Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems Massimo Tessarotto Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12/1, 34127 Trieste, Italy and Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezruˇcovo n´am.13, CZ-74601 Opava, Czech Republic Claudio Asci Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12/1, 34127 Trieste, Italy 7 (Dated: January 10, 2017) 1 0 Inthispapertheproblemisposedofdeterminingthephysically-meaningfulasymptoticorderings 2 holding for the statistical description of a large N−body system of hard spheres, i.e., formed by N ≡ 1 ≫1 particles, which are allowed toundergoinstantaneousand purelyelastic unary,binary n ε or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et a J al., 2013-2016] and the related discovery of an exact kinetic equation realized by Master equation whichadvancesintimethe1−bodyprobabilitydensityfunction(PDF)forsuchasystem. Asshown 7 inthepaperthetaskinvolvesintroducingappropriateasymptoticorderingsintermsofεforallthe physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful ] h asymptotic approximations applicable for the Master kinetic equation, together with their possible p relationshipswiththeBoltzmannandEnskogkineticequations,andholdinginappropriateasymp- - toticregimes. Thesecorrespondeithertodiluteordensesystemsandareformedeitherbysmall–size s orfinite-sizeidenticalhardspheres,thedistinctionbetweenthevariouscasesdependingonsuitable s a asymptotic orderings in terms of ε. l c PACSnumbers: 05.20.-y,05.20.Dd,05.20.Jj,51.10.+y . s c i s 1 - INTRODUCTION y h p Inaseriesofpapers[1–7](see alsoRefs. [8,9])anewkinetic equationhasbeenestablishedforhardspheresystems [ subject to elastic instantaneous collisions, denoted as Master kinetic equation. Its basic features are that, unlike the Boltzmann and Enskog kinetic equations [11–13], the same equation and its corresponding Master collision operator 1 v are exact, i.e., they hold for an arbitraryN−body hard-sphere system SN which is isolated, namely for which the 4 numberofparticles(N)isconstant,whilefurthermorethatitssolutionsareentropy-preserving[6]andgloballydefined 3 [7]. Concerning, in particular, the first feature this means that in such a context S is allowed to have an arbitrary N 8 finite number N of finite-size and finite-mass hard spheres, namely each one characterized by finite diameter σ > 0 1 and mass m > 0. In addition, by assumption S is immersed in a bounded configuration domain Ω, subset of the 0 N 1. Euclidean space R3 which has a finite canonical measure L3o ≡ µ(Ω) > 0 (Lo denoting a finite configuration-domain characteristic scale length) and is endowed with a stationary and rigid boundary ∂Ω. However, the total volume 0 7 occupied by hard-spheres cannot exceed the configuration-space volume. Hence, the parameters N,Lo and σ must 1 necessarily satisfy the inequality : v Nµ(Φ) 4πNσ3 i ∆≡ ≡ ≤1, (1) X µ(Ω) 3L3 o r a with µ(Φ) ≡ 4πσ3 denoting the volume of a single hard sphere and ∆ the global diluteness parameter. For such an 3 equationtheparticlecorrelationsappearingthroughthe2−bodyprobabilitydensityfunction(PDF)areexactlytaken into account by means of suitably-prescribed 1− and 2−body occupation coefficients which are position-dependent only. These peculiar features follow uniquely as a consequence of the new approach to classical statistical mechanics developed in Refs. [1–3] and referred to as ”ab