the symmetries observed in othermodels [35].Thus, given the canonical ball-piston energy distribution, which is the product ofGamma distributions of shape parameters respectively and , and common scaleparameter (the temperature) β − , P ( can ) β ( (cid:15) b , (cid:15) p ) = β / √ π(cid:15) p exp[ − β ( (cid:15) b + (cid:15) p )] , (35)the zeroth moment of the energy transfer rate, similar to f bp ( (cid:15) p ) , equation (31), butwithout the assumption (cid:15) b = − (cid:15) p , et al. δ = ����� ϵ = ������� - π - π � � π � π ���������������������������� α � ϵ ( α ) (a) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ������� - π - π � � π � π ���������������������������� α � ϵ ( α ) (b) (cid:15) = 0 . , δ = 0 . δ = ������ ϵ = ������� - π - π � � π � π ���������������������������� α � ϵ ( α ) (c) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ����� - π - π � � π � π �������������������������������� α � ϵ ( α ) (d) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ����� - π - π � � π � π �������������������������������� α � ϵ ( α ) (e) (cid:15) = 0 . , δ = 0 . δ = ������ ϵ = ����� - π - π � � π � π �������������������������������� α � ϵ ( α ) (f) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ���� - π - π � � π � π ���������������������������� α � ϵ ( α ) (g) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ���� - π - π � � π � π �������������������������������� α � ϵ ( α ) (h) (cid:15) = 0 . , δ = 0 . δ = ������ ϵ = ���� - π - π � � π � π ���������������������������� α � ϵ ( α ) (i) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ����� - π - π � � π � π ������������������������ α � ϵ ( α ) (j) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ����� - π - π � � π � π ������������������������ α � ϵ ( α ) (k) (cid:15) = 0 . , δ = 0 . δ = ������ ϵ = ����� - π - π � � π � π ������������������������ α � ϵ ( α ) (l) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ������� - π - π � � π � π �������������������� α � ϵ ( α ) (m) (cid:15) = 0 . , δ = 0 . δ = ����� ϵ = ������� - π - π � � π � π �������������������� α � ϵ ( α ) (n) (cid:15) = 0 . , δ = 0 . δ = ������ ϵ = ������� - π - π � � π � π �������������������� α � ϵ ( α ) (o) (cid:15) = 0 . , δ = 0 . Fig. 4: Histograms of the measured ingoing first hit velocity distributions obtained fromoutgoing distributions (33) with exponent n = 0 . The horizontal axes show the anglevalues α . The parts of the densities corresponding to σ = +1 are shown in solid bluelines and to σ = − in solid red lines (only for (cid:15) p < / ). Every curve is compared tothe corresponding density induced by the equilibrium measure (filled areas). Each rowin the figure corresponds to a fixed value of (cid:15) p and each column to a fixed value of δ ,decreasing from left to right. tochastic limit of a rarely interacting ball-piston gas 17 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ϵ = ������� ϵ = ������ ϵ = ����� ϵ = ���� ϵ = ����� ϵ = ������ ϵ = ������� ���� ���� ���� ���� ���� ������ × �� - � �� × �� - � �� × �� - � �� × �� - � �� × �� - � �� - � δ � � � (a) n = 0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ϵ = ������� ϵ = ������ ϵ = ����� ϵ = ���� ϵ = ����� ϵ = ������ ϵ = ������� ���� ���� ���� ���� ���� ������� × �� - � ��� × �� - � ��� × �� - � ��� × �� - � ��� × �� - � ��� × �� - � ��� × �� - � δ � � � (b) n = 1 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ϵ = ������� ϵ = ������ ϵ = ����� ϵ = ���� ϵ = ����� ϵ = ������ ϵ = ������� ���� ���� ���� ���� ���� ������ - � �� - � ���������� δ � � � (c) n = 5 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ϵ = ������� ϵ = ������ ϵ = ����� ϵ = ���� ϵ = ����� ϵ = ������ ϵ = ������� ���� ���� ���� ���� ���� ������ - � �� - � ���������� δ � � � (d) n = 10 Fig. 5: The Kullback-Leibler divergence (34) of the measured density of the ingoingvelocity distributions