Macroscopic irreversibility and decay to kinetic equilibrium of the 1-body PDF for finite hard-sphere systems
aa r X i v : . [ m a t h - ph ] D ec Macroscopic irreversibility and decay to kinetic equilibriumof the 1-body PDF for finite hard-sphere systems
Massimo Tessarotto
Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy andInstitute of Physics, Faculty of Philosophy and Science,Silesian University in Opava, Bezruˇcovo n´am.13, CZ-74601 Opava, Czech Republic
Claudio Cremaschini
Institute of Physics and Research Center for Theoretical Physics and Astrophysics,Faculty of Philosophy and Science, Silesian University in Opava,Bezruˇcovo n´am.13, CZ-74601 Opava, Czech Republic (Dated: December 5, 2018)The conditions for the occurrence of the so-called macroscopic irreversibility property and therelated phenomenon of decay to kinetic equilibrium which may characterize the 1-body probabilitydensity function (PDF) associated with hard-sphere systems are investigated. The problem is setin the framework of the axiomatic ”ab initio” theory of classical statistical mechanics developedrecently and the related establishment of an exact kinetic equation realized by the Master equationfor the same kinetic PDF. As shown in the paper the task involves the introduction of a suitablefunctional of the 1-body PDF, identified here with the
Master kinetic information . It is then provedthat, provided the same PDF is prescribed in terms of suitably-smooth, i.e., stochastic, solution ofthe Master kinetic equation, the two properties indicated above are indeed realized.
PACS numbers: 05.20.-y, 05.20.Dd, 05.20.Jj, 51.10.+y
1. INTRODUCTION
The axiomatic theory of Classical Statistical Mechanics (CSM) recently proposed in a series of papers (see Refs.[1–3, 6]), and referred to as ab initio theory of CSM , provides a self-consistent pathway to the kinetic theory of hard-spheresystems, as well as in principle also point particles subject to finite-range interactions [7]. Its theoretical basis andconditions of validity are indeed founded on a unique physical realization of the axioms which are set at the foundationsof CSM [1–3], a fact which permits the treatment of phase-space and kinetic probability density functions (PDF) whichare either realized by stochastic (i.e., ordinary) functions or distributions such as the N − body Dirac Delta (or certaintyfunction [8]). This feature is physically-based being due to the prescription of the collision boundary conditions (CBC,[2]), i.e., the relationship occurring at collision events between incoming and outgoing multi-body probability densityfunctions PDF. The choice of the appropriate CBC indicated in Ref. [2], denoted as modified collision boundarycondition (MCBC), is actually of crucial importance and departs from the customary realization/interpretation (ofthe same axioms) originally adopted in Boltzmann [9], Enskog [10, 11] and Grad [12] kinetic approaches (for a reviewof Grad’s kinetic theory based on CSM see also Cercignani [13, 14]). The same choice implies, in fact, a number oftheoretical and physically-relevant consequences. In particular, it follows that the new theory: • Unlike Enskog theory [3] applies also to finite N − body hard-sphere systems S N , namely systems formed by N like smooth hard-spheres of diameter σ and mass m , in which the parameters ( N, σ, m ) remain all constantand finite [3]. On the other hand, the same particles are assumed as usual: A) subject to instantaneous (unary,binary and multiple) elastic collisions which leave unchanged the particles angular momenta and B) immersedin a stationary bounded domain Ω of the Euclidean space R with finite canonical measure. • Has lead to the discovery [3] of an exact kinetic equation holding globally in time [4] (i.e., for all t ∈ I ≡ R )for these systems and denoted as Master kinetic equation (recalled in Appendix A). In other words the Masterequation is non-asymptotic in character with respect to the (finite) parameters ( N, σ, m ). In addition thesame equation holds under suitable maximal entropy conditions for the statistical treatment of the so-calledBoltzmann-Sinai classical dynamical system (CDS), which implies that initial (binary or multi-body) phase-space statistical correlations are assumed identically vanishing, while at the same time only suitable uniquely-prescribed configuration-space correlations can arise. As such the equation generalizes and extends the validityof the Boltzmann and Enskog kinetic equations and notably applies to arbitrary 1 − body PDFs which can berealized either in terms of stochastic functions or distributions. • Is time-reversal invariant [6], namely the Master kinetic equation is time-reversal (
T R− ) symmetric. In otherwords, the same equation is invariant with respect to the
T R− transformation τ ≡ t − t o → τ ′ ≡ t ′ − t o = − t + t o ≡ − τ, r → r ′ = r , v → v ′ = − v . (1)Thus, representing the absolute time as t = τ + t o , with t o being a prescribed (arbitrary) initial time, it followsthat the T R− transformation leaves invariant the initial time t o and the instantaneous position r = r ( t ≡ t o + τ )of an arbitrary particle, while reversing the signature (i.e., versus) of its velocity v = v ( t ≡ t o + τ ) . Accordingly,thanks to
T R− symmetry the two initial-value problems associated with the Master kinetic equation in thetwo cases are related in such a way that, respectively, the initial 1 − body PDF at time t o , ρ ( N )1 ( x , t o ) ≡ ρ ( N )1( o ) ( r , v ) , and the corresponding time-evolved PDF ρ ( N )1 ( x , t ) are carried into the T R− transformed 1 − bodyPDFs ρ ( N )1( o ) T R ( r ′ , v ′ ) and ρ ( N )1 T R ( r ′ , v ′ , t ′ ) respectively prescribed according to the law ( ρ ( N )1( o ) ( r , v ) ρ ( N )1 ( x , t ) ≡ ρ ( N )1 ( r , v , t o + τ ) → ( ρ ( N )1( o ) T R ( r ′ , v ′ ) ≡ ρ ( N )1( o ) ( r , − v ) ρ ( N )1 T R ( r ′ , v ′ , t ′ ) ≡ ρ ( N )1 ( r , − v , t o − τ ) . (2) • Conserves the corresponding Boltzmann-Shannon (BS) statistical entropy [6]. This is identified with the phase-space moment S ( ρ ( N )1 ( t )) ≡ − Z Γ d x ρ ( N )1 ( x , t ) ln ρ ( N )1 ( x , t ) A , (3)with ρ ( N )1 ( t ) ≡ ρ ( N )1 ( x , t ) being an arbitrary stochastic PDF solution of the Master kinetic equation and A anarbitrary positive constant such that the initial PDF ρ ( N )1 ( t o ) ≡ ρ ( N )1 ( x , t o ) = ρ ( N )10 ( x )is such that the corresponding BS functional S ( ρ ( N )1 ( t o )) is defined. As a consequence it follows that an arbitrarysmooth solution of the Master kinetic equation satisfies the constant H-theorem ∂∂t S ( ρ ( N )1 ( t )) = 0 (4)for all t ∈ I ≡ R (see again related discussion in Ref.[6]).Based on the ab initio theory of CSM, in this paper the problem is posed of the existence of two phenomena whichare expected to characterize the statistical description of finite N − body hard-sphere systems and therefore shouldlay at the very foundation of CSM and kinetic theory. These are related to the physical conditions for the possibleoccurrence of the so-called property of macroscopic irreversibility (PMI) and the consequent one represented by the decay to kinetic equilibrium (DKE) which characterize the 1 − body (kinetic) PDF in these N − body systems, i.e., when1 − body-factorized initial conditions are considered for the N − body Liouville equation [3]. The conjecture is that -in some sense in analogy with the ubiquitous character of the ergodicity property which characterizes hard-spheresystems and hence the S N − CDS [15, 16] - the occurrence of such phenomena should be independent of the number N of constituent particles of the system and therefore apply to actual physical systems for which the parameters( N , σ, m ) are obviously all finite.
