AARCHIMEDES’ PRINCIPLE FOR IDEAL GAS
KRZYSZTOF BURDZY AND JACEK MA(cid:32)LECKI
Abstract.
We prove Archimedes’ principle for a macroscopic ball in ideal gas consist-ing of point particles with non-zero mass. The main result is an asymptotic theorem,as the number of point particles goes to infinity and their total mass remains constant.We also show that, asymptotically, the gas has an exponential density as a function ofheight. We find the asymptotic inverse temperature of the gas. We derive an accurateestimate of the volume of the phase space using the local central limit theorem. Introduction
There seems to be no rigorous proof of Archimedes’ principle in the mathematicalliterature. The most likely reason for this omission is that Archimedes’ principle istrivial given a few natural assumptions. The principle can be easily derived usingthe divergence theorem, assuming that the formula for the pressure as a function ofheight is known. The “barometric formula” which says that the pressure in gas has anexponential density as a function of height can be easily derived from the ideal gas law.While this derivation of Archimedes’ principle is sufficient for the scientific applications,one could ask whether the principle can be derived from a more fundamental modelof the matter, as in Hilbert’s 6-th problem. This is what we will do in the presentarticle. Perhaps the most significant difference between our model and the derivationof Archimedes’ principle alluded to above is that the floating object is allowed to movein our case. We are not aware of an existing proof of Archimedes’ principle, rigorousor not, based on a model with a moving floating object.We will consider a container with a bounded base, vertical side walls and no top.The container will hold point particles (ideal gas) and a floating object in the shape ofa ball and no internal structure (the mass will be uniformly spread over the ball). Thespherical shape of the floating object allows us to avoid the discussion of the energygoing into rotation—the collisions of the ball with point particles and the walls of thecontainer will not induce rotation of the ball.The point particles and the ball will move according to Newton’s laws in a gravi-tational field with constant acceleration. We will assume conservation of energy andmomentum but this assumption does not have a unique interpretation in the case ofcollisions of point particles with the infinitely heavy walls of the container. We willconsider two types of reflections of point particles from the walls of the container: (i)specular reflections where the angle of reflection is equal to the angle of incidence,
Mathematics Subject Classification.
Key words and phrases.
Archimedes’ principle, ideal gas.KB’s research was supported in part by Simons Foundation Grant 506732. J. Ma(cid:32)lecki was supportedby the Polish National Science Centre (NCN) grant no. 2018/29/B/ST1/02030. a r X i v : . [ m a t h . P R ] F e b KRZYSZTOF BURDZY AND JACEK MA(cid:32)LECKI and (ii) random reflections according to the Lambertian distribution also known as theKnudsen law. We will assume that the system is in equilibrium so that its density isgiven by the microcanonical ensemble formula. We will prove that this distribution isthe unique invariant measure in case (ii). Simple examples show that there are multipleinvariant measures if we assume specular reflections.In our asymptotic results we will assume that the following objects and quantitiesare fixed: the container, the mass and radius of the macroscopic ball, the total massof the gas (all point particles), the total (potential and kinetic) energy of all movingobjects (point particles and the ball), and the gravitational acceleration. The numberof point particles will go to infinity so the mass of a single particle will go to zero.On the way to the main result, Archimedes’ principle, we will derive a few otherresults that may have independent interest.We will present an accurate formula for the volume of the phase space. Our calcula-tion is based on the local central limit theorem. This theorem was proved long time agobut the literature is hard to follow so we hope that our short review of that literaturewill help those readers who might need this result in their own research.The microcanonical ensemble formula is well known, see, e.g., [Rue99, Sect. 1.2]. Wecould not find a version of the formula needed here in the literature so a derivation wassupplied in a parallel project [BDG + Literature review.
A version of Archimedes’ principle was proved in [BCP11]but that model was completely different from the present one. The “gas” consistedof hard spheres with strictly positive radius. Their centers moved according to inde-pendent Brownian motions (except for the collisions). The number of “gas molecules”was constant and the asymptotic theorems were proved by sending the “gravitationalacceleration” to infinity.Our article is concerned with a model in which a macroscopic object interacts withmicroscopic molecules according to Newtonian mechanics. In this sense, our model isclosest to the “piston problem” proposed in [Lie99]. A large number of papers wereinspired by [Lie99] and devoted to the piston problem; see, for example, [Sin99, LPS00,CL02, CLS02, LSC02, NS04, Gor11, IS15] and references therein. Several differentmodels were considered in those papers. In one of the models, a piston moves along atube and is bombarded by microscopic molecules from both sides.1.2.
Limitations of the model.
Our model is, obviously, an oversimplified represen-tation of reality and there is no hope that it could be modified to be very realistic.Still, from the mathematical point of view, some aspects of the model might be worthgeneralizing in future research.(i) In our model, the macroscopic ball has no internal structure so it has negligiblysmall heat capacity. It might be possible to analyze a model in which the ball is replaced
RCHIMEDES’ PRINCIPLE 3 with a “balloon,” i.e., an infinitely thin sphere holding inside (mobile) point particleswith different masses than those of the outside particles.(ii) Since we assume that the gas is ideal, the point particles do not interact and,therefore, their collisions with the macroscopic ball are the only way in which theycan exchange energy between each other. This is the only way in which the energymay become approximately equidistributed in the stationary regime. To generalize ourresults to gases consisting of hard spheres with positive radius one would need accurateestimates of the volume of the phase space. The virial expansion might be useful inthis context, see [MM77, Sect. 8i] or [Rue99, Sect. 4.3]. If this approach works, it willrequire a whole new set of calculations.(iii) In our model, the container has a flat bottom and vertical side walls. Many ofour calculations depend on this assumption.1.3.
Organization of the paper.
The rigorous presentation of the model and thestatement of our main results are in Section 2. We will review known results on thelocal central limit theorem in Section 3. Section 4 contains a review of the results onthe microcanonical ensemble formula developed in a parallel paper. We will derive anaccurate estimate of the phase space volume in Section 5. The proof of Archimedes’principle will be given in Section 6. The inverse temperature will be identified inSection 7. The uniqueness of the stationary probability distribution under Lambertianreflections will be proved in Section 8.2.
Model and main results
We will consider n point particles (ideal gas) and one macroscopic hard ball in a d -dimensional container D with vertical walls, a bounded base, extending upward toinfinity in the vertical direction.Suppose that d ≥ D b ⊂ R d − denote the bottom of the container D = D b × [ , ∞) in R d . The radius of the ( n + ) -st (macroscopic) ball will be fixed anddenoted R >
0. Obviously, the point particles will have radii equal to 0. We will assumethat D b has a smooth boundary and satisfies the inner ball condition with radius R ,i.e., for every point x ∈ ∂D b , there is a unique ( d − ) -dimensional open ball with radius R inside D b whose boundary is tangent to ∂D b at x and, moreover, x is the only pointin the intersection of the ball boundary and ∂D b . We will also assume that a closedball of radius R fits in the interior of D , so that each point particle has room to movefrom below to above the ball, and vice versa.We will assume that the mass of the i -th point particle is m i = m / n > i = , . . . , n ,for some m >
0. When we let n → ∞ in our theorems, the total mass of point particleswill remain constant and equal to m . The mass of the macroscopic ball will be denoted M = m n + >
0. We will assume that the mass is evenly spread over the volume of themacroscopic ball. The walls of the container will be assumed to have infinite mass—thisassumption is a way of specifying the meaning of “totally elastic collisions” of pointparticles and the ball with the walls of the container.We will assume that the point particles and the ball are moving within a gravitationalfield with the constant acceleration g > KRZYSZTOF BURDZY AND JACEK MA(cid:32)LECKI will undergo totally elastic collisions with the ball. The collisions of the ball with thewalls of the container will be totally elastic.Time will be suppressed in the notation, except for Section 8. We will assume thatthe system is in equilibrium. Random objects will represent the state of the system attime 0 (or any other fixed time).Let ( X i , Y i ) ∈ R d denote the random position of the i -th point particle or the center ofthe macroscopic ball (for i = n + X i ∈ R d − represents the horizontal coordinatesand Y i ≥ Y n + ≥ R ). By V i ∈ R d we will denote the randomvelocity of the i -th point particle or the ball.Let x k = ( x , . . . , x k ) , y k = ( y , . . . , y k ) , v k = ( v , . . . , v k ) , (2.1) X k = ( X , . . . , X k ) , Y k = ( Y , . . . , Y k ) , V k = ( V , . . . , V k ) , (2.2)where x i ∈ R d − , y i ∈ R + , and v i ∈ R d . In the notation given above, upper case lettersrepresent random variables and lower case letters represent their values.We will consider ( X n + , Y n + , V n + ) to be a random vector distributed according tothe microcanonical ensemble formula (2.7) stated below, although we will give differentdistributions to these random vectors in some of the proofs.We will assume that the total energy of our system, E , is fixed. Hence, a.s., E = n + ∑ i = ( m i gY i + m i ∣∣ V i ∣∣ ) , (2.3)with the convention that the zero level represents zero potential energy. We will alwaysassume that E > M gR so that the ball and point particles cannot rest motionless atthe bottom.A ball in R d with center ( x, y ) and radius r will be denoted B (( x, y ) , r ) . Let D ′ b = { x ∈ D b ∶ B (( x, y ) , R ) ⊂ D for all y > R } , (2.4) D n = {( x n , x n + , y n , y n + , v n + ) ∈ D nb × D ′ b × R n + × [ R, ∞) × R nd ∶ (2.5) ( x k , y k ) ∈ D ∖ B (( x n + , y n + ) , R ) , k = , . . . n } , D En = {( x n + , y n + , v n + ) ∈ D n ∶ n + ∑ i = ( m i y i g + m i v i ) = E } . D En ( y ) = {( x n , x n + , y n ) ∈ D nb × D ′ b × R n + ∶ n + ∑ i = m i y i g ≤ E, (2.6) ( x k , y k ) ∈ D ∖ B (( x n + , y ) , R ) , k = , . . . n } . Let µ y n + ( d x ) denote the uniform probability measure on the sphere in R ( n + ) d (sothat the sphere is (( n + ) d − ) -dimensional), centered at the origin, with the radius ( E − ∑ n + i = m i y i g ) / . Consider the following measure P n on D En , P n ( d x n + d y n + d v n + ) (2.7) RCHIMEDES’ PRINCIPLE 5 = C ( E − n + ∑ i = m i y i g ) (( n + ) d − )/ µ y n + ( d v √ m , . . . , d v n + √ m n + ) d x n + d y n + , where C is the normalizing constant so that P n is a probability measure. The measure P n is a special case of the “microcanonical ensemble formula,” see, e.g., [Rue99, Sect. 1.2].We will say that a point particle undergoes a Lambertian reflection from a surface orthat the particle reflects according to the Knudsen law if the reflection occurs at a pointwhere the inner normal to the surface is uniquely defined, the probability density ofthe angle between the reflected trajectory and the normal vector is proportional to thecosine of the angle, and the law is invariant under rotations about the normal vector.Note that the Lambertian distribution of the outgoing velocity vector is independentfrom the incoming velocity vector, except that the two velocities have the same norm, sothat energy is conserved. This law for random reflections was considered by Lambert inthe context of light reflection from rough surfaces ([Lam60]) and by Knudsen ([Knu34])as a model for gas molecule reflections. By [BDG +
21, Cor. 3.2] (which can be derivedfrom [Pla12, Thm. 4.1] or [ABS13, Thm. 2.2]), the Lambertian reflection law is theunique random reflection law that does not depend on the angle of incidence and isconsistent with the standard (specular) reflection law from a rough surface consistingof small crystals with smooth reflecting surfaces.