initio” axiomatic approach. As shown in the same references (for a review see also Ref. [9]), this is based on the adoption of appropriate extended functional setting and physics-based modifiedcollision boundaryconditions (MCBC)[1,3]whichareprescribedinordertoadvanceintimeacrossarbitrary (unary, binary or multiple) collision events the N−body probability density function (PDF). The physical origin of MCBC follows from the requirement that the deterministic N−body Dirac delta is included among the physically- admissible N−body PDF’s for S [2]. Its physical interpretation is intuitive [9] being viewed as the jump condition N for the N−body PDF along the phase-space Lagrangian trajectory {x(t)} for an ensemble of N tracer particles [10] undergoing an arbitrary collision event. 2 Basedonthe discoveryofthe Masterkinetic equation,ahostofnewdevelopmentshaveopenedup. Theseconcern, firstofall,theinvestigationofnovelconceptualaspectsofthesameequation(foranextendeddiscussionseeRefs. [4– 8]). However,itgoesalmostwithoutsayingthatpossible applicationsofthe new equationarepotentially ubiquitous. Many of these applications typically concern large systems, i.e., which are formed by a large number N ≡ 1 ≫1 of ε particles. The goal of this paper is that of identifying the relevant physically-meaningful asymptotic approximations for the same kinetic equation which may correspond to physically-relevant practical applications of the theory. As shown below,this taskinvolvesthe adoptionofappropriateasymptotic orderingsinterms ofε for allthe physically-relevant parameters. These include, besides the configuration-space scale length L and the hard-sphere diameter σ also the o characteristic scale length L which is associated with the spatial variations of the 1−body PDF which is prescribed ρ so that 1 ∂lnρ(N)(x ,t) =sup 1 1 , ∀(x ,t)∈Γ ×I . (2) Lρ ((cid:12) ∂r1 (cid:12) 1 1(1) ) (cid:12) (cid:12) (cid:12) (cid:12) Notice that hereon ∂lnρ(1N)(x1,t) is assumed(cid:12)(cid:12)to be bounded f(cid:12)(cid:12)or all (x ,t) spanning the extended 1−body phase space ∂r1 1 Γ ×I (see notations below). In the present paper the issues are investigated which concern the classification of 1(1) N−body systems which are characterized respectively by suitable asymptotic orderings. More precisely: • A) σ−ordering regime: in which the diameter σ of the hard spheres is either small-size or finite size, in the sense that either σ ∼O(εα) with suitable α>0 or σ ∼O(ε0); • B) ∆−ordering regime: in which the parameter ∆ is either ∆ ∼ O(εβ) with β > 0 or ∆ ∼ O(ε0), namely the hard-sphere system is dilute or dense; A type of alternate asymptotic ordering equivalent to B) is provided also by: • C) K −ordering regime: in which a suitably-defined Knudsen number K (see below) may be either of order n n O(εγ),beingγ asuitablepositive,eithervanishingornegativerealnumber. Accordinglythehard-spheresystems will be denoted as weakly-collisional, collisional or strongly collisional. The topics indicated above include in particular: ISSUE #1: the search of possible asymptotic (i.e., approximate) kinetic equations holding in cases A, B and C; ISSUE #2: the possible asymptotic evaluation of the 1− and 2−body occupation coefficients in the same cases. The goalis alsoto displaythe possible relationshipofthe Master kinetic equationwith well-knownkinetic equations, i.e.,theBoltzmannandEnskogkineticequations[11–13]. Inparticular,althoughinthecaseoffinite-sizehardspheres the strict validity of both the Boltzmann and Enskog equations ”per se” is ruled out [3], this concerns: ISSUE #3: the determination of the asymptotic modifications which enter the Boltzmann equation in the case the particle diameter σ is suitably small. ISSUE #4: the identification of the relevant asymptotic parameter sub-domains in which the Enskog equation still applies, albeit in a suitable approximate sense. 