1A - Motivations and background
Both properties indicated above concern the statistical behavior of an ensemble S N of like particles which areadvanced in time by a suitable N − body classical dynamical system, identified here with the S N − CDS. Specificallythey arise in the context of the kinetic description of the same CDS, i.e., in terms of the corresponding 1 − body(kinetic) probability density function (PDF) ρ ( N )1 ( t ) ≡ ρ ( N )1 ( x , t ) . The latter is required to belong to the functionalclass n ρ ( N )1 ( x , t ) o of suitably smooth and strictly positive ordinary functions which are particular solutions of therelevant kinetic equation.In fact, PMI should be realized by means of a suitable, but still possibly non-unique, functional which should beglobally defined in the future (i.e., for all times t ≥ t o , being t o the initial time) bounded and non-negative, andtherefore to be identified with the notion of information measure. Most importantly, however, the same functional,to be referred to here as Master kinetic information (MKI), should also exhibit a continuously-differentiable andmonotonic, i.e., in particular decreasing, time-dependence.Regarding, instead, the second property of DKE this concerns the asymptotic behavior of the 1 − body PDF ρ ( N )1 ( t ) ≡ ρ ( N )1 ( x , t ) which, accordingly, should be globally defined and decay for t → + ∞ to a stationary and spatially-uniformMaxwellian PDF ρ ( N )1 M ( v ) = n o π / (2 T o /m ) / exp ( − m ( v − V o ) T o ) , (5)where { n o > , T o > , V o } are constant fluid fields.Both PMI and DKE correspond to physical phenomena which are actually expected to arise in disparate classical N − body systems. The clue for their realization is represented by the ubiquitous occurrence of kinetic equilibriaand consequently, in principle, also of the corresponding possible manifestation of macroscopic irreversibility anddecay processes. Examples of the former ones are in principle easy to be found, ranging from neutral fluids [31] tocollisional/collisionless and non-relativistic/relativistic gases and plasmas [32–34].However, the most notable example is perhaps provided by dilute hard-sphere systems (”gases”) characterized by alarge number of particles ( N ≡ ε ≫
1) and a small (i.e., infinitesimal) diameter σ ∼ O ( ε / ) of the same hard-spheres,for which the Boltzmann equation applies. Indeed the Boltzmann equation is actually specialized to the treatmentof dilute hard-sphere systems in the Boltzmann-Grad limit discussed in the Lanford theorem [17–19] (for a detaileddiscussion of the topics in the context of the ab initio-theory see also Ref.[6]). In such a case the 1 − body PDF canbe formally obtained by introducing the Boltzmann-Grad limit operator [6] L BG ≡ lim N ≡ ε →∞ σ ∼ O ( ε / ) , (6)whereby the limit function ρ ( x , t ) is denoted ρ ( x , t ) = L BG ρ ( N )1 ( x , t ) , (7)and ρ ( x , t ) identifies a particular solution of the Boltzmann kinetic equation.Historically, the property of irreversibility indicated above is known to be related to the Carnot’s second Law ofClassical Thermodynamics. More precisely, it is related to the first-principle-proof originally attempted by LudwigBoltzmann in 1872 [9]. Actually it is generally agreed that both phenomena lie at the very heart of Boltzmannand Grad kinetic theories [9, 12] and the related original construction of the Boltzmann kinetic equation (1872). Inparticular, the goal set by Boltzmann himself in his 1872 paper was the proof of Carnot’s Law providing at the sametime also a possible identification of thermodynamic entropy. This was achieved in terms of what is nowadays knownas Boltzmann-Shannon (BS) statistical entropy, which is identified with the phase-space moment M X E ( ρ ( t )) ≡ Z Γ d x ρ ( x , t ) X E ( x , t ) = − Z Γ d x ρ ( x , t ) ln ρ ( x , t ) A ≡ S ( ρ ( t )) . (8)Here X E ( x , t ) ≡ − ln ρ ( x ,t ) A , ρ ( x , t ) and A denote respectively the BS entropy density, an arbitrary particularsolution of the Boltzmann equation for which the same phase-space integral exists and an arbitrary positive constant.In fact, according to the Boltzmann H-theorem [9] the same functional should satisfy the entropic inequality ∂∂t S ( ρ ( t )) ≥ , (9)while, furthermore, the entropic equality condition ∂∂t S ( ρ ( t )) = 0 ⇔ ρ ( x , t ) = ρ ( N )1 M ( v ) (10)should hold. The latter equation implies therefore that, provided ρ ( t ) and S ( ρ ( t )) exist globally [28], then necessarily lim t → + ∞ ρ ( x , t ) = ρ M ( v ) , with ρ M ( v ) denoting the stationary and spatially-uniform Maxwellian PDF (5).In this reference, however, the question arises of the precise characterization of the concept of irreversibility, i.e.,whether it should be regarded as a purely macroscopic phenomenon (”macroscopic irreversibility”), i.e., affectingonly the BS entropy S ( ρ ( t )) through the Boltzmann H-theorem indicated above, or microscopic in the sense that thesame Boltzmann equation should be considered as irreversible (”microscopic irreversibility”). Thus, in principle, in thesecond case the further issue emerges of the possible physical origin of microscopic irreversibility in special referenceto the Lanford’s derivation of the Boltzmann equation and subsequent related comments discussed respectively byUffink and Valente and Ardourel in Refs. [20] and [21] (see also Drory [22]).However, as shown in Ref.[6], the Boltzmann equation is actually T R− symmetric. Such a conclusion is of basicimportance since it overcomes the so-called Loschmidt paradox, i.e., the objection raised by Loschmidt in 1876[23] regarding the original Boltzmann formulation of his namesake kinetic equation and H-theorem [9]. In fact,Loschmidt claimed that the Boltzmann H-theorem inequality should change sign under time reversal and thus violatethe microscopic time-reversibility of the underlying hard-sphere classical dynamical system. In his long-pondered replygiven in 1896 [24] Boltzmann himself introduced what was later referred to as the modified form of the BoltzmannH-theorem [25].The key implication is therefore that, in contrast to Boltzmann’s own statement and the traditional subsequentmainstream literature interpretation (see for example by Cercignani, Lebowitz in Refs. [26, 27] and more recently thereview given by Gallavotti [29]), the Boltzmann H-theorem indicated - together with the modified form indicated above- cannot be interpreted as an intrinsic irreversibility property occurring at the microscopic level, namely holding forthe Boltzmann equation itself. On the contrary, consistent with the physical interpretation of the Loschmidt paradoxprovided in Ref.[6], this must be regarded only as property of macroscopic irreversibility (or PMI) of the 1 − bodyPDF solution of the Boltzmann equation. In other words, the Boltzmann inequality (9) necessarily holds independentof the orientation of the time axis (arrow of time) and therefore cannot represent a true (i.e., microscopic) propertywhich as such should uniquely determine the arrow of time.Nevertheless, the possible realization of either PMI or DKE is more subtle. In fact they actually depend in a criticalway on the prescription of the functional class (cid:8) ρ ( N ) ( x , t ) (cid:9) , so that their occurrence is actually non-mandatory.Indeed, both cannot occur - in principle also for Boltzmann and Grad kinetic theories - if the N − body probabilitydensity function ρ ( N ) ( x , t ) is identified with the deterministic N − body PDF [1], namely the N − body phase-spaceDirac delta. This is defined as δ ( x − x ( t )) ≡ Q ,N δ ( x i − x i ( t )) , with x ≡ { x , ..., x N } denoting the state of the N − body system and x ( t ) ≡ { x ( t ) , ..., x N ( t ) } is the image of an arbitrary initial state x ( t o ) ≡ x o generated by thesame N − body CDS. That such a PDF necessarily must realize an admissible particular solution of the N − bodyLiouville equation follows, in fact, as a straightforward consequence of the axioms of classical statistical mechanics[1].Despite these premises, however, the case of a finite Boltzmann-Sinai CDS, which is characterized by a finitenumber of particles N and/or a finite-size of the hard spheres and/or a dense or locally-dense system, is more subtleand - as explained below - even unprecedented since it has actually remained unsolved to date. The reasons areas follows. First, Boltzmann and Grad kinetic theories are inapplicable to the finite Boltzmann-Sinai CDS. Second,the Boltzmann-Shannon entropy associated with an arbitrary particular solution ρ ( N ) ( t ) ≡ ρ ( N ) ( x , t ) of the Masterkinetic equation, i.e., the functional S ( ρ ( N )1 ( t )) ≡ M X E ( ρ ( N )1 ( t )) , in contrast to S ( ρ ( t )) ≡ M X E ( ρ ( t )) , is exactlyconserved in the sense that identically ∂S ( ρ ( N )1 ( t )) ∂t ≡ t → + ∞ the 1 − body PDF must decay uniformly to the stationary and spatially-uniform Maxwellian PDF (5).However, besides the construction of the kinetic equation appropriate for such a case, a further unsolved issue liesin the determination of the functional class n ρ ( N )1 ( x , t ) o for which both PMI and DKE should/might be realized. Inparticular, the possible occurrence of both PMI and DKE should correspond to suitably-smooth, but nonetheless stillarbitrary, initial conditions n ρ ( N )1 ( x , t o ) o . These should warrant that in the limit t → + ∞ , ρ ( N )1 ( x , t ) uniformlyconverges to the spatially-homogeneous and stationary Maxwellian PDF ρ M ( v ) (5). Such a result, however, ishighly non-trivial since it should rely on the establishment of a global existence theorem for the same 1 − body PDF ρ ( N )1 ( x , t ) - namely holding in the whole time axis I ≡ R , besides the same 1 − body phase space Γ - for the involvedkinetic equation which is associated with the S N − CDS. In the context of the Boltzmann equation in particular,despite almost-endless efforts this task has actually not been accomplished yet, the obstacle being intrinsically relatedto the asymptotic nature of the Boltzmann equation [4]. In fact for the same equation it is not known in satisfactorygenerality whether smooth enough solutions of the same equation exist which satisfy the H − theorem inequality anddecay asymptotically to kinetic equilibrium [26, 28].