Theorem 2.1. (i) The measure P n (microcanonical ensemble formula) is invariant forthe dynamical system defined above and two types of reflections:(a) totally elastic reflections between any pair of objects,(b) independent Lambertian reflections for point particles reflecting from the containerwalls (including the bottom) and totally elastic reflections between any other pair ofobjects.(ii) In case (b), P n is the unique non-degenerate stationary probability distributionfor the system of point particles and the macroscopic ball. The following are the onlyclasses of “degenerate” invariant distributions:(1) Invariant distributions such that no point particle ever hits a wall of the container.(2) Invariant distributions such that at least one point particle has no energy, so thatit is resting at the bottom of the container. The proof of Theorem 2.1 will be given in Sections 4 and 8.
Remark 2.2. (i) Part (ii) of Theorem 2.1 is not true under assumption (a); see Remark8.1 (i).(ii) We call invariant distributions in (1) degenerate because Lambertian reflectionsare never activated and the system has no opportunity to mix. See Remark 8.1 (ii) foran example of such a distribution.(iii) Invariant distributions in (2) are degenerate because in this case the number ofpoint particles is effectively less than n .We will often use integrals of the form ∫ D ∖ B ((̃ x,y ) ,R ) λe − λr d x d r . In cases like this (andsimilar), ̃ x should be interpreted as any point in D ′ b , d x will represent Lebesgue measurein R d − , and d r will represent Lebesgue measure in R . Note that the integral does notdepend on ̃ x as long as ̃ x ∈ D ′ b . KRZYSZTOF BURDZY AND JACEK MA(cid:32)LECKI
Theorem 2.3.
Fix d, D, R, M, m, g and E . Consider the equations M = m ∫ B ((̃ x,y ) ,R ) λe − λr d x d r ∫ D ∖ B ((̃ x,y ) ,R ) λe − λr d x d r , (2.8) dmg λ + mg ∫ D ∖ B ((̃ x,y ) ,R ) rλe − λr d x d r ∫ D ∖ B ((̃ x,y ) ,R ) λe − λr d x d r + M gy = E, (2.9) with unknowns y ≥ R and λ > .(i) There exists a unique λ ∗ which satisfies (2.9) with y = R .(ii) Suppose that M < m ∫ B ((̃ x,R ) ,R ) λ ∗ e − λ ∗ r d x d r ∫ D ∖ B ((̃ x,R ) ,R ) λ ∗ e − λ ∗ r d x d r . (2.10) Then (2.8) - (2.9) have a unique solution that will be denoted ( y A , λ A ) . Moreover R < y A < E /( M g ) . The proof of Theorem 2.3 will be given in Section 5.For ̃ x ∈ D ′ b , y ′ ≥ R and λ >
0, let ν ̃ x,y ′ ,λ be the probability distribution defined by ν ̃ x,y ′ ,λ ( d x, d y ) = D ∖ B ((̃ x,y ′ ) ,R ) ( x, y ) λe − λy ∫ D ∖ B ((̃ x,y ′ ) ,R ) λe − λr d z d r d x d y, x ∈ D b , y > . (2.11)Let δ ( X i ,Y i ) denote the probability measure on D which consists of a single atomat ( X i , Y i ) . The normalized (probability) empirical distribution Q n of the gas (pointparticles) is defined as Q n = n n ∑ i = δ ( X i ,Y i ) . (2.12) Theorem 2.4. (Archimedes’ principle) Fix d, D, R, M, m, g and E . Assume that thedistribution of ( X n + , Y n + , V n + ) is P n . Recall the notation from Theorem 2.3 andassume that (2.10) holds.(i) For every ε > , lim n →∞ P n (∣ Y n + − y A ∣ > ε ) = . (2.13) (ii) The marginal distribution of X n + under P n is uniform in D ′ b . Given { X n + = x } ,the conditional distribution of Q n converges to ν x,y A ,λ A weakly, in probability as n → ∞ . The proof of Theorem 2.4 will be given in Section 7.
Remark 2.5. (i) Theorem 2.4 is a mathematical representation of Archimedes’ prin-ciple. Part (ii) of the theorem identifies the limiting empirical distribution of gasmolecules when n → ∞ . The gas density is constant on the horizontal hyperplanesand it is exponential as a function of the height, outside the ball.Part (i) of Theorem 2.4 and (2.8) say that the ball is likely to float at the height suchthat the weight of the ball is equal to the weight of the displaced gas (i.e., (2.8) holds),assuming that the gas is distributed as in part (ii). RCHIMEDES’ PRINCIPLE 7 (ii) We need an extra equation (2.9) to identify y A and λ A uniquely. The equation isan expression of additivity and conservation of energy. The three terms on the left handside represent (asymptotically) the kinetic energy of the gas, the potential energy ofthe gas, and the potential energy of the ball. The kinetic energy of the ball is negligiblysmall asymptotically.The form of the first term on the left hand side of (2.9), i.e., dmg /( λ A ) , representingthe kinetic energy of the gas, is a manifestation of the virial theorem, which states howthe energy is distributed between potential and kinetic forms. The virial theorem is bynow a classical result, discovered by Claussius in 1870 (see, for example [Col78]).Formula (2.9) reduces to dmg λ A + mgλ A = E in the case when there is no ball, i.e., R = M =
0. In other words, the ratio ofpotential to kinetic energy is 2 / d for pure gas.Alternatively one could say that dmg /( λ A ) represents the kinetic energy, hence theheat energy, because λ A can be identified with the inverse temperature (see (7.1) for aprecise formula).(iii) The heuristic meaning of the first claim of Theorem 2.3 is that there is a uniqueasymptotic distribution of gas if the ball rests at the bottom of the container. Thesecond part says that if the weight of the ball is smaller than the weight of the gasdisplaced by the ball when it is placed at the bottom then, asymptotically, there is aunique level at which the ball will float and the corresponding unique distribution ofthe gas. 3. Local Central Limit Theorem
For any random variable A , let f A denote the density of A (if it exists). Supposethat random variables ξ k , k ≥
1, are i.i.d. with E ξ k = Eξ k = f ξ k . Let M = E ∣ ξ k ∣ and S n = ∑ nk = ξ k /(√ nσ ) . Let ϕ ( x ) denote the standard normal density,i.e., ϕ ( x ) = √ π e − x / .The following result is Theorem 1 in [Sv65]. Lemma 3.1.
There exists an absolute constant C such that if f ξ k ( x ) ≤ C for all x ∈ R then for all n ≥ , sup x ∈ R ∣ f S n ( x ) − ϕ ( x )∣ ≤ CM max ( , C )√ n . (3.1) Remark 3.2.
There is a considerable literature on the local central limit theorem butthe results are hard to extract from that literature for a number of reasons, includingabsence of some journals from accessible libraries and lack of proofs in a number ofpublications.The bound in Lemma 3.1 is given in Remark 5, Section 4, Chapter VII in [Pet75].The bound is given there without a proof and it is attributed to [ˇS66]. Unfortunately,[ˇS66] does not contain a proof. A similar bound is a special case of Theorem 1 in [ˇS71]but, once again, that paper does not contain a proof.
KRZYSZTOF BURDZY AND JACEK MA(cid:32)LECKI
One can derive Lemma 3.1 from Theorem 4 in [Sta65].The book [Pet75] contains several versions of the local central limit theorem, see,e.g., Theorem 15 in Chapter VII. However, in each of these theorems the error is givenin the form o ( /√ n ) . One could extract a bound of the form c /√ n with an explicitformula for c depending on the moments of the summands from the proof but thatwould be a very tedious task.4. Microcanonical ensemble formula
Proof of Theorem 2.1 (i).
The formula (2.7) is a version of a well known “microcanon-ical ensemble formula,” see, e.g., [Rue99, Sect. 1.2]. We could not find a rigorous proofthat this distribution is invariant in the contemporary literature. A proof has been givenin [BDG + +
21, Prop. 5.3] implies that f ( x n + , y n + , v n + ) ∶= ( E − n + ∑ i = m i y i g ) (( n + ) d − )/ d µ y n + d v n + ( v √ m , . . . , v n + √ m n + ) is the density of an invariant measure on the whole space R ( n + ) d .We claim that this density restricted to D n is invariant for our dynamical system withreflections. It suffices to show that the specular and Lambertian reflections betweenthe point particles, the ball and the walls of the container leave the measure invariant.For the specular reflections, the proof is essentially identical to that of the proof of[BDG +
21, Prop. 2.3]. The result can be extended to Lambertian reflections of pointparticles from the wall, as shown in [BDG +
21, Prop. 3.3, Remark 3.4] (see the remarkspreceding and following that result). (cid:3) Phase space volume
We will use two different sets of notations for some quantities. While this may createsome confusion, we will explain why this convention has some advantages.For n = , , , . . . , γ > y ≥ R we let (cid:104) n = (cid:104) n ( γ, y ) = ∫ D ∖ B ((̃ x,y ) ,R ) γr n e − γr d x d r, (5.1)where ̃ x ∈ D ′ b . The definition of and notation for (cid:104) n lead to the following very simpleformula, obtained using the dominated convergence theorem, ∂ (cid:104) n ∂γ = (cid:104) n γ − (cid:104) n + . (5.2)Some of the functions (cid:104) n have physical meaning, so we find it easier to memorizethem if different notation is used in some cases. Specifically, (cid:113) ( γ, y ) = ∫ D ∖ B ((̃ x,y ) ,R ) γe − γr d x d r = (cid:104) ( γ, y ) , (5.3) (cid:117) ( γ, y ) = (cid:113) ( γ, y ) ∫ D ∖ B ((̃ x,y ) ,R ) rγe − γr d x d r = (cid:104) ( γ, y ) (cid:104) ( γ, y ) , (5.4) (cid:119) ( γ, y ) = ∫ D ∖ B ((̃ x,y ) ,R ) γr e − γr d x d r = (cid:104) ( γ, y ) . (5.5) RCHIMEDES’ PRINCIPLE 9
We will prove in Lemma 5.2 that for given u > y ≥ R , the equation u = (cid:117) ( λ, y ) uniquely defines λ = λ ( u, y ) . We let h n ( u, y ) = (cid:104) n ( λ ( u, y ) , y ) , q ( u, y ) = (cid:113) ( λ ( u, y ) , y ) , w ( u, y ) = (cid:119) ( λ ( u, y ) , y ) . (5.6) Lemma 5.1.