relative to the equilibrium density (20) provides a quantitativemeasurement of the convergence of the former to the latter as the parameter δ → .Data obtained by sampling initial conditions with respect to the densities h ( n ) (cid:15) p , (a) n = 0 , (b) n = 1 , (c) n = 5 , (d) n = 10 are displayed as functions of the penetrationlength δ of the piston into the ball cell. The oscillatory behaviour observed for somevalues of the energies appears to be logarithmic with respect to δ . In panel (b), wecompare the numerically obtained values of the equilibrium distribution h (1) (cid:15) p to itsanalytic expression (20), and thus obtain a useful benchmark to gauge the accuracywithin which equilibrium statistics can be reached. The factor between the sets with (cid:15) p < / and (cid:15) p ≥ / is due to the fact that the phase space is divided into twice asmany cells when (cid:15) p < / compared to (cid:15) p ≥ / . f ( (cid:15) b , (cid:15) p ) ≡ (cid:90) d ζ W ( (cid:15) b , (cid:15) p | (cid:15) b − ζ, (cid:15) p + ζ ) , (36) = | ∂ Γ bp || Γ | (cid:40) π (cid:104) √ (cid:15) b − (cid:15) p + √ (cid:15) p arcsin (cid:113) (cid:15) p (cid:15) b (cid:105) , (cid:15) b > (cid:15) p , √ (cid:15) p , (cid:15) b ≤ (cid:15) p , the first moment, j ( (cid:15) b , (cid:15) p ) ≡ (cid:90) d ζ ζ W ( (cid:15) b , (cid:15) p | (cid:15) b − ζ, (cid:15) p + ζ ) , (37) et al. = | ∂ Γ bp || Γ | π (cid:104) (4 (cid:15) b − (cid:15) p ) √ (cid:15) b − (cid:15) p +3( (cid:15) b − (cid:15) p ) √ (cid:15) p arcsin (cid:112) (cid:15) p /(cid:15) b (cid:105) , (cid:15) b > (cid:15) p , ( (cid:15) b − (cid:15) p ) √ (cid:15) p , (cid:15) b ≤ (cid:15) p , and the second moment, h ( (cid:15) b , (cid:15) p ) ≡ (cid:90) d ζ ζ W ( (cid:15) b , (cid:15) p | (cid:15) b − ζ, (cid:15) p + ζ ) , (38) = | ∂ Γ bp || Γ | π (cid:110) (cid:15) b − (cid:15) p ) / + 15 (cid:104) − (cid:15) p ( (cid:15) b − (cid:15) p ) √ (cid:15) b − (cid:15) p + √ (cid:15) p ( (cid:15) b − (cid:15) b (cid:15) p + (cid:15) p ) arcsin (cid:112) (cid:15) p /(cid:15) b (cid:105)(cid:111) , (cid:15) b > (cid:15) p , √ (cid:15) p ( (cid:15) b − (cid:15) b (cid:15) p + (cid:15) p ) , (cid:15) b ≤ (cid:15) p , all satisfy the following identities, involving averages with respect to the canonicalmeasure (35): (cid:104) f ( (cid:15) b , (cid:15) p ) (cid:105) ( can ) β = β (cid:104) ( (cid:15) b − (cid:15) p ) j ( (cid:15) b , (cid:15) p ) (cid:105) ( can ) β = β (cid:104) h ( (cid:15) b , (cid:15) p ) (cid:105) ( can ) β = 1 √ πβ | ∂ Γ bp || Γ | . (39)In the limit δ → of rare interactions, this is lim δ → δ − (cid:104) f ( (cid:15) b , (cid:15) p ) (cid:105) ( can ) β = 1 √ πβ (cid:104) − λ − ρ (cid:0) π − λ (cid:1)(cid:105) − , (40)which provides an approximation of the heat conductivity of the ball-piston gas. Thiswill be the subject of a separate publication. In reference [9] a family of billiard models was introduced in the hope of provingsuitable for deriving the heat equation. It is moreover believed it will be possibleto determine the actual expression of the associated coefficient of heat conductivity.Such an achievement would bring to completion a programme aiming at explainingmacroscopic laws from deterministic microscopic assumptions, one of mathematicalphysics great outstanding challenges.These models, which combine the kinetics of gases of hard balls with the periodicstructure of crystalline solids, lend themselves to a systematic analysis, whose toolswere made available in no small part thanks to the pioneering works of David Ruelleand Yasha Sinai. The authors of [9] outlined a simple two-step strategy to attain theirgoal: (i) going from the microscopic scale to a mesoscopic one (micro-to-meso), and (ii)from that scale to the macroscopic one (meso-to-macro). Moreover, they also realisedtheir programme on the level of analytic