2 - DIMENSIONLESS REPRESENTATION OF THE MASTER KINETIC EQUATION The prerequisite for carrying out the tasks outlined above is setting the Master kinetic equation in dimensionless form. To this end let us first notice that by construction the 1−body PDF ρ(N)(x ,t) depends on the extended 1 1 Newtonian state (x ≡{r ,v } ,t). In particular this means that: 1 1 1 1 1)theNewtonianpositionvectorandvelocityvectorsr andv ,whichlabelthecenterofmasspositionandvelocity 1 1 of a representative particle, span the Euclidean configuration and velocity spaces Ω ⊂ R3 and U ≡ R3, while t 1(1) belongstotheGalileantimeaxisI ≡R.AsaconsequencetheGalileanstructureofΩ×I, i.e., theEuclideandistance in Ω and the time-interval in I, remains uniquely determined. 2) by construction ρ(N)(x ,t) is a scalar with respect to the group of Galilei transformation which preserves the 1 1 Galilean structure of the set Ω×I. Next, let us introduce the characteristic scale length L≡min{L ,L }, (3) o ρ 3 and a suitable constant characteristic time scale τ (whose definition remains in principle arbitrary). Then all the Newtonian variables (x ≡{r ,v },t) can be conveniently replaced with the corresponding dimensionless quan- 1 1 1 tities r = 1r ,v = τv and t = 1t. This implies that the phase-space volume element must transform as 1 L 1 1 L 1 τ dr dv = L6dr dv , while, in order to warrant the conservation of probability dr dv ρ(N)(r ,v ,t) = dr dv ρ(N), 1 1 τ3 1 1 1 1 1 1 1 1 1 1 the dimensionless form, of the PDF ρ(N) must be identified with ρ(N) = L6ρ(N)(r ,v ,t). Notice here, however, 1 1 τ3 1 1 1 that to preserve the scalar property of the transformed PDF ρ(N) with respect to the Galilei group the latter should 1 dependexplicitly ontheextendedNewtonianstate (r ,v ,t)ratherthenthe transformedstate(r ,v ,t). Infactdue 1 1 1 1 to their arbitrariness,the parameters L and τ change the Galilei structure of space-time, i.e., the Euclideandistance and the time interval. Hence ρ(N) still depends, as ρ(N)(r ,v ,t), on the same extended state (x ≡{r ,v } ,t), 1 1 1 1 1 1 1 1 and therefore is of the form ρ(N) ≡ ρ(N)(r ,v ,t). In terms of such a prescription the Master kinetic equation (see 1 1 1 1 Ref.[3]) can therefore be formally represented in the dimensionless form (N) (N) (N) L ρ =C (ρ ρ ), (4) 1 1 1 1 1 (cid:12) with L and C (ρ(N) ρ(N)) denoting respectively the free-stream(cid:12)(cid:12)ingand Master collisionoperators. Both are castin 1 1 1 1 the dimensionless rep(cid:12)resentation, i.e., so that L ≡ ∂ +v · ∂ and (cid:12) 1 ∂t 1 ∂r1 (cid:12) (−) C (ρ(N) ρ(N))=K dv dΣ . 