For all y ′ ≥ R and γ > , ( − e − / ) ≤ (cid:113) ( γ, y ′ )∣ D b ∣ ≤ , (5.7) 1200 ≤ γ (cid:117) ( γ, y ′ ) ≤ − e − / < , (5.8) 2 − e − ≤ γ ( (cid:119) ( γ, y ′ ) (cid:113) ( γ, y ′ ) − (cid:117) ( γ, y ′ ) ) ≤ − e − / , (5.9) 2 − < − e − /
16 1200 ≤ (cid:117) ( γ, y ′ ) (cid:119) ( γ, y ′ )/ (cid:113) ( γ, y ′ ) − (cid:117) ( γ, y ′ ) ≤ e ( − e − / ) < . (5.10) Proof.
Let B ∗ = B ((̃ x, y ) , R ) . Elementary geometry shows that if A = { x ∈ D b ∶ ∣ x − ̃ x ∣ < √ R } then ( D b ∖ A )×( , y ′ − R / ) ∈ D ∖ B ∗ . If σ ( r ) denotes the volume of a ( d − ) -dimensionalball then, using our assumption that d ≥ ∣ D b ∖ A ∣∣ D b ∣ = − ∣ A ∣∣ D b ∣ ≥ − σ (√ R / ) σ ( R ) = − ( √ ) d − ≥ − √ > . (5.11)We have (cid:113) = ∫ D ∖ B ∗ γe − γr d x d r ≥ ∫ D b d x ∫ ∞ y ′ + R γe − γr d r = ∣ D b ∣ exp (− γ ( y ′ + R )) . (5.12)By (5.11), (cid:113) = ∫ D ∖ B ∗ γe − γr d x d r ≥ ∫ D b ∖ A d x ∫ y ′ − R / γe − γr d r = ∣ D b ∖ A ∣( − exp (− γ ( y ′ − R / ))) (5.13) ≥ ∣ D b ∣( − exp (− γ ( y ′ − R / ))) . Recall that y ′ ≥ R . If γ ≤ / y ′ then (5.12) yields (cid:113) ≥ ∣ D b ∣ exp (− γ ( y ′ + R )) ≥ ∣ D b ∣ exp (−( / y ′ )( y ′ + R )) = ∣ D b ∣ exp (− − R / y ′ ) ≥ ∣ D b ∣ e − . (5.14)In the case when γ ≥ / y ′ , we use (5.13) to see that (cid:113) ≥ ∣ D b ∣( − exp (− γ ( y ′ − R / ))) ≥ ∣ D b ∣( − exp (−( / y ′ )( y ′ − R / )))= ∣ D b ∣( − exp (− + R /( y ′ ))) ≥ ∣ D b ∣( − exp (− + / )) = ∣ D b ∣( − e − / ) . Since e − > ( − e − / ) , the above estimate and (5.14) give (cid:113) ≥ ∣ D b ∣( − e − / ) . (5.15)This proves the lower bound in (5.7). The upper bound follows directly from thedefinition of (cid:113) .We use (5.15) to derive the upper bound in (5.8) as follows, (cid:117) = ∫ D ∖ B ∗ rγe − γr d x d r ∫ D ∖ B ∗ γe − γr d x d r = (cid:113) ∫ D ∖ B ∗ rγe − γr d x d r ≤ (cid:113) ∫ D rγe − γr d x d r (5.16) = (cid:113) ∣ D b ∣ γ ≤ ( − e − / ) γ . Let W = ( W , W ) be a random vector with the distribution ν = ν ̃ x,y ′ ,γ defined in(2.11), where W ∈ D b and W >
0. Note that (cid:119) / (cid:113) − (cid:117) = Var W . (5.17)In view of (5.11) and (5.7), the marginal density f W ( y ) satisfies f W ( y ) ≤ ∣ D b ∣ (cid:113) γe − γy ≤ − e − / γe − γy , y > , (5.18) f W ( y ) ≥ γe − γy , y ≥ y ′ + R, (5.19) f W ( y ) ≥ γe − γy , y ≤ y ′ − R / . (5.20)It follows from (5.18) that (cid:119) / (cid:113) − (cid:117) = Var W ≤ E W = ∫ ∞ y f W ( y ) d y ≤ ∫ ∞ y − e − / γe − γy d y = − e − / γ . This gives the upper bound in (5.9).Suppose that 1 / γ ≤ y ′ and let y ′′ = / γ . Since y ′ ≥ R and y ′′ ≤ y ′ , we have y ′′ / ≤ y ′ − R /
2. Hence, we can apply (5.20) to y ∈ ( , y ′′ / ) and obtain (cid:117) = E W ≥ ∫ y ′′ / yf W ( y ) d y ≥ ∫ y ′′ / y γe − γy d y = γ ( − e − y ′′ γ / ( y ′′ γ / + )) (5.21) = γ ( − ( / ) e − / ) > γ . Suppose that 1 / γ ≥ y ′ and let y ′′ = / γ . Since y ′ ≥ R and y ′′ ≥ y ′ , we have 2 y ′′ ≥ y ′ + R .Hence, we can apply (5.19) to y > y ′′ and obtain (cid:117) = E W ≥ ∫ ∞ y ′′ yf W ( y ) d y ≥ ∫ ∞ y ′′ yγe − γy d y = γ e − y ′′ γ ( y ′′ γ + )) = γ e − > γ . This and (5.21) yield the lower bound in (5.8).Suppose that 1 / γ ≤ y ′ and let y ′′ = / γ . Then for any a ∈ R there exists an interval ( a , a ) ⊂ ( , y ′′ / ) such a − a ≥ y ′′ / y ∈ ( a , a ) , we have ∣ y − a ∣ ≥ y ′′ / RCHIMEDES’ PRINCIPLE 11
We will apply this observation with a = E W . Since y ′ ≥ R and y ′′ ≤ y ′ , we have a ≤ y ′′ / ≤ y ′ − R /
2. Hence, we can apply (5.20) to y ∈ ( a , a ) and obtainVar W ≥ ∫ a a ∣ y − E W ∣ f W ( y ) d y ≥ ∫ a a ( y ′′ / ) γe − γy d y ≥ ( a − a )( y ′′ / ) γe − γa (5.22) ≥ ( y ′′ / ) ( / y ′′ ) e − γy ′′ = ( y ′′ ) − e − = − e − γ . Next suppose that 1 / γ ≥ y ′ and let y ′′ = / γ . Then for any b ∈ R there existsan interval ( b , b ) ⊂ ( y ′′ , y ′′ ) such b − b ≥ y ′′ and for any y ∈ ( b , b ) , we have ∣ y − b ∣ ≥ y ′′ . We will apply this observation with b = E W . Since y ′ ≥ R and y ′′ ≥ y ′ ,we have b ≥ y ′′ ≥ y ′ + R . Hence, we can apply (5.19) to y ∈ ( b , b ) and obtainVar W ≥ ∫ b b ∣ y − E W ∣ f W ( y ) d y ≥ ∫ b b ( y ′′ ) γe − γy d y ≥ ( b − b )( y ′′ ) γe − γb ≥ ( y ′′ ) ( / y ′′ ) e − γy ′′ = ( y ′′ ) e − = e − γ . Since e − > − e − , the above estimate, (5.17) and (5.22) imply that (cid:119) / (cid:113) − (cid:117) = Var W ≥ − e − γ . This yields the lower bound in (5.9).The bound in (5.10) follows from (5.8) and (5.9). (cid:3)
Lemma 5.2. (i) For any y ≥ R , the function λ → (cid:117) ( λ, y ) is strictly decreasing.(ii) For every u > and y ≥ R there exists a unique λ > satisfying u = (cid:117) ( λ, y ) .Proof. Recall (cid:104) n defined in (5.1) and formula (5.2). We have ∂∂λ ( ∫ D ∖ B ∗ rλe − λr d x d r ∫ D ∖ B ∗ λe − λr d x d r ) = ∂∂λ ( (cid:104) (cid:104) ) = (cid:104) − (cid:104) (cid:104) (cid:104) , which is strictly negative by the Cauchy-Schwarz inequality and, consequently, λ → (cid:117) ( λ, y ) = (cid:104) ( λ, y )/ (cid:104) ( λ, y ) is strictly decreasing for every y ≥ R . This proves (i).Note that (cid:104) ( λ, y ) = ∫ D ∖ B ∗ λre − λr d x d r ≤ ∫ D λre − λr d x d r = ∣ D b ∣ λ . This and (5.7) imply that (cid:117) ( λ, y ) = (cid:104) ( λ, y ) (cid:104) ( λ, y ) ≤ ∣ D b ∣ λ (cid:113) ≤ λ ( − e / ) → , as λ → ∞ . On the other hand, using (5.7) again, when λ → (cid:117) ( λ, y ) = (cid:104) ( λ, y ) (cid:104) ( λ, y ) = (cid:104) ( λ, y ) (cid:113) ( λ, y ) ≥ ∣ D b ∣ (cid:113) ∫ ∞ y + R λre − λr d r ≥ λ ∫ λ ( y + R ) se − s d s → ∞ . We conclude that for given u > y ≥ R there exists a unique λ = λ ( u, y ) > u = (cid:117) ( λ, y ) holds. (cid:3) Lemma 5.3.
Recall that λ ( u, y ) denotes the solution to u = (cid:117) ( λ, y ) , and recall (5.3) - (5.6) . We have ∂λ∂u = qqu − w , (5.23) ∂λ∂y = λu ∣ D b ∣ − qqu − w , (5.24) ∂q∂u = qλ ( − λu ) ∂λ∂u , (5.25) ∂q∂y = qλ ( − λu ) ∂λ∂y + λ (∣ D b ∣ − q ) , (5.26) ∂∂u ( qe λu λ ) = qe λu . (5.27) Proof.