calculations with precise physical meaning.Among realistic models to study Fourier’s law, billiard models are generally mostamenable to a rigorous derivation of both mesoscopic and macroscopic laws from de-terministic microscopic assumptions, however delicate their technical analysis. It hastherefore been a top priority of the community to provide a mathematically soundproof completing the approach outlined above. Our main goal here has been to suggest tochastic limit of a rarely interacting ball-piston gas 19 a new addition to the family of Gaspard-Gilbert models, for which a mathematicaltreatment of the micro-to-meso step is a distinctly realistic task.In this paper, we focused on the description of the ball-piston model and the com-putation of several quantities characterising its statistical properties. The limit of rareball-piston interactions provides a meaningful interpretation, both physically and math-ematically, of some of these properties at the level of a mesoscopic description. Namely,energy exchanges are described by a Markov jump process with a precise form of thetransition kernel. The complete mathematical discussion is postponed to subsequentpublications (as to the first of these, see [28]).We have, in addition, devised a statistical procedure, making use of the Kullback-Leibler divergence, to test quantitatively whether the limit of rare interactions of theminimal ball-piston model indeed possesses the Markov property. Our numerical resultsare affirmative and help shed new light on the approach to this limit. Our procedurecan be put to use in other models of the GG family and its application will be describedelsewhere.
A Collision volume
To determine the volume | Γ | of configuration space, note that when q > (1 − λ ) / ,the volume of all possible positions q and q is (cid:90) (1 − λ ) / d q (cid:20) − (cid:113) ρ − (cid:0) q − (cid:1) (cid:21) = 1 − λ − ρ (cid:0) π − λ (cid:1) . (A.1)When q < (1 − λ ) / , we must subtract from the above area the quantity (cid:90) (1 − λ ) / q d q (cid:20) − (cid:113) ρ − (cid:0) q − (cid:1) (cid:21) = 12 − λ − q − − q (cid:112) ρ − (1 − q ) + ρ (cid:34) arctan λ − arctan 1 − q (cid:112) ρ − (1 − q ) (cid:35) . (A.2)We therefore obtain the total volume of configuration space (10) by multiplying equa-tion (A.1) by λ + 2 δ and subtracting the integral of equation (A.2) over q from (1 − λ ) / − δ to (1 − λ ) / .To compute the collision surface integral, note that the position coordinates on ∂ Γ bp are bounded according to (1 − λ ) − δ ≤ q = q ≤ (1 − λ ) , − + (cid:113) ρ − ( q − ) ≤ q ≤ − (cid:113) ρ − ( q − ) (A.3)Its projection on the ( q , q ) plane is the area (A.2) evaluated at q = (1 − λ ) / − δ .Since the surface itself makes an angle π/ with respect to the ( q , q ) plane, we obtainfor | ∂ Γ bp | the expression given by equation (11). et al. B Ball-wall and piston-wall return times
Wall collision return times of the piston and ball can be computed in ways similar toequation (9), τ pw = 4 | Γ || ∂ Γ pw | ,τ bw = 4 | Γ || ∂ Γ bw | , (B.1)where | ∂ Γ pw | and | ∂ Γ bw | are the areas of piston-wall and ball-wall collisions. Theformer corresponds to the area of all positions q and q such that q = (1 ± λ ) / ± δ ,which is parallel to the ( q , q ) plane and twice the area (A.1) minus the projectionof the collision surface | ∂ Γ bp | (11) on this plane, and the latter to the positions q , q such that ( q ± ) + ( q ± ) = ρ while q < q , with q integrated over the interval(1). That is, | ∂ Γ pw | = 2 (cid:104) − λ − ρ (cid:0) π − λ (cid:1)(cid:105) − √ | ∂ Γ bp | , (B.2)and | ∂ Γ bw | = ρ ( λ + 2 δ ) (cid:16) − λ √ ρ − arcsin λ + 2 δ ρ + arcsin λ ρ (cid:17) + ρ (cid:2) − (cid:112) − δ ( λ + δ ) (cid:3) . (B.3) C Restriction of the invariant measure on M bp to fixed energyconfigurations Substituting the parametrisation (17) of