1B - Goals and organization of the paper
Based on these premises, the crucial new results that we intend to display in this paper concern the proof-of-principle of two phenomena which are expected to characterize the statistical description of finite N − body hard-sphere systemsand therefore should lay at the very foundation of classical statistical mechanics and kinetic theory alike. These arerelated to the physical conditions for the possible occurrence of both PMI and the consequent one represented by thepossible occurrence of DKE which should characterize the kinetic PDF in these systems. These phenomena are wellknown to occur in the case of dilute hard-sphere systems, i.e., in the Boltzmann-Grad limit. In particular, for anexhaustive treatment of the related issues which arise in the context of the ab initio theory we refer to discussionsreported in Ref. [6]. Nevertheless, as indicated above, their existence in the case of finite hard-sphere systems is partlymotivated by a previous investigation dealing with the kinetic description incompressible Navier-Stokes granular fluids[5].Therefore, main goal of the paper is to show that these properties actually emerge as necessary implications of theab initio theory of CSM. Incidentally, in doing so, the Master kinetic equation must be necessarily adopted. In fact,the finiteness requirement on the S N − CDS rules out for further possible consideration either the Boltzmann or theEnskog kinetic equations, these equations being inapplicable to the treatment of systems of this type [3]. Specifically,in the following the case
N > − body occupation coefficients arise (see related notations which are applicable for N > N = 2 is nevertheless briefly discussed inAppendix D.For this purpose, first, in Section 2, the MKI functional is explicitly determined. We display in particular itsconstruction method (see No. ). Based on the theory of the Master kinetic equation earlierdeveloped [3] and suitable integral and differential identities (see Appendices A, B and C), the properties of theMKI functional are investigated. These concern in particular the establishment of appropriate inequalities holdingfor the same functional (THM.1, subsection 2A), the signature of the time derivative of the same functional (THM.2,subsection 2B) and the property of DKE holding for a suitable class of 1 − body PDFs (THM.3, subsection 2C). In thesubsequent Sections 3 and 4, the issue of the consistency of the phenomena of PMI and DKE with microscopic dynamicsis posed together with the physical interpretation and implications of the theory. The goal is to investigate therelationship of the DKE-theory developed here with the microscopic reversibility principle and the Poincar´e recurrencetheorem. Finally in Section 5 the conclusions of the paper are drawn and possible applications/developments of thetheory are pointed out. In view of the considerations given above in this section the problem is posed of the explicit realization of the MKIfunctional in terms of suitable axiomatic prescriptions. The same functional, denoted I M ( ρ ( N )1 ( t )) , should depend onthe 1 − body PDF ρ ( N )1 ( t ) , with ρ ( N )1 ( t ) ≡ ρ ( N )1 ( x , t ) being identified with a particular solution of the Master kineticequation (see Eq.(69) in Appendix A holding for N > N = 2).Unlike Boltzmann kinetic equation, the Master kinetic equation actually deals with the treatment of finite hard-sphere N − body systems, i.e., in which both the number of particles N and their diameter σ remain finite [3]. To achievesuch a goal suitably-prescribed physical collision boundary conditions (CBC) of the N − body PDF need to be adopted.More precisely, this concerns the prescription for arbitrary collision events of the relationship between incoming ( − )and outgoing (+) PDFs, i.e., respectively the left and right limits ρ ( ± )( N ) ( x ( ± ) ( t i ) , t i ) = lim t → t ( ± ) i ρ ( N ) ( x ( t ) , t ) , with x ( ± ) ( t i ) = lim t → t ( ± ) i x ( t ) denoting the corresponding incoming ( − ) and outgoing (+) states. In particular, uponinvoking due to causality the assumption of left-continuity, i.e., the requirement ρ ( − )( N ) ( x ( − ) ( t i ) , t i ) ≡ ρ ( N ) ( x ( − ) ( t i ) , t i ) , (12)the incoming PDF is required to coincide with the same N − body PDF evaluated in terms of the incoming state andtime [1, 3]. Hence, as recalled in Appendix C (see also Ref. [2]) from Eq.(91) if follows that the so-called causal formof the modified collision boundary condition (MCBC [2]) ρ (+)( N ) ( x (+) ( t i ) , t i ) = ρ ( N ) ( x (+) ( t i ) , t i ) (13)is mandatory. A further important requirement concerns precisely setting also the related functional class of admissiblesolutions n ρ ( N )1 ( x , t ) o in such a way that, besides ρ ( N )1 ( t ), also the same functional I M ( ρ ( N )1 ( t )) exists globally forarbitrary t ∈ I ≡ R . For definiteness, we shall consider for this purpose the case of 1 − body PDFs which satisfy theinitial condition ρ ( N )1 ( t o ) ≡ ρ ( N )1 ( x , t o ) = ρ ( N )1( o ) ( x ) , (14)with ρ ( N )1( o ) ( x ) belonging to the functional class of stochastic − body PDFs n ρ ( N )1 ( t o ) o . For a generic t ∈ I belongingto the time axis I ≡ R this is the ensemble of 1 − body PDFs ρ ( N )1 ( t ) which are respectively: A) smoothly differentiable;B) strictly positive; C) summable, in the sense that the velocity - or phase-space - moments for the same PDF existwhich correspond either to arbitrary monomial functions of v (or its components v i , for i = 1 , ,