Comparing (5.1) and (5.3)-(5.6), we see that q = h , uh = h and w = h .Using (5.2) and differentiating the identity u (cid:104) ( λ ( u, y ) , y ) = (cid:104) ( λ ( u, y ) , y ) (5.28)with respect to u we obtain (cid:104) + u ( (cid:104) λ − (cid:104) ) ∂λ∂u = ( (cid:104) λ − (cid:104) ) ∂λ∂u ,∂λ∂u = (cid:104) ( (cid:104) λ − (cid:104) − u (cid:104) λ + u (cid:104) ) − = qqu − w , where we used the facts that w = h and uq = uh = h . This proves (5.23).We prove (5.25) as follows, ∂q∂u = ∂ (cid:104) ∂λ ∂λ∂u = ( (cid:104) λ − (cid:104) ) ∂λ∂u = qλ ( − λu ) ∂λ∂u . We use the above formula in the following proof of (5.27), ∂∂u ( qe λu λ ) = e λu λ ( q ( − λu ) ∂λ∂u + qλ ( λ + u ∂λ∂u ) − q ∂λ∂u ) = qe λu . Since ∫ ∞ λr n e − λr d r = n ! / λ n and the volume of the ( d − ) -dimensional sphere withradius ρ is π ( d − )/ ρ d − / Γ ( d + ) , (cid:104) n ( λ, y ) = ∣ D b ∣ n ! λ n − π ( d − )/ Γ ( d + ) ∫ y + Ry − R λr n e − λr ( R − ( y − r ) ) ( d − )/ d r = ∣ D b ∣ n ! λ n − π ( d − )/ Γ ( d + ) e − λy ∫ R − R λ ( s + y ) n e − λs ( R − s ) ( d − )/ d s. Consequently, by the dominated convergence theorem, we obtain ∂ (cid:104) n ∂y = λ ( ∣ D b ∣ n ! λ n − (cid:104) n ) − n ( ∣ D b ∣( n − ) ! λ n − − (cid:104) n − ) . (5.29) RCHIMEDES’ PRINCIPLE 13
In particular, ∂ (cid:104) ∂y = λ (∣ D b ∣ − (cid:104) ) , ∂ (cid:104) ∂y = (∣ D b ∣ − λ (cid:104) ) − (∣ D b ∣ − (cid:104) ) = (cid:104) − λ (cid:104) . (5.30)Differentiation of (5.28) with respect to y gives u ( ∂ (cid:104) ∂λ ∂λ∂y + ∂ (cid:104) ∂y ) = ∂ (cid:104) ∂λ ∂λ∂y + ∂ (cid:104) ∂y ,∂λ∂y = ( ∂ (cid:104) ∂y − u ∂ (cid:104) ∂y ) ( u ∂ (cid:104) ∂λ − ∂ (cid:104) ∂λ ) − . We now use (5.2) and (5.30) to see that ∂λ∂y = λu ∣ D b ∣ − qqu − w . This proves (5.24).We apply (5.2) and (5.30) once again to prove (5.26), ∂q∂y = ∂ (cid:104) ∂λ ∂λ∂y + ∂ (cid:104) ∂y = qλ ( − λu ) ∂λ∂y + λ (∣ D b ∣ − q ) . (cid:3) Proof of Theorem 2.3.
In view of (5.3)-(5.4), the equations (2.8)-(2.9) can be writtenin this form, M = K ( λ, y ) ∶= m ∣ D b ∣ − (cid:113) ( λ, y ) (cid:113) ( λ, y ) = m ( ∣ D b ∣ (cid:113) ( λ, y ) − ) , (5.31) G ( λ, y ) ∶= dmg λ + mg (cid:117) ( λ, y ) + M gy = E. (5.32)It is easy to see that all partial derivatives of any order of the functions K ( λ, y ) and G ( λ, y ) exist and are continuous.Consider any y ≥ R such that M gy < E . For sufficiently small λ > dmg /( λ ) > E ,so for some λ > G ( λ, R ) > E . Our assumption that M gy < E and (5.8) implythat for very large λ , G ( λ, R ) < E . By continuity of G ( λ, y ) , there exists λ such that G ( λ, y ) = E . Let λ y denote the smallest λ with this property.Part (i) of the lemma holds true because we have assumed that M gR < E . Therefore,we can take λ ∗ = λ R .We have lim y ↑ E /( Mg ) λ y = ∞ because the term M gy in the formula for G ( λ, y ) ap-proaches E , so the first term, dmg /( λ y ) , must go to 0.By the assumptions of part (ii) of the lemma, K ( λ R , R ) = K ( λ ∗ , R ) > M . It is easyto see that the function q ( λ, y ) converges to ∣ D b ∣ when λ → ∞ , no matter how y and λ are related. Hence lim y ↑ E /( Mg ) K ( λ y , y ) =
0. By continuity of K ( λ, y ) , there exists y such that K ( λ y , y ) = M . Let y A be the smallest y with this property and let λ A = λ y A .Note that R < y A < E /( M g ) .It remains to prove uniqueness of the solution ( y, λ ) to (2.8)-(2.9). Let z ( y ) = E − Mgymg . By Lemma 5.2, the function κ y ( u ) ∶= u + d λ ( u, y ) − z ( y ) is a strictly increasing function of u . By (5.8), κ y ( + ) = − z ( y ) and κ y ( z ( y )) >
0, so forevery given y ≥ R there exists a unique u = u ( y ) such that u ( y ) = z ( y ) − d λ ( u ( y ) , y ) . (5.33)Comparing this formula to (5.32), we see that we must have (cid:117) ( λ, y ) = u ( y ) . It willsuffice to show that there is at most one y such that (cid:113) ( λ ( u ( y ) , y ) , y ) = q ( u ( y ) , y ) satisfies (5.31).Assuming that u ( y ) satisfies (5.33),d u d y = − Mm + d λ ( ∂λ∂u ∂u∂y + ∂λ∂y ) . Hence, d u d y ( − d λ ∂λ∂u ) = − Mm + d λ ∂λ∂y , (5.34) d u d y = − Mm + d λ ∂λ∂y − d λ ∂λ∂u . We use this formula and (5.25)-(5.26) to see thatdd y q ( u ( y ) , y ) = ∂q∂u ∂u∂y + ∂q∂y = qλ ( − λu ) ( ∂λ∂u d u d y + ∂λ∂y ) + λ (∣ D b ∣ − q ) (5.35) = qλ ( − λu ) ⎛⎜⎜⎜⎝ ∂λ∂u − Mm + d λ ∂λ∂y − d λ ∂λ∂u + ∂λ∂y ⎞⎟⎟⎟⎠ + λ (∣ D b ∣ − q ) . Lemma 5.2 (i) implies that ∂λ / ∂u < − d λ ∂λ∂u >
0. Since we are interested only inthe sign of d q / d y , it will suffice to analyze A ∶= d q d y ( − d λ ∂λ∂u ) . Multiplying both sidesof (5.35) by 1 − d λ ∂λ∂u , we obtain A = qλ ( − λu ) ( ∂λ∂u (− Mm + d λ ∂λ∂y ) + ∂λ∂y ( − d λ ∂λ∂u )) + λ (∣ D b ∣ − q ) ( − d λ ∂λ∂u )= qλ ( − λu ) (− Mm ∂λ∂u + ∂λ∂y ) + λ (∣ D b ∣ − q ) ( − d λ ∂λ∂u )= qλ ( w − qu ) (( − λu ) ( M qm + q − λu ∣ D b ∣) + d (∣ D b ∣ − q )) + λ (∣ D b ∣ − q )= q ∣ D b ∣ λ ( w − qu ) (( − λu ) + ( − λu ) ( q ∣ D b ∣ ( + M / m ) − ) + d ( − q ∣ D b ∣ )) + λ (∣ D b ∣ − q ) . RCHIMEDES’ PRINCIPLE 15 If ( q /∣ D b ∣)( + M / m ) − = A = q ∣ D b ∣ λ ( w − qu ) (( − λu ) + d ( − q ∣ D b ∣ )) + λ (∣ D b ∣ − q ) . According to (5.17), w − qu >
0. Since we assume that R > ∣ D b ∣ − q >
0. It follows that A > q / d y > ( q /∣ D b ∣)( + M / m ) − =
0. In other words, d q / d y > q ( u ( y ) , y ) = ∣ D b ∣/( + M / m ) . A smooth function cannot cross a level multipletimes if its derivative is strictly positive at every crossing point. We have proved thatthere is at most one y such that q ( u ( y ) , y ) satisfies (5.31). (cid:3) Recall (2.1) and for u > y ≥ R and x ∈ D ′ b let D n ( y, u ) = ⎧⎪⎪⎨⎪⎪⎩( x n , x n + , y n ) ∈ D nb × D ′ b × R n ∶ n n ∑ i = y i = u, ( x k , y k ) ∈ D ∖ B (( x n + , y ) , R ) , k = , . . . n ⎫⎪⎪⎬⎪⎪⎭ , D n ( x, y, u ) = ⎧⎪⎪⎨⎪⎪⎩( x n , y n ) ∈ D nb × R n ∶ n n ∑ i = y i = u, (5.36) ( x k , y k ) ∈ D ∖ B (( x, y ) , R ) , k = , . . . n ⎫⎪⎪⎬⎪⎪⎭ . Proposition 5.4.
Consider u > , y ≥ R and x ∈ D ′ b . Let Vol n ( x, y, u ) be the ( nd − ) -dimensional volume of D n ( x, y, u ) and Vol n ( y, u ) be the (( n + ) d − ) -dimensionalvolume of D n ( y, u ) . Let λ = λ ( u, y ) > be the solution to u = (cid:117) ( λ, y ) and recall q = q ( u, y ) defined in (5.3) and (5.6) . Then for some absolute constants < c , c , c , c < ∞ and n , for all n ≥ n , c ∣ D b ∣ n − n ( qe λu λ ) n − ≤ Vol n ( x, y, u ) ≤ c ∣ D b ∣ n − n ( qe λu λ ) n − , (5.37) c ∣ D ′ b ∣ ⋅ ∣ D b ∣ n − n ( qe λu λ ) n − ≤ Vol n ( y, u ) ≤ c ∣ D ′ b ∣ ⋅ ∣ D b ∣ n − n ( qe λu λ ) n − . (5.38) Proof.