1 1 1 n 2 21 (cid:12) UZ Z (cid:12) 1(2) ρ(N)(x(2)((cid:12)+),t)−ρ(N)(x(2),t) |v ·n |Θ∗. (5) 2 2 21 21 h i In addition, in the Eq. (4) K identifies the Knudsen number n (N −1)σ2 K ≡ , (6) n L2 while ρ(N)(x(2),t) is the dimensionless 2− body PDF given by 2 ρ(N)(r ,v ,r ,v ,t)≡f(r ,r ,t)× 2 1 1 2 2 1 2 ρ(N)(r ,v ,t)ρ(N)(r ,v ,t), (7) 1 1 1 1 2 2 with f(r ,r ,t) denoting the position-dependent dimensionless weight-factor 1 2 k(N)(r ,r ,t) f(r ,r ,t)≡ 2 1 2 . (8) 1 2 k(N)(r ,t)k(N)(r ,t) 1 1 1 2 The remaining notations are standard [3]. Thus U ≡ R3 is the 1−body velocity space for the k−th particle, 1(k) (−) the symbol dΣ denotes integration on the subset of the solid angle of incoming particles namely for which 21 ∗ ∗ v ·n <0, Θ denotes Θ ≡Θ r − σn − σ , with Θ(x) being the strong theta function, while everywhere in 12 12 R 2 2 2 2 the operator C1(ρ(1N) ρ(1N)), r2 is(cid:0)id(cid:12)entified b(cid:12)y con(cid:1)struction with r2 =r1+σn21. Furthermore k1(N)(r1,t),k1(N)(r2,t) (cid:12) (cid:12) and k(N)(r ,r ,t) ide(cid:12)ntify the dimensionless 1− and 2−body occupation coefficients, whose definitions in terms of 2 1 2 (cid:12) the dimensionless 1−b(cid:12)ody PDF are respectively:  k1(N)(r1,t)=Γ1R(2) dx2ρk(11(NN))((xr22,,tt))k2(N)(r1,r2,t), (9)  k2(N)(r1,r2,t)=Γ1R(3) dx3ρk(11(NN))((xr33,,tt))...Γ1R(N) dxNρk(11(NN))((xrNN,,tt)), where once the position of particle 1 is assumed prescribed, Γ ,Γ ,...Γ are the admissible subsets of the 1(2) 1(3) 1(N) 1−body phase spaces of particles 2,3,..,N, Γ ,Γ ,...Γ obtained by subtracting respectively the forbidden 1(2) 1(3) 1(N) subsets Φ (from Γ ), Φ ∪Φ (from Γ ), Φ (from Γ ). 12 1(2) 13 23 1(3) iN 1(N) i=2,N−1 [ 4 A remark is in order concerning the comparison with the analogous dimensionless representation introduced orig- inally by Grad [14] for the BBGKY hierarchy and the Boltzmann equation in particular (see also Ref.[15]). The basic departure of Eq. (4) with respect to the latter equation lies of course in the different realization of the collision operator. However, another major difference arises because of the explicit introduction of the characteristic scale length L in the definition of the Knudsen number given in Eq. (6). Such a choice is actually required in order to permit the distinction between different asymptotic ordering regimes (see next Section), while, in contrast, Grad’s approach dealt only with the so-called Boltzmann-Grad limit. Indeed, as shown below, it was actually appropriate for the treatment of the so-called dilute-gas asymptotic ordering only, namely for the case in which both the scale length L and L are considered of order O(ε0). o 3 - CLASSIFICATION OF THE ASYMPTOTIC ORDERING REGIMES Inthissectiontherelevantasymptoticorderingsaredeterminedwhichareapplicableinthecaseoflargehard-sphere systems, i.e., for which N ≡ 1 ≫ 1 and subject to the validity of the volume constraint inequality (1). Based on ε the prescription of the Knudsen number (6) we are now able to show that the classification in terms of K pointed n out above corresponds to suitably prescribe the magnitude of the ratio dimensionless σ/L. More precisely weakly- collisional, collisional or strongly collisional asymptotic regimes are obtained requiring that σ/L be of order O(εγ−21) withγ beingrespectivelyγ <0,γ =0andγ >0.Fordefiniteness letusinitially focusonthe collisionalK −ordering n regime. In this case the following two possible dilute-gas ordering regimes (3A and 3B) can be distinguished. 3A - Dilute-gas small-size σ−ordering regime Let us consider a first possible realization of the ”small-size” σ−ordering regime. This is obtained assuming that the diameterofthe hard-spheresσ is considered≪1inasuitablesense,whilethe scale-lengthLisorderedaccording to the prescriptions that σ/L∼O(ε12) and letting σ ∼O(εα) L∼O(εα−12)  (10) L∼L ∼L  α∈ o0,1 .