3) or to the entropydensity ln ρ ( N )1 ( t ) , thus yielding the Boltzmann-Shannon (BS) entropy evaluated in terms of ρ ( N )1 ( t ).Concerning the choice of the setting n ρ ( N )1 ( t o ) o the following remarks are in order. As a first remark, the previousrequirements A), B) and C) for n ρ ( N )1 ( t o ) o , together with validity of MCBC (13), actually should warrant that thecorresponding solution of the Master kinetic equation ρ ( N )1 ( t ) exists globally in the extended phase-space ( x , t ) ∈ Γ × I and that for all t ∈ I the same PDF belongs to the class of stochastic PDFs n ρ ( N )1 ( x , t ) o indicated above and alsofulfills identically the constant H-theorem (11). Indeed, one can show [4] that global existence of solutions for theMaster kinetic equation follows in elementary way from the N − body Liouville equation. Indeed, an arbitrary 1 − bodyPDF which is a particular solution of the Master kinetic equation realizes by construction also a particular factorizedsolution of the N − body Liouville equation, i.e., of the N − body PDF [3]. The same PDF evolves uniquely in timealong arbitrary phase-space Lagrangian trajectories, its Lagrangian time evolution being determined at arbitrarycollision times by MCBC (13) [4]. As a second remark, the validity of assumptions A) and B) for ρ ( N )1( o ) ( x ) and ρ ( N )1 ( x , t ) implies also suitableassumptions to apply all t ∈ I for the local characteristic scale-length L ρ which characterize the same PDF ρ ( N )1 ( x , t ) . More precisely, this is associated with the spatial variations of the 1 − body PDF prescribed as L ρ ( t ) = inf x ∈ Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ln ρ ( N )1 ( x , t ) ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − , (15)which necessarily assumed non-zero at all time t ∈ I . Hence ∂ ln ρ ( N )1 ( x ,t ) ∂ r is assumed to be bounded for all ( x , t )spanning the extended 1 − body phase space Γ × I .Finally, as a third remark (see also the further related discussion in THM.2 below), the previous requirements areexpected to warrant also the global existence of the MKI functional I M ( ρ ( N )1 ( t )), so that n ρ ( N )1 ( x , t ) o effectivelyrealizes the functional class of admissible solutions indicated above.Given these premises let us pose now the problem of the identification of the functional I M (cid:16) ρ ( N )1 o ( x ) (cid:17) , basedon the introduction of ’ad hoc’ physical requirements, to be referred to here as MKI Prescriptions No. . Theprescriptions are as follows: • MKI Prescription No. the first one is that the functional I M ( ρ ( N )1 ( t )) should be determined in such a waythat the existence of I M (cid:16) ρ ( N )1 o ( x ) (cid:17) at a suitable initial time t o ∈ I should warrant also that I M ( ρ ( N )1 ( t )) mustnecessarily exist globally in the future, i.e., for all t ≥ t o . As a consequence the functional class n ρ ( N )1 ( x , t ) o must be suitably prescribed. • MKI Prescription No. second, we shall require that I M ( ρ ( N )1 ( t )) is real, non-negative and bounded in thesense that 0 ≤ I M ( ρ ( N )1 ( t )) ≤ . (16)This implies that I M ( ρ ( N )1 ( t )) can be interpreted as an information measure associated with the 1 − body PDF ρ ( N )1 ( t ) ≡ ρ ( N )1 ( x , t ). For this reason the previous inequalities will be referred to as information-measureinequalities . • MKI Prescription No. third, for consistency with the property of macroscopic irreversibility, I M ( ρ ( N )1 ( t )) isprescribed in terms of a smoothly time-differentiable and monotonically time-decreasing functional in the sensethat in the same time-subset the inequality: ∂∂t I M ( ρ ( N )1 ( t )) ≤ ∀ t ≥ t o , so that0 ≤ I M ( ρ ( N )1 ( t )) ≤ I M (cid:16) ρ ( N )1 o ( x ) (cid:17) ≤ . (18)This implies that I M ( ρ ( N )1 ( t )) is also globally defined for all t ∈ I ≡ R with t & t o . In addition, if ∂∂t I M ( ρ ( N )1 ( t )) (cid:12)(cid:12)(cid:12) t = t o = 0, without loss of generality its initial value I M (cid:16) ρ ( N )1 o ( x ) (cid:17) can always be set such that I M (cid:16) ρ ( N )1 o ( x ) (cid:17) = 1 . (19) • MKI Prescription No. fourth, in order to warrant the existence of DKE we shall require the functional I M ( ρ ( N )1 ( t )) to be prescribed in such a way that at an arbitrary time t ∈ I, with t & t o , the vanishing of both I M ( ρ ( N )1 ( t )) and its time derivative ∂∂t I M ( ρ ( N )1 ( t )) should occur if and only if the 1 − body PDF solution of theMaster kinetic equation coincides with kinetic equilibrium. As a consequence, for the functional I M ( ρ ( N )1 ( t ))the following propositions should be equivalent ( I M ( ρ ( N )1 ( t )) = 0 ∂∂t I M ( ρ ( N )1 ( t )) = 0 ⇔ ρ ( N )1 ( x , t ) ≡ ρ ( N )1 M ( v ) , (20)with ρ ( N )1 M ( v ) being a kinetic equilibrium PDF of the form (5).The immediate obvious implication of the previous prescriptions is that - provided a non-trivial realization of theMKI can be found in the functional class n ρ ( N )1 o ( x ) o - the existence of both PMI and DKE for the Master kineticequation is actually established. In the sequel the goal is to show, in particular, that the MKI functional can beidentified by means of the prescription I M ( ρ ( N )1 ( t ) , b ) ≡ K M ( ρ ( N )1 ( t ) , b ) K Mo , (21)where K M ( ρ ( N )1 ( t ) , b ) and K Mo denote respectively a suitable (and possibly non-unique) moment-dependent phase-space functional and an appropriate normalization constant to be chosen in such a way to satisfy all the MKIprescriptions indicated above. In particular, as shown below, an admissible choice for K M ( ρ ( N )1 ( t ) , b ) and K Mo isprovided by K M ( ρ ( N )1 ( t ) , b ) = − R Γ d x Θ ( ∂ Ω)1 ( r ) M ( v , b ) ρ ( N )1 ( x ,t ) b ρ ( N )1 ( x ,t ) ∂ b ρ ( N )1 ( x ,t ) ∂ r · ∂ r ,K Mo = sup n , K M ( ρ ( N )1 o ( x ) , b ) o , (22)while M ( v , b ) denotes the directional kinetic energy (along the unit vector b ) carried by particle 1, namely thedynamical variable M ( v , b ) ≡ ( v · b ) , (23)with b denoting a still arbitrary constant unit vector. Hence, M ( v , v , b ) = 12 [ M ( v , b ) + M ( v , b )] (24)identifies the corresponding total directional kinetic energy carried by particles 1 and 2. Here the remaining notationis standard. Thus, ρ ( N )1 ( t ) ≡ ρ ( N )1 ( x , t ) , ρ ( N )1 o ( x ) and b ρ ( N )1 ( t ) ≡ b ρ ( N )1 ( x , t ) are respectively the 1 − body PDF solutionof the initial problem associated with the Master kinetic equation (see Eq.(69) in Appendix A), the initial PDF andthe renormalized 1 − body PDF b ρ ( N )1 ( x , t ) ≡ ρ ( N )1 ( x , t ) k ( N )1 ( r , t ) , (25)while furthermore k ( N )1 ( r , t ) is the 1 − body occupation coefficient recalled in Appendix B (see Eq.(78)). As a con-sequence in the previous equation it follows that ρ ( N )1 ( x ,t ) b ρ ( N )1 ( x ,t ) ≡ k ( N )1 ( r , t ). Furthermore, Θ ( ∂ Ω)1 ( r ) is the boundarytheta-function given by Eq.(74) (see Appendix A). Finally, regarding the initial value K M ( ρ ( N )1 o ( x ) , b ) it follows thatif respectively K M ( ρ ( N )1 o ( x ) , b ) ≥ ≤ K M ( ρ ( N )1 o ( x ) , b ) < , (26)then correspondingly one obtains, consistent with (18), that the initial value of MKI functional I M ( ρ ( N )1 o , b ) is I M ( ρ ( N )1 o , b ) = (cid:26) ,K M ( ρ ( N )1 o ( x ) , b ) . (27)