First note that (5.38) is an immediate corollary of (5.37) so it will suffice toprove (5.37).Let B ∗ = B (( x, y ) , R ) , H = {( z , t , . . . , z n , t n ) ∶ ( t + ⋅ ⋅ ⋅ + t n )/ n = u ; t k > , z k ∈ D b , for 1 ≤ k ≤ n } ,H ∗ = H ∩ ( D ∖ B ∗ ) n . Let ( Z , T ) , . . . , ( Z n , T n ) be i.i.d., with Z i ∈ R d − and T i > ≤ i ≤ n . Assumethat T i and Z i are independent, Z i has the uniform distribution in D b , and T i has theexponential distribution with parameter λ .Let f ( z , t , . . . , z n , t n ) be the density of (( Z , T ) , . . . , ( Z n , T n )) . We have f ( z , t , . . . , z n , t n ) = ∣ D b ∣ − n λ n exp (− λ ( t + ⋅ ⋅ ⋅ + t n )) , (5.39) for z i ∈ D b , t i >
0, 1 ≤ i ≤ n . Note that the density f is constant on H .Let ( Z , T ) , . . . , ( Z n , T n ) be i.i.d., with ( Z i , T i ) being distributed as ( Z i , T i ) con-ditioned by {( Z i , T i ) ∉ B ∗ } . By the definition of q and (5.39), the density f of (( Z , T ) , . . . , ( Z n , T n )) is given by f ( z , t , . . . , z n , t n ) = ( q ∣ D b ∣ ) − n ∣ D b ∣ − n λ n exp (− λ ( t + ⋅ ⋅ ⋅ + t n )) (5.40)on the set ( D ∖ B ∗ ) n .Let S n = n ∑ nk = T k = ∑ nk = ( T k / n ) . The distribution of T k / n is exponentialwith mean ( nλ ) − , so the distribution of S n is gamma with the density f S n ( s ) =(( nλ ) n / Γ ( n )) s n − e − nλs . Hence, f S n ( u ) = ( nλ ) n Γ ( n ) u n − e − nλu . (5.41)Let S n = n ∑ nk = T k and T j = T j − u for 1 ≤ j ≤ n . It follows from the fact that u = (cid:117) ( λ, y ) that E T j = u and E T j =
0. Let σ = Var T j = Var T j and S n = n / σ ∑ nk = T k = n / σ − ( S n − u ) . We have f S n ( s ) = σ − n / f S n ( σ − n / ( s − u )) . By Lemma 3.1, f S n ( u ) = σ − n / ( √ π + A ) , (5.42)where ∣ A ∣ ≤ C ( E ∣ T j / σ ∣ ) max ( , C )√ n , (5.43) C = sup u ∈ R f T j / σ ( u ) . (5.44)In view of (5.39), (5.40), (5.41) and (5.42),Vol ( H ∗ ) Vol ( H ) = q n f S n ( u )∣ D b ∣ n f S n ( u ) = q n σ − n / ( √ π + A )∣ D b ∣ n ( nλ ) n Γ ( n ) u n − e − nλu . (5.45)The volume of a regular n -simplex with unit side length is √ n /(( n − ) !2 ( n − )/ ) . Thevolume of a regular n -simplex with the side length √ (√ ) n − √ n /(( n − ) !2 ( n − )/ ) =√ n /( n − ) !. Hence Vol ( H ) = ∣ D b ∣ n u n − √ n /( n − ) !. This and (5.45) imply thatVol ( H ∗ ) = Vol ( H ) q n σ − n / ( √ π + A )∣ D b ∣ n ( nλ ) n Γ ( n ) u n − e − nλu = ∣ D b ∣ n u n − √ n ( n − ) ! q n σ − n / ( √ π + A )∣ D b ∣ n ( nλ ) n Γ ( n ) u n − e − nλu (5.46) = q n σ − n ( √ π + A )( nλ ) n e − nλu = ( qe λu λ ) n [ n − n σ − ( √ π + A )] . RCHIMEDES’ PRINCIPLE 17
In this proof, for any functions a and b of any number of variables, we will write a ≈ b to indicate that there exist universal constants 0 < c ′ , c ′′ < ∞ such that c ′ ≤ a / b ≤ c ′′ forall values of the arguments.By Lemma 5.1, q ≈ ∣ D b ∣ , λu ≈ λσ = λ √ w / q − u ≈
1, so (5.46) implies thatVol ( H ∗ ) ≈ ( qe λu λ ) n − [∣ D b ∣ n − n ( √ π + A )] . (5.47)Next we will find an upper bound for A defined in (5.42) using (5.43)-(5.44).We use the bound f T j ( y ) ≤ ∣ D b ∣ q λe − λy and the substitution λr = s to obtain E ∣ T j / σ ∣ ≤ σ − ∣ D b ∣ q ∫ ∞ ∣ r − u ∣ λe − λr d r = ( λσ ) ∣ D b ∣ q ∫ ∞ ∣ s − uλ ∣ e − s d s. We have already pointed out that λσ ≈ ∣ D b ∣/ q ≈ uλ ≈
1. The last formula showsthat E ∣ T j / σ ∣ ≤ c < ∞ , where c is a universal constant.Recall that λσ ≈ ∣ D b ∣/ q ≈ c < ∞ , f T j / σ ( s ) ≤ q − σλe − λ ( sσ + u ) ≤ σλq − ≤ c ∣ D b ∣ . This bound and E ∣ T j / σ ∣ ≤ c imply that ∣ A ∣ ≤ c / (√ n min ( , ∣ D b ∣ )) where c is anabsolute constant.Consider the case ∣ D b ∣ =
1. Then ∣ A ∣ ≤ c /√ n . Let n be so large that for n ≥ n ,12 √ π < √ π + c √ n < √ π . This and (5.47) prove (5.37) in the case ∣ D b ∣ = ∣ D b ∣ ≠
1, we use scaling. If x, y, D, R and u are multiplied by c ∗ >
0, it iselementary to verify that Vol n ( x, y, u ) , q , λ and u are rescaled by powers of c ∗ suchthat (5.37) remains true. (cid:3) Remark 5.5.
We believe that Proposition 5.4 holds for all n . One could prove thebounds for n < n using elementary estimates similar to those in the proof of Lemma5.1. We do not provide a proof because it is not needed for our main theorem.6. Archimedes’ principle
In this section, for any functions a ( ⋅ ) and a ( ⋅ ) of any arguments we will write a ( ⋅ ) ≈ a ( ⋅ ) ⇐⇒ ∃ < c , c < ∞ ∶ c a ( ⋅ ) ≤ a ( ⋅ ) ≤ c a ( ⋅ ) . (6.1)The constants c and c may depend only on “fixed” parameters in our model: d, D, D b , R, M, m, g and E . Proposition 6.1.
Recall the notation from Theorem 2.3 and assume that (2.10) holds.Recall (5.4) and let u A = (cid:117) ( λ A , y A ) . For every ε > , lim n →∞ P n (∣ Y n + − y A ∣ > ε or ∣ n ∑ i = Y i − u A ∣ > ε ) = . Proof.
Recall notation from (2.6). The following formula for the marginal density f Y n + ( y ) of Y n + follows from (2.7),(6.2) f Y n + ( y ) = Z n ∫ D En ( y ) ( E − mg n n ∑ i = y i − M gy ) (( n + ) d − )/ d x n + d y n , where Z n is the normalizing constant. Let z ( y ) = ( E − M gy )/( mg ) and note that z ( y ) > y ∈ [ R, E /( M g )) because we have assumed that E > M gR . Recall notationfrom Proposition 5.4. The proposition implies that f Y n + ( y ) = Z n ∫ z ( y ) ( E − M gy − mgu ) (( n + ) d − )/ Vol n ( y, u ) d u (6.3) ≈ Z ′ n ∣ D b ∣ n − n ∫ z ( y ) ( z ( y ) − u ) (( n + ) d − )/ ( q ( u, y ) e λ ( u,y ) u λ ( u, y ) ) n − d u (6.4)where Z ′ n = ∣ D b ∣ n − n ∫ E /( Mg ) R d y ∫ z ( y ) ( z ( y ) − u ) (( n + ) d − )/ ( q ( u, y ) e λ ( u,y ) u λ ( u, y ) ) n − d u. The integral in (6.4) can be written as(6.5) ∫ z ( y ) α n − y ( u )( z ( y ) − u ) d − d u, where α y ( u ) = ( z ( y ) − u ) d / qe λu / λ .Let β y ( u ) = λ ( u, y ) − d ( z ( y ) − u ) . (6.6)By (5.27),(6.7) ∂∂u α y ( u ) = ( z ( y ) − u ) d / qe λu λ ( λ − d ( z ( y ) − u ) ) = α y ( y ) β y ( u ) . By Lemma 5.2, the function u → λ ( u, y ) is strictly decreasing. Hence β y ( u ) is strictlydecreasing. When u ↓ λ → ∞ by (5.8). Thus β y ( + ) = ∞ for y ∈ [ R, E /( M g )) .The estimate (5.8) implies that β y ( z ( y ) − ) = −∞ . We conclude that there is a unique u = u ( y ) ∈ ( , z ( y )) such that β y ( u ) = β y ( y ) is positive on ( , u ) and negativeon ( u , z ( y )) . In view of (6.7), α y ( u ) attains its only maximum at u on the interval ( , u ) , and the maximum is strict. Note that(6.8) β y ( u ) = β y ( u ) − β y ( u ) = λ ( u, y ) − λ ( u , y ) + d u − u ( z ( y ) − u )( z ( y ) − u ) . This, and the facts that the function u → λ ( u, y ) is decreasing and z ( y ) ≥ z ( y ) − u > u ∈ ( , z ( y )) , imply that ∣ β y ( u )∣ = − β y ( u ) ≥ d z ( y ) ( u − u ) = d z ( y ) ∣ u − u ∣ , u ∈ ( u , z ( y )) , (6.9) ∣ β y ( u )∣ = β y ( u ) ≥ d z ( y ) ( u − u ) = d z ( y ) ∣ u − u ∣ , u ∈ ( , u ) . (6.10) RCHIMEDES’ PRINCIPLE 19
To simplify notation, we will write k = n −