ρ 3 Introducing the dimensionless parameter  (cid:3) (cid:3) 4πNσ3 ∆ ≡ (11) L 3L3 this implies necessarily that ∆L ∼ O(ε21) and hence, due to the inequalities L ≤ Lo, and ∆ ≤ ∆L also ∆ . O(ε12). Therefore the orderings (10) necessarily equivalently identify a dilute-gas ordering, which can therefore be characterized as a collisional, dilute-gas and small-size σ−ordering regime. In particular, when α= 1 and L∼L it 2 o follows that ∆L ∼ ∆ ∼ O(ε21) and Kn ∼ O(ε0), so that the customary dilute-gas ordering considered originally by Grad [14] (see also Refs. [3, 7]) is recovered. This is obtained requiring Nσ2 ∼O(ε0) (12) L∼L ∼O(ε0). o (cid:26) 3B - Dilute-gas finite-size σ−ordering regime Another type of ordering regime is obtained requiring σ to be finite while prescribing again the scale-length L in such a way to satisfy the requirement K ∼O(ε0). Let us require n σ ∼O(ε0) L∼O(ε−21) (13)  L∼L .L .  ρ o  5 Noticethatagain∆L ∼O(ε12)andhence∆.O(ε12).Thereforetheordering(13)correspondstoadilute-gasordering which will be referred to as collisional, dilute-gas and finite-size σ−ordering regime. Finally, for completeness we point out possible realizations of dense-gas ordering regimes. 3C - Dense-gas ordering regimes Let us require for this purpose that the parameter ∆ is of order O(ε0), i.e., that the hard-sphere system is dense. This happens in the case in which σ/Lo ∼ O(ε31). Since by construction due to the inequality L ≤ Lo also ∆ ≤ ∆L it follows necessarily that it must be ∆∼∆L and hence L∼Lo too. This means that Kn ∼O(ε−13) which therefore corresponds necessarily to a strongly-collisional regime. Let us assume for this purpose that the following orderings apply: σ ∼O(εα)  L∼O(εα−31) (14) L∼L ∼L .  α∈[0ρ,∞].o Therefore this implies that necessarily ∆ ∼ ∆L ∼O(ε0). In particular, if α > 0 this corresponds to a small-size σ−ordering regime, to be referred to as strongly-collisional, dense-gas and small-size σ−ordering regime. A special case is provided by the choice α=0, which corresponds instead to a finite-size σ−ordering regime for which σ ∼O(ε0) L∼O(ε−13), (15)  L∼L ∼L .  ρ o This can therefore be characterized as strongly-collisional, dense-gas and finite-size σ−ordering.  4 - ASYMPTOTIC APPROXIMATIONS OF THE MASTER KINETIC EQUATION In this section we intend to pose the problem of the construction of the asymptotic approximations of the Master kinetic equation which are appropriate for the treatment of most of the asymptotic regimes discussed in subsections 3A, 3B and 3C. In detail we intend to show that: • Firstasymptotic approximation: intheorderingregime(10)tolowestorderinO(ε)theMasterequationreduces to the Boltzmann kinetic equation, with the Master collision operator being approximated in this case by the collision operator Nσ2 (+) C (ρ(N) ρ(N))= dv dΣ f(r ,r =r ,t). 1MB 1 1 L2 2 21 1 2 1 (cid:12) UZ Z (cid:12) 1(2) ρ(N)(r ,v(.