2A - Proof of the non-negativity of the MKI information measure
The strategy adopted for the proof of the MKI Prescriptions No. I M ( ρ ( N )1 ( t ) , b ) cannot acquire negative values forarbitrary t ≥ t o . The result is established by the following theorem. THM. 1 - Non-negativity of K M ( ρ ( N )1 o ( x ) , b ) , K M ( ρ ( N )1 ( t ) , b ) and I M ( ρ ( N )1 ( t ) , b ) Let us assume that ρ ( N )1 ( x , t ) is an arbitrary stochastic and suitably smoothly-differentiable, particular solution ofthe Master kinetic equation (69) prescribed so that the integral (22) expressed in terms of the initial PDF, namely K M ( ρ ( N )1 o ( x ) , b ) , is non-vanishing. Then, it follows necessarily that: • Proposition P1 : K M ( ρ ( N )1 o ( x ) , b ) > . (28) • Proposition P1 : the corresponding time-evolved functional K M ( ρ ( N )1 ( t ) , b ) for all t ∈ I with t > t o is suchthat K M ( ρ ( N )1 ( t ) , b ) ≥ . (29) • Proposition P1 : for all t ∈ I with t > t o the functional I M ( ρ ( N )1 ( t ) , b ) fulfills the inequality I M ( ρ ( N )1 ( t ) , b ) ≥ . (30) • Proposition P1 : the following necessary and sufficient condition holds at a given time t ∈ I with t ≥ t o : K M ( ρ ( N )1 ( t ) , b ) = 0 ⇔ ρ ( N )1 ( x , t ) ≡ ρ ( N )1 M ( v ) . (31) Proof -
One first notices that K M ( ρ ( N )1 ( t ) , b ) can be equivalently written in the form K M ( ρ ( N )1 ( t ) , b ) ≡ − Z Γ d x Θ ( ∂ Ω)1 ( r ) M ( v , b ) k ( N )1 ( r , t ) ∂ b ρ ( N )1 ( x , t ) ∂ r · ∂ r , (32)where in order that the same functional exists it is obvious that the renormalized 1 − body PDF b ρ ( N )1 ( x , t ) must beof class C (2) . Integrating by parts and noting that the gradient term ∂ Θ ( ∂ Ω)1 ( r ) ∂ r gives a vanishing contribution to thephase-space integral, this yields equivalently K M ( ρ ( N )1 ( t ) , b ) ≡ Z Γ d x Θ ( ∂ Ω)1 ( r ) M ( v , b ) ∂k ( N )1 ( r , t ) ∂ r · ∂ b ρ ( N )1 ( x , t ) ∂ r . (33)Therefore, upon invoking Eq.(85) reported in Appendix B, direct substitution delivers K M ( ρ ( N )1 ( t ) , b ) = ( N − Z Γ d x Θ ( ∂ Ω)1 ( r ) M ( v , b ) ∂ b ρ ( N )1 ( x , t ) ∂ r · Z Γ d x n × δ ( | r − r | − σ ) Θ ( ∂ Ω)2 ( r ) b ρ ( N )1 ( x , t ) k ( N )2 ( r , r , t ) . (34)Next, invoking the identity n δ ( | r − r | − σ ) = − ∂∂ r Θ ( | r − r | − σ ) and noting again that ∂∂ r Θ ( ∂ Ω)2 ( r ) givesvanishing contribution, one can perform a further integration by parts with respect to r . This permits to cast therhs of previous equation in the form K M ( ρ ( N )1 ( t ) , b ) ≡ K (1) M ( ρ ( N )1 ( x , t ) , b ) + ∆ K (1) M ( ρ ( N )1 ( x , t ) , b ) . (35)Here the two terms on the rhs of Eq.(35) are defined as follows: 1) the first term K (1) M ( ρ ( N )1 ( t ) , b ) is symmetric andnon-negative, so that it can be expressed so to carry the total directional kinetic energy M ( v , v , b ) of particles 1and 2 (see Eq.(24)). Hence, it takes the form K (1) M ( ρ ( N )1 ( x , t ) , b ) = ( N − R Γ d x R Γ d x Θ ( ∂ Ω)1 ( r )Θ ( ∂ Ω)2 ( r ) × ∂ b ρ ( N )1 ( x ,t ) ∂ r · ∂ b ρ ( N )1 ( x ,t ) ∂ r k ( N )2 ( r , r , t ) M ( v , v , b )Θ ( | r − r | − σ ) . (36)2) The second term ∆ K (1) M ( ρ ( N )1 ( x , t ) , b ) reads instead∆ K (1) M ( ρ ( N )1 ( x , t ) , b ) ≡ ( N − Z Γ d x Θ ( ∂ Ω)1 ( r ) ∂ b ρ ( N )1 ( x , t ) ∂ r · Z Γ d x × Θ ( ∂ Ω)2 ( r ) M ( v , b )Θ ( | r − r | − σ ) b ρ ( N )1 ( x , t ) ∂∂ r k ( N )2 ( r , r , t ) , (37)where ∂∂ r k ( N )2 ( r , r , t ) is given by the differential identity (86) reported in Appendix B. Thus, upon invoking theidentity n δ ( | r − r | − σ ) = − ∂∂ r Θ ( | r − r | − σ ), one notices that an integration by parts can be performed alsowith respect to r . This means that a procedure analogous to the one used for the calculation of ∂k ( N )1 ( r ,t ) ∂ r can beinvoked and iterated at all orders, i.e., up to the ( N − − body occupation coefficient (see Eq.(87) in Appendix B). Asa consequence the functional K M ( ρ ( N )1 ( t ) , b ) can be represented in terms of a finite sum of the form K M ( ρ ( N )1 ( t ) , b ) ≡ P j =1 ,N − K ( j ) M ( ρ ( N )1 ( x , t ) , b ) in which each term of the sum K ( j ) M ( ρ ( N )1 ( x , t ) , b ) is non-negative and symmetric. This0implies therefore that the same functional K M ( ρ ( N )1 ( t ) , b ) can be cast in the form K M ( ρ ( N )1 ( t ) , b ) = Z Γ d x Z Γ d x Θ ( ∂ Ω)1 ( r )Θ ( ∂ Ω)2 ( r ) M ( v , v , b ) × F ( r , r , t ) ∂ b ρ ( N )1 ( x , t ) ∂ r · ∂ b ρ ( N )1 ( x , t ) ∂ r Θ ( | r − r | − σ ) , (38)with M ( v , v , b ) ≥ F ( r , r , t ) a suitable real scalar kernel whichis symmetric in the variables r and r . Hence K M ( ρ ( N )1 ( t ) , b ) actually defines a non-negative functional. This provesthe validity of the inequality (30) (Proposition P1 ).In a similar way also the remaining Propositions can be established. In fact, invoking Eq.(27) it follows that theinequalities (29) and (30) - and hence also Propositions P1 and P1 - manifestly hold too. Finally, regarding theproof of Proposition P1 , one notices that K M ( ρ ( N )1 ( t ) , b ) ≡ ∂∂ r b ρ ( N )1 ( x , t ) ≡
0. Since b ρ ( N )1 ( x , t ) is by construction a solution of the Master kinetic equation it follows that this requires necessarily that b ρ ( N )1 ( x , t ) must coincide with the local Maxwellian ρ ( N )1 M ( v ) (see Eq.(5)) and hence Eq.(31) must hold too under thesame realization (Proposition P1 ). Q.E.D.
The conclusion is therefore that the definition of the MKI functional (21) given above in terms of K M ( ρ ( N )1 ( t ) , b )and K Mo (see Eqs.(22)) is indeed consistent with the physical prerequisites represented by the MKI PrescriptionsNo.