1. From (6.9)-(6.10), we obtain for u ∉[ u − z ( y ) k − / , u + z ( y ) k − / ] ,2 √ kz ( y )∣ β y ( u )∣ d = d z ( y )∣ β y ( u )∣ z ( y ) k − / ≥ d z ( y )∣ β y ( u )∣∣ u − u ∣ ≥ . This, (6.5) and (6.7), and the bound z ( y ) − u < z ( y ) imply that (∫ u − z ( y ) k − / + ∫ z ( y ) u + z ( y ) k − / ) α ky ( u )( z ( y ) − u ) d − d u (6.11) ≤ (∫ u − z ( y ) k − / + ∫ z ( y ) u + z ( y ) k − / ) α ky ( u )( z ( y ) − u ) d − √ kz ( y )∣ β y ( u )∣ d d u = z d ( y )√ kd (∫ u − z ( y ) k − / + ∫ z ( y ) u + z ( y ) k − / ) α k − y ( u ) α y ( u ) β y ( u ) d u = z d ( y ) d √ k ( α ky ( u − z ( y )√ k ) + α ky ( u + z ( y )√ k ) − α ky ( ) − α ky ( z ( y ))) ≤ d √ k z d ( y ) α ky ( u ) . We combine this estimate with the observation that z ( y ) ≤ E / mg and the followingbound(6.12) ∫ u + z ( y ) k − / u − z ( y ) k − / α ky ( u )( z ( y ) − u ) d − d u ≤ z d ( y )√ k α ky ( u ) , to arrive at(6.13) ∫ z ( y ) α ky ( u )( z ( y ) − u ) d − d u ≤ ( + d ) z d ( y )√ k α ky ( u ) . Since β y ( u ) ≤ u ∈ [ u , z ( y )) , (6.6) implies that, for u ∈ [ u , z ( y )) , u ≥ z ( y ) + d uλ ( u,y ) . This and (5.8) yield, u ≥ z ( y ) + d , u ∈ [ u , z ( y )) . (6.14)Since β y ( u ) =
0, we obtain from (6.6), u ( y ) = z ( y ) + d u ( y ) λ ( u ( y ) ,y ) , and, by (5.8), u ( y ) ≤ z ( y ) + d / ≤ z ( y ) . (6.15)Using (5.23), (5.8), (5.9) and (6.14), we get for u ∈ [ u , z ( y )) , ∣ ∂λ∂u ∣ = − ∂λ∂u = ( w / p − u ) λ ( λu ) u ≤ e ⋅ ( + d ) z ( y ) = c z ( y ) , (6.16) where c depends only on d .For u ∈ ( u , u + z ( y ) k − / ) and k ≥ = z ( y )( z ( y ) − u )( z ( y ) − u ) ≤ z ( y )( z ( y ) − z ( y )) ( z ( y ) − z ( y ) − z ( y ) k − / )≤ ( / )( / ) ≤ . This estimate, (6.8) and (6.16) show that for u ∈ ( u , u + z ( y ) k − / ) and k ≥ − β y ( u ) = λ ( u , y ) − λ ( u, y ) + d u − u ( z ( y ) − u )( z ( y ) − u )≤ u − u z ( y ) ( c + d ) ≤ c z ( y )√ k , (6.17)where c > d . For u ∈ ( u , u + z ( y ) k − / ) and k ≥ z ( y ) − u ≥ z ( y ) − z ( y ) − z ( y ) k − / ≥ z ( y )/ . This implies that, for u ∈ ( u , u + z ( y ) k − / ) and k ≥ ∫ z ( y ) α ky ( u )( z ( y ) − u ) d − d u (6.18) ≥ α ky ( u ) ∫ u + z ( y ) k − / u ( z ( y ) − u ) d − ( − α y ( u ) − α u ( u ) α y ( u ) ) k d u ≥ z d − ( y ) α ky ( u ) ( / ) d − ∫ u + z ( y ) k − / u ( − α y ( u ) − α u ( u ) α y ( u ) ) k d u. Recall that u is the maximum of α y , and also (6.7) and (6.17). There exists ˜ u ∈( u , u + z ( y ) k − / ) such that, α y ( u ) − α u ( u ) α y ( u ) = − α y ( ˜ u ) α y ( u ) β y ( ˜ u )( u − u ) ≤ c z ( y )√ k ( u − u ) ≤ c k . Hence, for u ∈ ( u , u + z ( y ) k − / ) and k ≥ ( − α y ( u ) − α u ( u ) α y ( u ) ) k ≥ c . We combine this with (6.18) to see that(6.19) ∫ z ( y ) α ky ( y )( z ( y ) − u ) d − d u ≥ c √ k z d ( y ) α ky ( u ) . Let λ = λ ( u ( y ) , y ) , q = q ( u ( y ) , y ) and ψ ( y ) = ( / d ) d / α y ( u ) = ( / d ) d / ( z ( y ) − u ( y )) d / q e λ u λ . (6.20) RCHIMEDES’ PRINCIPLE 21
Since u ( y ) makes the right hand side of (6.6) equal to 0, ψ ( y ) = ( / d ) d / ( z ( y ) − u ( y )) d / q e λ u λ = q e λ u λ d / + . (6.21)Recall that k = n −
1. It follows from (6.4), (6.5), (6.13), (6.19) and (6.20) that(6.22) f Y n + ( y ) ≈ Z ′′ n ψ n − ( y ) z d ( y ) , with the normalizing constant Z ′′ n = ∫ E /( Mg ) R ψ n − ( y ) z d ( y ) d y .Comparing (5.33) and (6.6), we see that we can apply (5.34) to u ( y ) , i.e.,(6.23) d u d y ( − d λ ∂λ∂u ) = − Mm + d λ ∂λ∂y . We use this formula, (5.25), (5.26) and (6.21) in the following calculation, λ d / + e − λ u d ψ d y = λ ( ∂q∂u d u d y + ∂q∂y ) + λ q d u d y + q ( λ u − d + ) ( ∂λ∂u d u d y + ∂λ∂y )= d u d y ( λ ∂q∂u + q ( λ u − d + ) ∂λ∂u + λ q ) + λ ∂q∂y + q ( λ u − d + ) ∂λ∂y = d u d y ( λ q λ ( − λ u ) ∂λ∂u + q ( λ u − d + ) ∂λ∂u + λ q )+ λ ( q λ ( − λ u ) ∂λ∂y + λ (∣ D b ∣ − q )) + q ( λ u − d + ) ∂λ∂y = λ q d u d y ( − d λ ∂λ∂u ) + λ (∣ D b ∣ − q ) − dq ∂λ∂y = λ q (− Mm + d λ ∂λ∂y ) + λ (∣ D b ∣ − q ) − dq ∂λ∂y = λ q m ( m ∣ D b ∣ − q q − M ) . Recall the usual sign function that takes values − , ( d ψ ( y ) d y ) = sign ( m ∣ D b ∣ − q ( y ) q ( y ) − M ) = sign ⎛⎝ m ∫ B ((̃ x,y ) ,R ) λ e − λ r d x d r ∫ D ∖ B ((̃ x,y ) ,R ) λ e − λ r d x d r − M ⎞⎠ . (6.24)According to (6.6) and the definition of u ( y ) , λ ( u ( y ) , y ) − d ( z ( y ) − u ( y )) = . (6.25)This is equivalent to (2.9) (see also (5.32) and (5.33)). In view of (2.8)-(2.10), (6.24)and (6.25), by Theorem 2.3 (ii), there exists a unique y A ∈ ( R, E /( M g )) such that d ψ ( y ) d y ∣ y = y A = ψ ( y ) attains its maximum at y A and it isstrictly increasing on ( R, y A ) and strictly decreasing on ( y A , E /( M g )) .For later reference we note that the above argument also shows that λ = λ ( u ( y A ) , y A ) = λ A , with λ A as defined in Theorem 2.3, and, therefore, u ( y A ) = u A ,with u A as defined in the statement of the present theorem.According to (6.22), for some c and any ε ∈ ( , min ( y A − R, E /( M g ) − y A )) , P n (∣ Y n + − y A ∣ ≥ ε ) ≤ c Z ′′ n (∫ y A − εR + ∫ E /( Mg ) y A + ε ) ψ n ( y ) z d / ( y ) d y. Since Z ′′ n ≥ ∫ y A + ε / y A ψ n ( y ) z d / ( y ) d y + ∫ y A y A − ε / ψ n ( y ) z d / ( y ) d y ≥ ε ψ n ( y A + ε / ) z d / ( y A + ε / ) + ε ψ n ( y A − ε / ) z d / ( y A ) , and (∫ y A − εR + ∫ E /( mg ) y A + ε ) ψ n ( y ) z d / ( y ) d y ≤ ( E /( M g ) − R ) z d / ( R )( ψ n ( y A − ε ) + ψ n ( y A + ε )) , we have P n (∣ Y n + − y A ∣ ≥ ε ) ≤ ( E /( M g ) − R ) z d / ( R )( ψ n ( y A − ε ) + ψ n ( y A + ε )) ε ψ n ( y A + ε / ) z d / ( y A + ε / ) + ε ψ n ( y A − ε / ) z d / ( y A )≤ EεM g ( z ( R ) z ( y A + ε / ) ) d / [( ψ ( y A + ε ) ψ ( y A + ε / ) ) n + ( ψ ( y A − ε ) ψ ( y A − ε / ) ) n ] . The right-hand side goes to 0 as n → ∞ , i.e.,lim n →∞ P n (∣ Y n + − y A ∣ ≥ ε ) = . (6.26)A calculation analogous to (6.11) gives for δ > (∫ u − δ + ∫ z ( y ) u + δ ) α ky ( u )( z ( y ) − u ) d − d u (6.27) ≤ z d ( y ) d √ k ( α ky ( u − δ ) + α ky ( u + δ ) − α ky ( ) − α ky ( z ( y )))≤ z d ( y ) d √ k ( α ky ( u − δ ) + α ky ( u + δ )) . Recall that k = n − n ∑ ni = y i = u . The following remarkmade above about ψ applies also to α y because of (6.20): “ ψ ( y ) attains its maximumat y A and it is strictly increasing on ( R, y A ) and strictly decreasing on ( y A , E /( M g )) .”Combining (6.19) and (6.27), we obtain for every fixed y ,lim sup n →∞ P n (∣ n n ∑ i = Y i − u ( y )∣ > δ ∣ Y n + = y ) (6.28) RCHIMEDES’ PRINCIPLE 23 ≤ lim sup k →∞ z d ( y ) d √ k ( α ky ( u − δ ) + α ky ( u + δ )) c √ k z d ( y ) α ky ( u ) = . Using the fact that u ( y ) is a continuous function of y (see, e.g., (6.23)), (6.26), (6.28)and applying the dominated convergence theorem to the indicator function of the event {∣ n ∑ ni = Y i − u ( y )∣ > ε } we obtain from (6.26) and (6.28),lim n →∞ P n (∣ Y n + − y A ∣ > ε or ∣ n ∑ i = Y i − u ( y A )∣ > ε ) = . It remains to recall that we have shown that u ( y A ) = u A . (cid:3) Recall definitions (2.2) of ( X n , Y n ) , (2.11) of ν x,y,λ , and (5.36) of D n ( x, y, u ) . Lemma 6.2.
Fix any ̂ x ∈ D ′ b , u > and ̂ y ≥ R , and let ̂ B = B ((̂ x, ̂ y ) , R ) . Let P ̂ x, ̂ y,un denote the uniform distribution on D n (̂ x, ̂ y, u ) . Suppose that ( X n , Y n ) has the distri-bution P ̂ x, ̂ y,un .(i) When n → ∞ then for every fixed j ≥ , the distribution of ( X j , Y j ) convergesto ν ̂ x, ̂ y,λ , where λ is the solution to u = (cid:117) ( λ, y ) . Moreover, for any j ≠ j , the jointdistribution of (( X j , Y j ) , ( X j , Y j )) converges to ν ̂ x, ̂ y,λ × ν ̂ x, ̂ y,λ .(ii) The convergence in (i) is uniform in the sense that for any bounded d -dimensionalrectangular parallelepipeds A , A ⊂ R d with non-empty interior, any < u < u < ∞ ,and any ε > , there exists n such that for all n ≥ n , ̂ x ∈ D ′ b , y > R and u ∈ [ u , u ] , ∣ P ̂ x,y,un ((( X j , Y j ) , ( X j , Y j )) ∈ A × A ) − ν ̂ x,y,λ × ν ̂ x,y,λ ( A × A )∣ < ε. (6.29) Proof.