−),t)ρ(cid:12)(N)(r ,v(−),t)−ρ(N)(r ,v ,t)ρ(N)(r ,v ,t) |v ·n |, (16) 1 1 1 1 1 2 1 1 1 1 1 2 21 21 wheref(r ,r ,t) ihsthe strictly-positiveweight-factorprescribedbyEq. (8). Inthie expressionofsame equation 1 2 the occupation coefficients are here approximated as follows: k(N)(r ,t)∼=1− N dr n(N)(r ,t), 1 1 2 2 1 2 Φ12 (17) k(N)(r ,r ,t)∼=1− 3N dr n(NR)(r ,t)− 3N dr n(N)(r ,t),  2 1 2 4 2 1 2 4 3 1 3 Φ12 Φ23 R R with Φij denoting thehard-sphere interior domain Φij ≡ {rj :|rj −ri|<σ}. This yields therefore the asymp- totic approximation 1− 3N dr n(N)(r ,t)− 3N dr n(N)(r ,t) 4 2 1 2 4 3 1 3 f(r ,r ,t)∼= Φ12 Φ23 . (18) 1 2 1− N R dr n(N)(r ,t)− N Rdr n(N)(r ,t) 2 2 1 2 2 3 1 3 Φ12 Φ23 R R 6 The following remarks are in order regarding the collision operator C (ρ(N) ρ(N)). First one notices that 1MB 1 1 it provides a generalization of the Boltzmann collision operator (issue #2). In p(cid:12)articular, one can readily show (cid:12) (see the proof reported below) that to order O(ε0) it coincides by construction w(cid:12)ith the customary Boltzmann collision operator, since then the weight-factor f(r ,r ,t) can be approximated with unity. The asymptotic 1 2 approximate formula (18) for the weight-factor f(r ,r ,t) given by Eq. (8) retains, instead, also leading- 1 2 order corrections which are produced by the 1− and 2−body occupation coefficients (issue #3). Second, the structure of the collision operator (16) has also formal analogies with the one introduced by Enskog in his namesake equation. The key feature in this case lies in the prescription of the weight-factor f(r ,r ,t) which 1 2 is here provided by Eq. (18) while remaining in principle undetermined in the context of the Enskog kinetic equation. Thus, provided, the same prescription indicated above is made for f(r ,r ,t), the collision operator 1 2 (16) can be viewed as realizing also an approximate representation of the Enskog collision operator (issue #4). • Second asymptotic approximation: invalidityoftheorderingregimes(13)theMasterequationreduces,instead, to an asymptotic Master kinetic equation determined by the collision operator Nσ2 (−) C (ρ(N) ρ(N))= dv dΣ f(r ,r ,t). 1 1 1 L2 2 21 1 2 (cid:12) UZ Z (cid:12) 1(2) ρ(N)(r ,v(.−),t)ρ(N(cid:12))(r ,v(−),t)−ρ(N)(r ,v ,t)ρ(N)(r ,v ,t) |v ·n |Θ∗ (19) 1 1 1 1 2 2 1 1 1 1 2 2 21 21 h i in which the weight-factor f(r ,r ,t) is expressed in terms of the asymptotic estimate (18) and due to the 1 2 requirement that σ remains finite, so that necessarily r =r +σn . This implies that although Eq. (19) has 2 1 21 formalanalogieswiththecustomaryformoftheEnskogcollisionoperator,twomajordifferencesarise. Thefirst one lies in the prescriptionof the weight-factoritself, which in the presentcase is determined by Eq. (18) while it choice remains unspecified in the context of the Enskog statistical approach. The second follows because of the adoptionofMCBC requiringthatinEqs. (19)the solid-angleintegrationmustbe carriedoutonthe subset (−)dΣ of incoming particles for which v ·n <0 instead on the complementary set (−)dΣ as done in 21 12 12 21 the Enskog collision operator (issue #4). R R • Third asymptotic approximation: in the ordering regime (14) subject to the requirement α > 0 to leading orderinO(ε) the Masterequationreduces to the asymptotic Master kinetic equation,expressedin terms of the collision operator which to leading order in O(ε) reads Nσ2 (+) C (ρ(N) ρ(N))= dv dΣ f(r ,r =r ,t)× 1 1 1 L2 2 21 1 2 1 (cid:12) UZ Z (cid:12) 1(2) ρ(N)(r ,v(.−),t(cid:12))ρ(N)(r ,v(−),t)−ρ(N)(r ,v ,t)ρ(N)(r ,v ,t) |v ·n |, (20) 1 1 1 1 1 2 1 1 1 1 1 2 21 21 h i where f(r ,r ,t) is prescribedagainby Eq. (8). However,now in difference with the two cases indicated above 1 2 the asymptotic approximations (17) and (18) do not hold, so that the occupation coefficients k(N)(r ,t) and 1 1 k(N)(r ,r ,t) need to be determined iteratively in terms of Eqs. (9). 