2B - Proof of PMI for the Master kinetic equation
The next step is to prove that the functional I M ( ρ ( N )1 o , b ) defined above (see Eq.(27)) indeed exhibits a monotonictime-decreasing behavior which is consistent with the MKI Prescriptions No. • the time derivative inequality (17) and the conditions of existence of kinetic equilibrium (20); • the validity of the inequality I M ( ρ ( N )1 ( t )) ≤ M ( v , v , b ) (see Eq.(24)), namely thephase-space scalar function ∆ M ( v , v , b ) ≡ M ( v (+)1 , v (+)2 , b ) − M ( v v , b ). One obtains∆ M ( v , v , b ) = b · n (cid:12)(cid:12)(cid:12) n · v (+)12 (cid:12)(cid:12)(cid:12) v (+)12 · b − ( b · n ) (cid:16) n · v (+)12 (cid:17) , (39)the rhs being expressed in terms of the outgoing particle velocities ( v (+)1 , v (+)2 ) only. Then, the following propositionholds. THM. 2 - Property of macroscopic irreversibility ( Master equation PMI theorem ) Let us assume that ρ ( N )1 ( x , t ) ≡ ρ ( N )1 ( r , v , t ) is an arbitrary stochastic particular solution of the Master kineticequation (69) with initial condition ρ ( N )1 o ( x ) such that the integral K M ( ρ ( N )1 o ( x ) , b ) exists and is non-vanishing. Thenit follows that • Proposition P2 : one finds that for all t ≥ t o : ∂∂t K M ( ρ ( N )1 ( t ) , b ) = − ( N − σ Z U d v Z U d v Z Ω d r Θ ( ∂ Ω)1 ( r ) Z ( − ) d Σ (cid:12)(cid:12)(cid:12) v (+)12 · n (cid:12)(cid:12)(cid:12) ( b · n ) (cid:16) n · v (+)12 (cid:17) ∂ b ρ ( N )1 ( r , v (+)1 , t ) ∂ r · ∂∂ r b ρ ( N )1 ( r = r + σ n , v (+)2 t ) k ( N )2 ( r , r = r + σ n , t ) ≤ . (40)1 • Proposition P2 : the inequality ∂∂t I M ( ρ ( N )1 ( t ) , b ) ≤ holds globally (i.e., identically for all t ≥ t o ) so that necessarily I M ( ρ ( N )1 ( t ) , b ) is globally defined too, being alsoprescribed so that I M ( ρ ( N )1 ( t ) , b ) ≤ . (42) • Proposition P2 : one finds that a given time t ∈ I with t ≥ t o : ∂∂t K M ( ρ ( N )1 ( t ) , b ) = 0 ⇔ ρ ( N )1 ( x , t ) ≡ ρ ( N )1 M ( v ) . Proof -
Consider first the proof of proposition P2 which requires evaluation of the partial time derivative ∂∂t K M ( ρ ( N )1 ( t ) , b ) . Upon invoking the first form of the Master kinetic equation (see Eq.(66) in Appendix A),explicit differentiation of K M ( ρ ( N )1 ( t ) , b ) delivers ∂∂t K M ( ρ ( N )1 ( t ) , b ) = − Z Γ d x M ( v , b )Θ ( ∂ Ω)1 ( r ) k ( N )1 ( r , t ) (cid:18) − v · ∂∂ r (cid:19) ∂ b ρ ( N )1 ( x , t ) ∂ r · ∂ r − Z Γ d x M ( v , b )Θ ( ∂ Ω)1 ∂ b ρ ( N )1 ( x , t ) ∂ r · ∂ r (cid:18) ∂∂t (cid:19) k ( N )1 ( r , t ) , (43)namely, upon integration by parts in the first integral on the rhs, ∂∂t K M ( ρ ( N )1 ( t ) , b ) = − Z Γ d x M ( v , b )Θ ( ∂ Ω)1 ∂ b ρ ( N )1 ( x , t ) ∂ r · ∂ r (cid:18) ∂∂t + v · ∂∂ r (cid:19) k ( N )1 ( r , t ) . (44)Hence, thanks to the differential identity (88) it follows: ∂∂t K M ( ρ ( N )1 ( t ) , b ) = − ( N − Z Γ d x M ( v , b )Θ ( ∂ Ω)1 ( r ) ∂ b ρ ( N )1 ( x , t ) ∂ r · ∂ r Z Γ d x v · n × δ ( | r − r | − σ ) k ( N )2 ( r , r , t ) b ρ ( N )1 ( x , t ) . (45)Performing an integration by parts with respect to r and upon invoking the first differential identity (90)reported in Appendix B one obtains therefore: ∂∂t K M ( ρ ( N )1 ( t ) , b ) = W M ( ρ ( N )1 ( t ) , b ) , (46) W M ( ρ ( N )1 ( t ) , b ) ≡ Z Γ d x M ( v , b )Θ ( ∂ Ω)1 ( r ) ∂ b ρ ( N )1 ( x , t ) ∂ r ( N − × Z Γ d x v · n ∂∂ r [ δ ( | r − r | − σ )] k ( N )2 ( r , r , t ) b ρ ( N )1 ( x , t ) , (47)where ∂∂ r [ δ ( | r − r | − σ )] = − ∂∂ r [ δ ( | r − r | − σ )]. Hence performing a further integration by parts withrespect to r and using the second differential identity on Eq. (90) (see Appendix B) the previous equationfinally yields W M ( ρ ( N )1 ( t ) , b ) = ( N − Z Γ d x Θ ( ∂ Ω)1 ( r ) Z Γ d x v · n M ( v , v , b ) × δ ( | r − r | − σ ) k ( N )2 ( r , r , t ) ∂ b ρ ( N )1 ( x , t ) ∂ r · ∂ b ρ ( N )1 ( x , t ) ∂ r , (48)2where the symmetry property with respect to the exchange of states ( x , x ) has been invoked. In the previousequation the integration on the Dirac delta can be performed at once letting Z Γ d x Θ ( ∂ Ω)1 Z Γ d x δ ( | r − r | − σ ) = σ Z U d v Z U d v Z Ω d r Θ ( ∂ Ω)1 ( r ) "Z (+) d Σ | v · n | − Z ( − ) d Σ | v · n | , (49)where the solid-angle integrations in the two integrals on the rhs are performed respectively on the outgoing (+)and incoming ( − ) particles. Furthermore, it is obvious that thanks to the causal form of MCBC (see Eq.(93) inAppendix C) the integral on outgoing particles R (+) d Σ can be transformed to a corresponding integration onincoming ones, namely R ( − ) d Σ . Thus, the contributions in the two phase-space integrals only differ becauseof the variation ∆ M ( v , v , b ) of the total directional kinetic energy of particles 1 and 2 . This implies that W M ( ρ ( N )1 ( t ) , b ) = ( N − σ Z U d v Z U d v Z Ω d r Θ ( ∂ Ω)1 ( r ) Z ( − ) d Σ ×| v · n | ∆ M ( v , v , b ) ∂ b ρ ( N )1 ( r , v (+)1 , t ) ∂ r · ∂ b ρ ( N )1 ( r = r + σ n , v (+)2 , t ) ∂ r × k ( N )2 ( r , r = r + σ n , t ) , (50)where the solid-angle integration is performed on the incoming particles whereas ∆ M ( v , v , b ) is evaluatedin terms of the outgoing particles (+) and therefore must be identified with the second equation on the rhsof Eq.