We will reuse some ideas from the proof of Proposition 5.4.Let ( Z , T ) , . . . , ( Z n , T n ) be i.i.d., with Z i ∈ R d − and T i > ≤ i ≤ n . Assumethat T i and Z i are independent, Z i has the uniform distribution in D b , and T i has theexponential distribution with parameter λ .Let f ( z , t , . . . , z n , t n ) be the density of (( Z , T ) , . . . , ( Z n , T n )) . We have f ( z , t , . . . , z n , t n ) = ∣ D b ∣ − n λ n exp (− λ ( t + ⋅ ⋅ ⋅ + t n )) , (6.30)for z i ∈ D b , t i >
0, 1 ≤ i ≤ n .Let ( Z , T ) , . . . , ( Z n , T n ) be i.i.d., with ( Z i , T i ) being distributed as ( Z i , T i ) con-ditioned by {( Z i , T i ) ∉ ̂ B } . Hence, ( Z i , T i ) has the distribution ν ̂ x, ̂ y,λ .By the definition of q and (6.30), the density f of (( Z , T ) , . . . , ( Z n , T n )) is givenby f ( z , t , . . . , z n , t n ) = ( q ∣ D b ∣ ) − n ∣ D b ∣ − n λ n exp (− λ ( t + ⋅ ⋅ ⋅ + t n )) (6.31)on the set ( D ∖ ̂ B ) n .Let S n = n ∑ nk = T k . Let (( Z , T ) , . . . , ( Z n , T n )) be the sequence (( Z , T ) , . . . , ( Z n , T n )) conditioned by { S n = u } . Note that the distribution of (( Z , T ) , . . . , ( Z n , T n )) is the uniform distribution on D n (̂ x, ̂ y, u ) . Hence, the distri-bution of ( X j , Y j ) under P ̂ x, ̂ y,un is the same as the distribution of ( Z j , T j ) . We willshow that the distribution of ( Z j , T j ) converges to that of ( Z j , T j ) as n → ∞ . This is equivalent to the weak convergence of P ̂ x, ̂ y,un to ν ̂ x, ̂ y,λ since the distribution of ( Z i , T i ) has the distribution ν ̂ x, ̂ y,λ .Fix j ≥ n > j . Let S ,jn = n − ∑ k = ,...,n ; k ≠ j T k .Since ( Z , T ) , . . . , ( Z n , T n ) is an i.i.d. sequence, the density f ( Z j ,T j ) has the form f ( Z j ,T j ) ( z, t ) = c f ( Z j ,T j ) ( z, t ) f S ,jn ( u + n − ( u − t )) , (6.32)where c is the normalizing constant.Let σ = Var T i and T i = ( T i − u )/( σ √ n − ) . Note that, since u = (cid:117) ( λ, y ) , E T i = S ,jn = ∑ k = ,...,n ; k ≠ j T k = σ √ n − ∑ k = ,...,n ; k ≠ j ( T k − u ) = √ n − σ ( S ,jn − u ) . Hence f S ,jn ( u + s ) = √ n − σ f S ,jn ( √ n − σ s ) . By Lemma 3.1, f S ,jn ( u + n − ( u − t )) = √ n − σ f S ,jn ( √ n − σ n − ( u − t )) (6.33) = √ n − σ f S ,jn ( σ √ n − ( u − t )) = √ n − σ ( φ ( σ √ n − ( u − t )) + A ) . Since the random variable T i has the same distribution as the random variable withthe same name in the proof of Proposition 5.4, the estimates (5.43)-(5.44) for A , andthose at the end of the proof of Proposition 5.4 apply in the present case. Thus ∣ A ∣ ≤ c √ n − , (6.34)where c depends only on ∣ D b ∣ .Fix any u >
0. Then, by (5.10) and (5.17), there exists c > u ≥ u , σ ≥ c . (6.35)Fix any u ∈ ( u , ∞) . It follows from (6.33), (6.34) and (6.35) that for any fixed0 ≤ t < t < ∞ , u ∈ [ u , u ] and ε > n such that for n ≥ n , t , t ∈ [ t , t ] ,and t ∉ [ t , t ] , 1 − ε < f S ,jn ( u + n − ( u − t )) f S ,jn ( u ) < + ε, (6.36) f S ,jn ( u + n − ( u − t )) f S ,jn ( u ) < . (6.37)This, (5.18) and (6.32) imply that the distribution of ( Z j , T j ) converges to that of ( Z j , T j ) as n → ∞ . Moreover, for any bounded d -dimensional rectangular parallelepiped RCHIMEDES’ PRINCIPLE 25 A ⊂ R d with non-empty interior, any 0 < u < u < ∞ , and any ε >
0, there exists n such that for all n ≥ n , ̂ x ∈ D ′ b , y > R and u ∈ [ u , u ] , ∣ P ̂ x,y,un (( X j , Y j ) ∈ A ) − ν ̂ x,y,λ ( A )∣ < ε. A completely analogous argument shows that for any j ≠ j , the distribution of (( Z j , T j ) , ( Z j , T j )) converges to that of (( Z j , T j ) , ( Z j , T j )) as n → ∞ , and alsopart (ii) of the lemma holds true. (cid:3) Recall ν ̃ x,y,λ defined in (2.11), the empirical measure Q n defined in (2.12), and D n ( x, y, u ) defined in (5.36). Lemma 6.3.
The marginal distribution of X n + under P n is uniform in D ′ b . Given { X n + = x } , the conditional distribution of Q n converges to ν x,y A ,λ A weakly, in probabilityas n → ∞ .Proof. It follows easily from the microcanonical ensemble formula (2.7) that the mar-ginal distribution of X n + under P n is uniform in D ′ b .The same formula (2.7) implies that one can represent P n as follows. Let P x,y,un denotethe uniform distribution on D n ( x, y, u ) . Then there exists a probability measure µ n on D ′ b × [ R, ∞) × R + such that P n = ∫ P x,y,un d µ n ( x, y, u ) . Let P xn = ∫ P x,y,un µ n ( x, d y, d u ) . Let E xn and E x,y,un denote expectations corresponding to P xn and P x,y,un .Fix any x ∈ D ′ b and write ν ′ x = ν x,y A ,λ A .Fix any bounded d -dimensional rectangular parallelepiped A ⊂ R d with non-emptyinterior. We have E xn ( Q n ( A ) − ν ′ x ( A )) = E xn ( n n ∑ i = A ( X i , Y i ) − ν ′ x ( A )) (6.38) = E xn ( n n ∑ i = ( A ( X i , Y i ) − ν ′ x ( A ))) = n n ∑ i = E xn ( A ( X i , Y i ) − ν ′ x ( A )) + n n − ∑ i = n ∑ j = i + E xn [( A ( X i , Y i ) − ν ′ x ( A )) ( A ( X j , Y j ) − ν ′ x ( A ))]= n E xn ( A ( X , Y ) − ν ′ x ( A )) + ( n − n ) n E xn [( A ( X , Y ) − ν ′ x ( A )) ( A ( X , Y ) − ν ′ x ( A ))]= n E xn ( A ( X , Y ) − ν ′ x ( A )) + ( n − n ) n ( E xn [ A ( X , Y ) A ( X , Y )] − E xn [ A ( X , Y ) ν ′ x ( A )]− E xn [ ν ′ x ( A ) A ( X , Y )] + ( ν ′ x ( A )) ) . It is easy to see that the function ( x, y, u ) → P x,y,un is continuous in the weak topology.This, Proposition 6.1 and (6.29) imply thatlim n →∞ E xn [ A ( X , Y ) A ( X , Y )] (6.39) = lim n →∞ ∫ E x,y,un [ A ( X , Y ) A ( X , Y )] µ n ( x, d y, d u ) = ( ν ′ x ( A )) . For the same reason, lim n →∞ E xn [ A ( X k , Y k )] = ν ′ x ( A ) , k = , . (6.40)The bound E xn ( A ( X , Y ) − ν ′ x ( A )) ≤
1, (6.38), (6.39) and (6.40) imply thatlim n →∞ E xn ( Q n ( A ) − ν ′ x ( A )) = . The lemma follows because this statement holds for every A . (cid:3) Proof of Theorem 2.4.
The theorem follows from Proposition 6.1 and Lemma 6.3. (cid:3) Inverse temperature
Proposition 7.1.
The inverse temperature of the gas is asymptotically proportional to λ A defined in Theorem (2.3) . More precisely, for every fixed j ≥ , lim n →∞ E n ( m ∥ V j ∥ ) = lim n →∞ E n ( n n ∑ i = m ∥ V i ∥ ) = dmg λ A . (7.1) Proof.
Since the total energy E is fixed, we have1 n n ∑ i = mgY i ≤ E, M gY n + ≤ E, n + n + ∑ i = m ∥ V i ∥ ≤ E. Hence, these sequences of random variables are tight. Recall P x,y,un and µ n from theproof of Lemma 6.3. By Proposition 6.1,lim n →∞ E n ( n n ∑ i = mgY i ) = lim n →∞ ∫ E x,y,un ( n n ∑ i = mgY i ) d µ n ( x, y, u ) (7.2) = lim n →∞ ∫ u d µ n ( x, y, u ) = mgu A = mg ∫ D ∖ B ((̃ x,y A ) ,R ) yλ A e − λ A y d x d y ∫ D ∖ B ((̃ x,y A ) ,R ) λ A e − λ A y d x d y . Proposition 6.1 and tightness also imply thatlim n →∞ E n M gY n + = M gy A . (7.3)This and (7.2) show that the expected value of the potential energy of the point par-ticles and the ball converges to a fixed number. Since the total energy E is fixed, theexpectation of the kinetic energy converges weakly to a fixed number as well, i.e., forsome σ >
0, lim n →∞ E n ( n + n + ∑ i = m ∥ V i ∥ ) = σ . (7.4) RCHIMEDES’ PRINCIPLE 27
The total energy is fixed so (7.2), (7.3) and (7.4) show that σ + mg ∫ D ∖ B ((̃ x,y A ) ,R ) yλ A e − λ A y d x d y ∫ D ∖ B ((̃ x,y A ) ,R ) λ A e − λ A y d x d y + M gy A = E. Since y A and λ A solve (2.9), we must have σ = dmg λ A .It remains to note that due to the symmetry of µ y n + in (2.7),lim n →∞ E n ( n n ∑ i = m ∥ V i ∥ ) = lim n →∞ E n ( n + n + ∑ i = m ∥ V i ∥ ) = σ = dmg λ A . (cid:3) Uniqueness of the stationary distribution
Proof of Theorem 2.1 (ii).
The idea of the proof is the following. If there were morethan one invariant measure, at least two of them would be mutually singular byBirkhoff’s ergodic theorem ([Sin94]). Given any two starting configurations we willexhibit two deterministic trajectories meeting at the same point in the phase space atsome time t >
0. Then we will argue that due to the random nature of some reflections,both processes have densities that are strictly positive in some neighborhood of thatpoint in the phase space. Hence, there are no mutually singular invariant measures.Much of the proof will be presented in a very informal way. This is because ourargument is totally elementary but it would be extremely tedious to write (or read) ina fully rigorous way.