2 1 2 Finally, we mention that the case representedby the ordering (15) must be treated separately,in the sense that no approximation is actually possible on the functional form of the Master collision operator (5). 4A - Proof of the first asymptotic approximation Let us first prove the validity of Eqs. (16) and (18). For this purpose one first notices that thanks to the ordering regime (10) the 1−body PDFs ρ(N)(r ,v(+),t) and ρ(N)(r ,v ,t) can be approximated in terms of ρ(N)(r ,v(+),t) 1 2 2 1 2 2 1 1 2 and ρ(N)(r ,v ,t) respectively. For the same reason the occupation coefficients k(N)(r ,t) and k(N)(r ,r ,t) can 1 1 2 1 2 2 1 2 be approximated in terms of k(N)(r ,t) and k(N)(r ,r ≡ r ,t). Second, again thanks to Eqs.(10), in the Master 1 1 2 1 2 1 collision operator the solid-angle integrationon the sub-domain v ·n <0 (namely (−)dΣ ) can be equivalently 12 12 21 exchanged with the corresponding complementary subset v · n ≥ 0, i.e., (+)dΣ , while the domain theta 12 12 R 21 R 7 ∗ ∗ function Θ (x ) becomes Θ (x ) ≡ Θ(|r |) so that its contribution to the collision integral is ignorable. Third, to 2 2 1 provetheasymptoticestimate(18),letusnoticethatinvalidityoftheordering(10)itfollowsthat dr n(N)(r ,t)∼ 2 1 2 Φ12 n(N)(r∗,t)4πσ3 ≡ σ3x, with x a suitable mean value such that x ∼ O(ε0). As a consequence fromREqs.(17) in order 1 2 3L3 L3 of magnitude it follows that b b b k(N)(r ,t)∼1− Nσ3xb 1 1 2L3 (21) (k2(N)(r1,r2,t)∼1− 3N2Lσ33xb which implies Nσ3x f(r ,r ,t)∼1− . (22) 1 2 2L3 b The proofof the asymptotic estimates (21) is straightforward. In fact, Eq.(9) then requires,basedon the mean-value theorem, implies that N k(N)(r∗,t) ∼ dx ρ(N)(x ,t) dx ρ(N)(x ,t) dx ρ(N)(x ,t)≡ 1 1 2 1 2 3 1 3 N 1 N (cid:16) (cid:17) Γ1Z(2) Γ(Z1) Γ(Z1c) 1(3) 1(N) ≡ dr n(N)(r ,t) dr n(N)(r ,t) dr n(N)(r ,t), (23) 2 1 2 3 1 3 N 1 N Z Z Z Ω1(2) Ω(11()3) Ω(11(cN)) with k(N)(r∗,t) a suitable mean-value. Therefore the same equation yields the asymptotic estimate 1 1 N σ3x σ3x σ3x k(N)(r∗,t) ∼ 1− 1−2 ... 1−(N −1) (24) 1 1 (cid:16) (cid:17) (cid:18) L3 (cid:19)(cid:18) L3 (cid:19) (cid:18) L3 (cid:19) b b b which is manifestly consistent with Eq.(21). The proof of the asymptotic estimates for k(N)(r ,r ,t) and (22) is 2 1 2 analogous,thus yielding the consistency of the asymptotic approximations (17) and (18). 4B - Proof of the second asymptotic approximation The proof of Eq.(19) is similar as far as the asymptotic estimate (22) is concerned. Now, however, due to the finite size of the hard spheres (see Eqs. (13)) the correctspatial dependences must be retained in the 1−body PDF’s ρ(N)(r ,v(+),t) and ρ(N)(r ,v ,t) which must both be evaluated at the position r = r +σn . As a consequence 1 2 2 1 2 2 2 1 21 the corresponding asymptotic approximation(19) manifestly holds for the Master collision operator. 4C - Proof of the third asymptotic approximation The proof of Eq. (20) is similarly straightforward. In fact, first one notices that in close analogy with case 4A, thanks to the small-size assumption introduced for σ , the 1−body PDFs ρ(N)(r ,v(+),t), ρ(N)(r ,v ,t) as well as 1 2 2 1 2 2 the occupationcoefficientsk(N)(r ,t)andk(N)(r ,r ,t)canallbeapproximatedreplacingr →r .Asaconsequence 1 2 2 1 2 2 1 (−) again the solid-angle integration dΣ can be equivalently be evaluated in terms of the outgoing-particle subset 21 (+) ∗ dΣ while the contribution of the theta function Θ is ignorable. Finally, due to the dense-gas asymptotic 21 R ordering included in (14) no obvious asymptotic approximationis available for the occupation coefficients k(N)(r ,t) R 1 1 and k(N)(r ,r =r ,t). Therefore their exact expression following from Eqs. (9) must be retained in Eq. (20). 