(39). Consider now the dependences in terms of the outgoing particle velocities v (+)1 and v (+)2 in theprevious phase-space integral. The velocity dependences contained in the factors | v · n | and ∂ b ρ ( N )1 ( r , v (+)1 ,t ) ∂ r · ∂ b ρ ( N )1 ( r , v (+)2 t ) ∂ r are symmetric with respect to the variables v (+)1 and v (+)2 . On the other hand, as a whole, thesame integral should remain unaffected with respect to the exchange of the outgoing particle velocities v (+)1 ⇔ v (+)2 . This means that the only term in ∆ M ( v , v , b ) which gives a (possibly) non-vanishing contribution is − ( b · n ) (cid:16) n · v (+)12 (cid:17) . As a consequence it is found that ∂∂t K M ( ρ ( N )1 ( t ) , b ) ≡ W M ( ρ ( N )1 ( t ) , b ) = − ( N − σ Z U d v Z U d v × Z Ω d r Θ ( ∂ Ω)1 ( r ) Z ( − ) d Σ ∂ b ρ ( N )1 ( r , v (+)1 , t ) ∂ r · ∂ b ρ ( N )1 ( r = r + σ n , v (+)2 , t ) ∂ r × (cid:12)(cid:12)(cid:12) v (+)12 · n (cid:12)(cid:12)(cid:12) ( b · n ) (cid:16) n · v (+)12 (cid:17) k ( N )2 ( r , r = r + σ n , t ) ≤ , (51)and hence ∂∂t K M ( ρ ( N )1 ( t ) , b ) is necessarily negative or null, the second case occurring only if ∂ b ρ ( N )1 ( r , v (+)1 ,t ) ∂ r ≡ ∂ b ρ ( N )1 ( r = r + σ n , v (+)2 t ) ∂ r ≡ follows in a similar way. In fact, first, one notices that thanks to the global validityof the 1 − body PDF [4] the 1 − body PDF ρ ( N )1 ( t ) necessarily belongs to the functional class of stochastic PDFs n ρ ( N )1 ( x , t ) o prescribed so that also the local characteristic scale-length defined above L ρ ( t ) (see Eq. (15)) is largerthan zero and finite . As a consequence it follows that both the functional K M ( ρ ( N )1 ( t ) , b ) and I M ( ρ ( N )1 ( t ) , b ) (seeEqs.(21)) are globally defined too. Consider in fact the representation of K M ( ρ ( N )1 ( t ) , b ) achieved in THM.1 andgiven by Eq.(34). Next, let us notice that thanks to Eq.(15) the characteristic scale length L µ, min ≡ inf (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ b ρ ( N )1 ( x , t ) ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (52)3is necessarily strictly positive. Then, upon noting that k ( N )2 ( r , r , t ) ≤ R Γ d x δ ( | r − r | − σ ) b ρ ( N )1 ( x , t ) ≤ sup (cid:16)b n ( N )1 ( r , t ) (cid:17) < + ∞ , with b n ( N )1 ( r , t ) being the velocity moment b n ( N )1 ( r , t ) = R U d v b ρ ( N )1 ( r , v , t ), it followsthat K M ( ρ ( N )1 ( t ) , b ) ≤ ( N − L µ, min sup (cid:16)b n ( N )1 ( r , t ) (cid:17) Z Γ d x Θ ( ∂ Ω)1 ( r ) M ( v , b ) b ρ ( N )1 ( x , t ) , (53)where the integral on the rhs is necessarily bounded. This happens because ρ ( N )1 ( x , t ) belongs to the functional class n ρ ( N )1 ( t ) o and therefore b n ( N )1 ( r , t ) is bounded, while, at the same time, the phase-space moments indicated abovenecessarily exist. Furthermore, since ∂∂t I M ( ρ ( N )1 ( t ) , b ) ≡ K Mo ∂∂t K M ( ρ ( N )1 ( t ) , b ) , the inequality (51) implies Eq.(41)and (42) too. Finally, since b ρ ( N )1 ( r , v , t ) is a solution of the Master kinetic equation ∂∂ r b ρ ( N )1 ( x , t ) ≡ b ρ ( N )1 ( x , t ) coincides with a Maxwellian kinetic equilibrium of the type (5). This result proves therefore alsoProposition P2 . Q.E.D.
The implication of THM.2 is therefore that provided the initial value K M ( ρ ( N )1 o ( x ) , b ) is non-vanishing then nec-essarily: • the functional K M ( ρ ( N )1 ( t ) , b ) is monotonically decreasing and thus K M ( ρ ( N )1 ( t ) , b ) ≤ K M ( ρ ( N )1 o ( x ) , b ); • similarly the MKI functional I M ( ρ ( N )1 ( t ) , b ) is monotonically decreasing too, i.e., I M ( ρ ( N )1 ( t ) , b ) ≤ I M ( ρ ( N )1 o ( x ) , b ); • both K M ( ρ ( N )1 ( t ) , b ) and I M ( ρ ( N )1 ( t ) , b ) are non-negative.
2C - Proof of the DKE property for the Master kinetic equation
Let us now show that in validity of THMs. 1 and 2 the time-evolved ρ ( N )1 ( x , t ) necessarily must decay asymptoticallyfor τ ≡ t − t o → + ∞ to kinetic equilibrium, i.e., that the limit function lim τ → + ∞ ρ ( N )1 ( x , t ) ≡ ρ ( N )1 ∞ ( x ) exists and itnecessarily coincides with a Maxwellian kinetic equilibrium of the type (5). In this regard the following propositionholds. THM. 3 - Asymptotic behavior of I M ( ρ ( N )1 ( t ) , b ) ( Master equation-DKE theorem ) Let us assume that the initial condition ρ ( N )1 o ( x ) ∈ n ρ ( N )1 o ( x ) o is such that the corresponding functional K M ( ρ ( N )1 o ( x ) , b ) is non-vanishing, i.e., in view of THM.1 necessarily > . Then it follows that the correspond-ing time-evolved solution of the Master kinetic equation ρ ( N )1 ( x , t ) in the limit τ ≡ t − t o → + ∞ necessarily mustdecay to kinetic equilibrium, i.e., lim τ → + ∞ ρ ( N )1 ( x , t ) = ρ ( N )1 M ( v ) . (54) Proof -
In order to reach the thesis it is sufficient to prove that necessarilylim τ → + ∞ ∂∂t I M ( ρ ( N )1 ( t ) , b ) = 0 . (55)In fact, let us assume ” ad absurdum ” that ∂∂t I M ( ρ ( N )1 ( t ) , b ) ≤ − k with k > ) requires that lim τ → + ∞ I M ( ρ ( N )1 ( t ) , b ) ≤ − lim τ → + ∞ ( t − t o ) k = −∞ , (56)4a result which contradicts THM.1. This proves the validity of Eq.(55). Furthermore, by construction ∂∂t I M ≡ K Mo ∂∂t K M and furthermore ∂∂t K M is identified with the functional W M ( ρ ( N )1 ( t ) , b ) ≤ W M ( ρ ( N )1 ( t ) , b ) , the identitylim τ → + ∞ ∂∂t I M ( ρ ( N )1 ( x , t ) , b ) = W M ( ρ ( N )1 ∞ ( x ) , b ) (57)holds, where, thanks to global existence of the 1 − body PDF (see Ref.[4]), the limit functionlim τ → + ∞ ρ ( N )1 ( x , t ) ≡ ρ ( N )1 ∞ ( x ) (58)necessarily exists. As a consequence Eq.(55) requires also the equation W M ( ρ ( N )1 ∞ ( x ) , b ) = 0 (59)to hold. Upon invoking proposition P2 of THM.2 this implies that necessarily ρ ( N )1 ∞ ( x ) = ρ ( N )1 M ( v ) so the thesis (54)is proved. Incidentally, thanks to THM.1, this requires also thatlim τ → + ∞ I M ( ρ ( N )1 ( t ) , b ) = I M ( ρ ( N )1 ∞ ( x ) , b ) = 0 . (60) Q.E.D .
2D - Remarks
A few remarks are worth being pointed out regarding the results presented above.1.