Step 1 . Assume that the initial condition of the system does not belong to any ofthe families (1) and (2) described in the statement of the theorem. We will construct asingle trajectory of the system. The trajectory will respect the laws of elastic collisionswhen they are assumed, i.e., for all collisions of the ball with the point particles and thewalls of the container. For any reflection of a point particle from a wall of the container,we will choose a direction after the reflection from all possible directions in a way thatmeets the needs of the argument.Recall that “walls” of the container include its bottom so point particles reflectaccording to the Lambertian law from the side walls and the bottom of the container.Fix distinct points z , . . . , z n + in D b , such that the distance of z n + from the side wallof the container is greater than R .We let the system evolve according to the original dynamics until one of the pointparticles hits a wall at a time s . We assume that the particle that hit the wall islabeled 1, since the labeling of point particles is irrelevant.The first point particle can reflect in any direction, including directions arbitrarilyclose to the boundary. So it can stay arbitrarily close to the boundary for an arbitrarilylong time and move towards any point on the boundary, with the only limitation beingits constant energy (the sum of potential and kinetic energies), see Figs. 1-2. We letthe system evolve according to the original dynamics after time s , except that the firstpoint particle will follow its own trajectory, constructed independently. Let s be thenext time when a particle different from the first one hits a wall. We choose a trajectory Figure 1.
Side view of a point particle trajectory reflecting from thebottom of the container. z j Figure 2.
View from above of a point particle trajectory reflecting fromthe bottom of the container.for the first particle very close to the wall and moving towards z in such a way that itavoids a collision with the ball on [ s , s ] (recall that point particles do not interact).We proceed by induction. Suppose that, for some j < n , a deterministic trajectory ofthe system has been constructed on [ , s j ] , including the trajectories of point particleslabeled 1 , . . . , j . These point particles stay close to the walls from the time of the firsthit of a wall until s j . Particle k moves towards z k from the first time it hits a walluntil s j , for k = , . . . , j . Given this inductive assumption, we let the system evolveaccording to the original dynamics after time s j , except that point particles labeled1 , . . . , j will follow their own trajectories, constructed independently. Let s j + be thenext time when a particle different from 1 , . . . , j hits a wall. We will call this particle j +
1. We choose a trajectory for each of the particles 1 , . . . , j + k -th particle is moving towards z k in such a way that it avoids a collision with theball on [ s j , s j + ] , for k = , . . . , j + [ , s n ] . Wewill continue after discussing a delicate point in the next step. Step 2 . It is possible that fewer than n point particles hit the walls of the container.This can happen only if some particles always reflect from the top part of the ball. Inthis case, we let one of the point particles that are staying close to the boundary movetowards the point where the ball reflects from the bottom of the container and thenwe let the point particle hit the ball slightly off center. That will nudge the ball off itstrajectory. The result will be that the point particles formerly reflecting from the topof the ball will move to the side and eventually hit a wall. RCHIMEDES’ PRINCIPLE 29
Figure 3.
Side view of the container. A point particle collides with theball.
Step 3 . At this step of the construction of the trajectory of the system, point particleswill come close to the point at the bottom where the ball reflects and they will hit theball, one at a time, see Fig. 3. All other point particles will keep close to the bottomand stay away from the ball.There are two major goals of the construction that can be achieved by this procedure.First, we can achieve equipartition of the energy. Second, we can put the ball above z n + and make it move vertically.We will explain how we can arrange for equipartition of energy between the pointparticles and the ball, so that ( E − M gR )/( n + ) of energy is given to each point particleand ( E − M gR )/( n + ) + M gR of energy is given to the ball. Note that the minimalamount of energy a point particle can have is 0, assuming that it is sitting motionlessat the bottom. For the ball, the minimal amount of energy is
M gR .The point particles do not interact with each other so the only way to transfer theenergy between them is via collisions with the macroscopic ball. We let point particlesapproach the point where the ball reflects from the bottom, one at a time. Then wemake the direction of the velocity of the point particle close to vertical (or at leastconsiderably different from the horizontal; see Fig. 3). We make the point particle hitthe ball either when the ball is moving up or down. In the first case, the point particlewill lose energy and in the second case it will acquire energy. By manipulating theplace of the collision and the velocity direction of the point particle before the collision,and by repeating the procedure, if necessary, many times, we can partition the energybetween particles and the ball in an arbitrary way.We need to add a few words clarifying the algorithm described above. If a particleand the ball have the same amount of kinetic energy and n is large then the speed of theparticle is much larger than the speed of the ball. Hence, we can start by transferringenergy to the ball from all point particles that have more than ( E − M gR )/( n + ) ofenergy. Then the energy can be transferred from the ball to the particles that had lessthan ( E − M gR )/( n + ) of energy, one by one. Step 4 . We make the n -th point particle collide with the ball as depicted in Fig. 3to change the trajectory of the ball so it reflects vertically at z n + ∈ D b . Figure 4.
Side view of the container. A point particle hits the ball atthe lowest point on the surface.After this is done, energy might not be equidistributed. If necessary, we induceenergy transfer between the n -th particle and the ball by collisions of the particle withthe bottom of the ball, as in Fig. 4.Fix some time t greater than the duration of the trajectory constructed so far, suchthat the ball hits z n + at time t . Make all point particles follow trajectories such thatthey all hit their own base points z k , k = , . . . , n , at time t . For this to be possible, itmay be necessary to move t to one of the later times when the ball hits z n + .For future reference, let the deterministic trajectory constructed above be calledΓ = { Γ ( t ) , ≤ t ≤ t } . Step 5 . To finish the proof, we will argue as follows. We have shown that the systemcan get to the same configuration at time t for every initial configuration (the time t may depend on the initial conditions—this is not a problem). The trajectory that weconstructed is deterministic but it “agrees” with the dynamics of the system, includingLambertian reflections. Lambertian reflections introduce randomness. They make thestate of the system at time t random, with a density. The densities for different initialconfigurations overlap so there is only one stationary measure.The subtle point is that the state density at time t is not with respect to Lebesguemeasure on R nd but on a hypersurface of dimension 2 nd − Z j ( t ) = (( X ( t ) , . . . , X j ( t )) , ( Y ( t ) , . . . , Y j ( t )) , ( V ( t ) , . . . , V j ( t ))) , Z j + ( t ) = (( X j ( t ) , . . . , X n + ( t )) , ( Y j ( t ) , . . . , Y n + ( t )) , ( V j ( t ) , . . . , V n + ( t ))) , z j = (( x , . . . , x j ) , ( y , . . . , y j ) , ( v , . . . , v j )) , z j + = (( x j , . . . , x n + ) , ( y j , . . . , y n + ) , ( v j , . . . , v n + )) , where x k ∈ R d − , y k ∈ R and v k ∈ R d .For arbitrarily thin tube around Γ there is a strictly positive probability that thesystem with random reflections Z n + will stay inside the tube until time t . Consider atube so thin that point particles undergoing random reflections in the tube collide withthe ball in the same order as along the deterministic trajectory Γ. RCHIMEDES’ PRINCIPLE 31
We will assume without loss of generality that the point particles exchange energywith the ball along Γ in the order 1 , . . . , n .Let u be the last time the first particle hits a wall of the container before startingthe process of exchanging the energy with the ball. If the tube is very thin, the firstparticle will not collide with the ball after exchanging the energy with the ball. Let u ′ be the last time the first ball collides with the ball before time t . We claim that forsome small neighborhood U of Γ ( t ) , some c > z n + = ( z , z + ) ∈ U , P n ( Z ( t ) ∈ d z ∣ Z n + ( u ))/ d z ≥ c , (8.1) P n ( Z n + ( t ) ∈ d z n + ∣ Z n + ( u ′ ))/ d z n + ≥ c P n ( Z + ( t ) ∈ d z + ∣ Z n + ( u ′ ))/ d z + . (8.2)The claim (8.1) is true because the first particle acquires a random amount of energy,and its position and velocity direction are random due to Lambertian reflections follow-ing u . The claim (8.2) is a form of independence for conditional processes (conditionedto stay in separate tubes).We proceed by induction. Suppose 2 ≤ j ≤ n −
1. Let u j be the last time the j -th particle hit a wall of the container before starting the process of exchanging theenergy with the ball. Let u ′ j be the last time the j -th particle collided with the ballbefore time t . Then for some small neighborhood U j ⊂ U j − of Γ ( t ) , some c j > z n + = ( z j , z j + ) ∈ U j , P n ( Z j ( t ) ∈ d z j ∣ Z n + ( u j ))/ d z j ≥ c j , P n ( Z n + ( t ) ∈ d z n + ∣ Z n + ( u ′ j ))/ d z n + ≥ c j P n ( Z ( j + )+ ( t ) ∈ d z ( j + )+ ∣ Z n + ( u ′ j ))/ d z ( j + )+ . We apply this claim to j = n − P n ( Z n + ( t ) ∈ d z n + ∣ Z n + ( u ′ n − ))/ d z n + ≥ c P n ( Z n + ( t ) ∈ d z n + ∣ Z n + ( u ′ n ))/ d z n + . At this point it remains to analyze the interaction of the n -th particle and the ball.The position of the ball and its energy and velocity direction can be all made to havea joint density by collisions with the n -th point particle after time u n .Finally, the position and velocity direction of the n -th point particle have a con-ditional density given everything else, because of its Lambertian reflections from thebottom of the container. The energy of the n -th particle cannot be adjusted but thisis fine because the same energy conservation principle applies to systems starting fromother initial conditions. (cid:3) Remark 8.1. (i) There are many elementary examples of invariant measures for ourdynamical system if we assume specular reflections of point particles from the walls ofthe container. To construct one of them, place all point particles on disjoint verticallines which do not intersect the macroscopic ball at time 0. Make initial velocitiesvertical for the ball and all point particles. Each of these objects will stay on a verticalline forever. To construct an invariant measure, use the ergodic theorem.(ii) The following example of an invariant distribution illustrates family (1) in Theo-rem 2.1. Suppose that the macroscopic ball has a vertical initial velocity and all pointparticles are located on the vertical line passing through the ball’s center, they areplaced above the ball, and they all have vertical initial velocities. In this case the cen-ter of the ball and the point particles will remain on the same vertical line forever and their velocities will also remain vertical (although the positions and velocities will notremain constant). For this initial condition, point particles will not hit the walls of thecontainer so there will be no opportunity for the random reflections to cause mixingin the system. Just like in part (i) of the remark, one can use the ergodic theorem toconstruct an invariant measure.9.
Acknowledgments
We are grateful to Shuntao Chen, Persi Diaconis, Martin Hairer, Robert Ho(cid:32)lyst,Werner Krauth, Mathew Penrose and David Ruelle for the most useful advice.
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KB: Department of Mathematics, Box 354350, University of Washington, Seattle,WA 98195
Email address : [email protected] JM: Department of Mathematics, Wroc(cid:32)law University of Science and Technology,ul. Wybrze˙ze Wyspia´nskiego 27, 50-370 Wroc(cid:32)law, Poland
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