the velocity vector v ∈ S and evaluating itsscalar product with the normal vector (6), the velocity integral in equation (19) splitsinto two contributions, integrated over an arc-length proportional to the angle alongthe arcs: (cid:90) × S : v · n ≥ d v ( v · n ) = 1 √ (cid:90) S : √ (cid:15) p ≥√ − (cid:15) p cos α d α ( √ (cid:15) p − √ − (cid:15) p cos α )+ 1 √ (cid:90) S : √ (cid:15) p ≤−√ − (cid:15) p cos α d α ( −√ (cid:15) p − √ − (cid:15) p cos α ) ; (C.1)see figure 6 for a graphical representation. Two separate regimes arise.On the one hand, when the piston’s energy is less than the ball’s, the condition √ (cid:15) p ≥ √ − (cid:15) p cos α in the first of the two integrals on the right-hand side of equa-tion (C.1) is equivalent to arccos (cid:114) (cid:15) p − (cid:15) p ≤ α ≤ π − arccos (cid:114) (cid:15) p − (cid:15) p . (C.2) tochastic limit of a rarely interacting ball-piston gas 21 Fig. 6: Equation (C.1) has either one ( / ≤ (cid:15) p ≤ ) or two ( ≤ (cid:15) p < / ) con-tributions, given by the integrals along the arc-circles on the hemisphere of velocitycoordinates whose projection along the normal to the collision surface (6) is positive.Here (cid:15) p = 1 / and the two arc-circles at v = ± contributing to equation (C.1) arethe portions (in green) of the corresponding full circles above the plane tangent to thecollision surface (the excluded parts of those circles are shown in white).Performing the integral, we obtain the contribution, √ (cid:90) S : √ (cid:15) p ≥√ − (cid:15) p cos α d α ( √ (cid:15) p − √ − (cid:15) p cos α )= 2 √ (cid:15) p (cid:32) π − arccos (cid:114) (cid:15) p − (cid:15) p (cid:33) + 2 (cid:113) − (cid:15) p . (C.3)Likewise, the condition √ (cid:15) p ≤ −√ − (cid:15) p cos α in the second of the two integralson the right-hand side of equation (C.1) is equivalent to π − arccos (cid:114) (cid:15) p − (cid:15) p ≤ α ≤ π + arccos (cid:114) (cid:15) p − (cid:15) p . (C.4)Thus the second integral yields the contribution et al. √ (cid:90) S : √ (cid:15) p ≤−√ − (cid:15) p cos α d α ( −√ (cid:15) p − √ − (cid:15) p cos α )= − √ (cid:15) p arccos (cid:114) (cid:15) p − (cid:15) p + 2 (cid:113) − (cid:15) p . (C.5)The contribution to equation (19) from the velocity integral in the correspondingenergy interval is thus given by the sum of equations (C.3) and (C.5).On the other hand, when the piston’s energy is larger than the ball’s, the condi-tion √ (cid:15) p ≥ √ − (cid:15) p cos α in the first of the two integrals on the right-hand side ofequation (C.1) holds true for all angles α . The result of the integration, √ (cid:90) S : √ (cid:15) p ≥√ − (cid:15) p cos α d α ( √ (cid:15) p − √ − (cid:15) p cos α ) = 2 π √ (cid:15) p , (C.6)yields the only contribution to the velocity integral in equation (19). Acknowledgements
The authors gratefully acknowledge fruitful discussions with TamásTasnády. They are also grateful to Makiko Sasada for communicating her results prior topublishing. TG and PN wish to acknowledge the hospitality of the Institute of Mathematicsat the Budapest University of Technology and Economics where part of this work was con-ducted. TG also wishes to acknowledge stimulating discussions with the DinAmicI communityduring their 2015 workshop, held in Corinaldo, Italy. The authors are grateful to the ErwinSchrödinger Institute, Vienna, for their hospitality on the occasion of the workshop
MixingFlows and Averaging Methods during which this work was finalized. Finally, they also expresstheir gratitude to the referees for valuable comments. PB, DSz and IPT acknowledge the fi-nancial support of the Hungarian National Foundation for Scientific Research (OTKA): grantT104745 and of Stiftung Aktion Österreich-Ungarn: grant 87öu6. TG is financially supportedby the (Belgian) FRS-FNRS.
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