2 1 2 1 5 - CONCLUSIONS In this paper the problem has been addressed of identifying possible physically-meaningful asymptotic approxi- mations of the Master kinetic equation which apply to , i.e., formed by N ≡ 1 ≫ 1 hard-spheres. The statistical ε 8 approachhasbeenbasedonthe”ab initio”axiomaticstatisticaltheoryrecentlydeveloped[1–9]. Asaresult,oncethe Master kinetic equation is cast in dimensionless form, the existence of multiple asymptotic ordering regimes for the same equationhas been pointedout whichhold forlargeN−body systems. These regimes correspondto appropriate prescriptionsoftherelevantphysicalparametersofthesameequationandinclude,asaparticularpossiblerealization, the customary dilute-gas ordering originally introduced by Grad [14] for his construction of the Boltzmann kinetic equation. The new ordering regimes encompass either small or finite-size hard-spheres as well as dilute or dense, collisional or strongly-collisionalparticle systems. In particular possible realizations include: • the dilute-gas small-size σ−ordering regime (prescribed by the ordering Eqs. (10)); • the dilute-gas finite-size σ−ordering regime (in the sense of Eqs. (13)); • the dense-gas ordering regime (see Eqs. (14) in the case in which α>0). Corresponding asymptotic approximations have been determined for the Master collision operator, displaying also their relationship/difference with respect to the Boltzmann and Enskog collision operators. The present results are believed to be crucialboth in kinetic theory and fluid dynamics. Indeed, regardingpossible challenging future developments one should particularly mention possible applications both of the Master kinetic equation itself as well as of the asymptotic approximations here pointed out for the first time. The hard-sphere kinetic statistical treatment based on these equations is expected to successfully apply to a variety of complex fluid- dynamics systems as well as to neutral and/or ionized gases of interest for laboratory researchand astrophysics. ACKNOWLEDGMENTS Workdevelopedwithintheresearchprojects: A)theAlbertEinsteinCenterforGravitationandAstrophysics,Czech ScienceFoundationNo. 14-37086G;B)thegrantNo. 02494/2013/RRC“kinetick´ypˇr´ıstupkproud˘en´ıtekutin”(kinetic approach to fluid flow) in the framework of the “Research and Development Support in Moravian-Silesian Region”, Czech Republic: C) the research projects of the Czech Science Foundation GACˇR grant No. 14-07753P. Initial frameworkand motivations of the investigationwere based on the researchprojects developedby the Consortiumfor MagnetofluidDynamics(UniversityofTrieste,Italy)and,inreferencetoissues#2and#3,theMIUR(ItalianMinistry forUniversitiesandResearch)PRINResearchProgram“Problemi Matematici delle Teorie Cinetiche eApplicazioni”, University of Trieste, Italy. The authors are grateful to the International Center for Theoretical Physics (Miramare, Trieste, Italy) for the hospitality during the preparation of the manuscript. 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