Remark
The choice of the MKI functional considered here (see Eq.(21)) is just one of the infinite particularadmissible realizations which meet the complete set of MKI-prescriptions indicated above. In particular, thechoice of the velocity moment M ( v , b ) considered here (see Eq.(23)) remains in principle arbitrary, since | v · b | can be equivalently replaced, for example, by any factor of the form | v · b | n , with n ≥ . Furthermoreit is obvious that M ( v , b ) can be replaced by any function of the form M ( v , b )+∆ M ( v , b ), being ∆ M ( v , b )prescribed in such a way that its contribution to ∂∂t I M vanishes identically so that the validity of the inequality(41) in THM. 2 is preserved. This implies in turn that the prescription of the MKI functional I M ( ρ ( N )1 ( t ))remains in principle non-unique.2. Remark
A possible issue is related to the requirement that the renormalized 1 − body PDF, as the 1 − bodyPDF itself, are strictly positive at all times and are non-vanishing. Here it is sufficient to state that an elementaryconsequence of the theory of the Master kinetic equation developed in Ref.[3] is that, provided the correspondinginitial N − body PDF set at a prescribed initial time t o is strictly positive in the whole N − body phase-space,both the corresponding renormalized 1 − body PDF, as the 1 − body PDF remain necessarily strictly positiveglobally in time too and everywhere in the 1 − body phase-space.3. Remark
It must be stressed that the signature of the time derivative ∂∂t I M ( ρ ( N )1 ( t ) , b ) actually dependscrucially on the adoption of the causal form of MCBC (i.e., see Eq.(91) or (93) in Appendix C) rather thanthe anti-causal one (given instead by Eq.(92)). The first choice is mandatory in view of the causality principle.Indeed, it is immediate to prove that ∂∂t I M ( ρ ( N )1 ( t ) , b ) changes signature if the anti-causal MCBC Eq.(92) isinvoked.4. Remark
THM.2 warrants that macroscopic irreversibility, namely the inequality ∂∂t I M ( ρ ( N )1 ( t ) , b ) ≤ b − directional total kinetic energy which occurs atarbitrary binary collision events; b) the occurrence of a velocity-space anisotropy in the 1 − body PDF, i.e., thefact that the same PDF may not coincide with a local Maxwellian PDF.5. Remark
The existence of the limit function lim t → + ∞ ρ ( N )1 ( x , t ) = ρ ( N )1 ∞ ( x ) follows uniquely as a conse-quence of the global existence theorem holding for the Master kinetic equation [4].56. Remark
Last but not least, the fact that the same limit function may coincide or not with the Maxwelliankinetic equilibrium (5) depends crucially on the functional setting prescribed for the same PDF ρ ( N )1 ( x , t ) . More precisely, DKE can only occur provided ρ ( N )1 ( x , t ) is a suitably-smooth stochastic PDF such that theMKI functional exists for the corresponding initial PDF at time t o , i.e., ρ ( N )1 ( x , t o ) = ρ ( N )1 o ( x ) . THMs 1-3 represent the main results reached in the paper of what may be referred to as the
PMI/DKE theory forfinite hard-sphere systems and which have concerned the axiomatic formulation in such a context of the notion ofmacroscopic irreversibility and the related one of decay to kinetic equilibrium.
The crucial problem which arises in the context of the ab initio-theory is in some sense analogous to that occurringin the Boltzmann and Grad kinetic theories. The question is in fact whether these phenomena are actually consis-tent with the fundamental symmetry properties of the underlying Boltzmann-Sinai CDS. The problem posed in thepresent section concerns, more precisely, the consistency with the time-reversible, energy-conserving, evolution of theunderlying N − body Boltzmann-Sinai classical dynamical system S N − CDS.1.
First issue: consistency with the microscopic reversibility principle - This is related to the famous objectionraised by Loschmidt to the Boltzmann equation and Boltzmann H-theorem: i.e., whether and possibly also how it may be possible to reconcile the validity of the reversibility principle for the S N − CDS with the manifestationof a decay of the 1 − body PDF to kinetic equilibrium, i.e., the uniform Maxwellian PDF of the form (5), aspredicted by the above Master equation-DKE Theorem. That a satisfactory answer to this question is actuallypossible follows from considerations which are based on the axiomatic (ab initio) statistical description realizedby the Master kinetic equation. In this regard it is worth recalling the discussion reported above concerning therole of MCBC regarding the functional ∂∂t I M ( ρ ( N )1 ( t )) . In particular, it is obvious that the signature depends onwhether the causal (or anti-causal) form of MCBC is invoked (see Appendix C). Such a choice is not arbitrarysince, for consistency with the causality principle, it must depend on the microscopic arrow of time, i.e., theorientation of the time axis chosen for the reference frame. Based on these premises, consistency between theoccurrence of macroscopic irreversibility associated with the DKE phenomenon and the principle of microscopicreversibility can immediately be established. Indeed, it is sufficient to notice that when a time-reversal or avelocity-reversal is performed on the S N − CDS the form of the collision boundary conditions (i.e., in the presentcase the MCBC provided by Eq.(91) in Appendix C) must be changed, replacing them with the correspondinganti-causal ones, i.e., Eq.(92). This implies that MKI functional decreases in both cases, i.e., after performingthe time-reversal, so that no contradiction can possibly arise in this case between THM.3 and the microscopicreversibility principle.2.
Second issue: consistency with Poincar´e recurrence theorem -
Similar considerations concern the consistencywith the recurrence theorem due to Poincar´e as well as the conservation of total (kinetic) energy for the S N − CDS. In fact, first, as it follows from Ref.[3], by construction the Master collision operator admits the custom-ary Boltzmann collisional invariants, including total kinetic energy of colliding particles. Hence, total energyconservation is again warranted for S N − CDS. Second, regarding Poincar´e recurrence theorem, it concerns theLagrangian phase-space trajectories of the S N − CDS, i.e., the fact that almost all of these trajectories returnarbitrarily close - in a suitable sense to be prescribed in terms of a distance defined on the N − body phase-space- to their initial condition after a suitably large ”recurrence time”. Incidentally, its magnitude depends stronglyboth on the same initial condition and the notion of distance to be established on the same phase-space. Nev-ertheless, such a ”recurrence effect” influences only the Lagrangian time evolution of the N − body PDF whichoccurs along the same Lagrangian N − body phase-space trajectories. Instead, the same recurrence effect hasmanifestly no influence on the time evolution of the Eulerian 1 − body PDF which is advanced in time in termsof the Eulerian kinetic equation represented by the Master kinetic equation. Therefore the mutual consistencyof DKE and Poincar´e recurrence theorem remains obvious.Hence, in the framework of the axiomatic ab initio-theory based on the Master kinetic equation the full consistencyis warranted with the microscopic dynamics of the underlying Boltzmann-Sinai CDS.6 Let us now investigate the physical interpretation and main implications emerging from the PMI/DKE theorydeveloped here. The first issue is related to the physical mechanism at the basis of the PMI/DKE phenomenology.It is well known that in the context of Boltzmann kinetic theory the property of macroscopic irreversibility aswell as the occurrence of the DKE-phenomenon are both ascribed to the Boltzmann H-theorem, both in its originalformulation [9] and in its modified form introduced by Boltzmann himself while attempting to reply [24] to Loschmidtobjection [23] (see also Refs. [26, 27] together with different views on the matter given in Refs. [20, 22]). As recalledabove, this is expressed in terms of the production rate for the Boltzmann-Shannon entropy ∂∂t S ( ρ ( t )) , with S ( ρ ( t ))being interpreted as a measure of the ignorance associated with a solution of the Boltzmann equation. In fact thecustomary interpretation is that they arise specifically because of the validity of the entropic inequality (9), i.e.,the monotonic increase of S ( ρ ( t )), and the corresponding entropic equality (10) stating a necessary and sufficientcondition for kinetic equilibrium. Such a theorem is actually intimately related with the equation itself. In fact boththe theorem and the equation generally hold only for stochastic PDFs ρ ( t ) = ρ ( x , t ) which are suitably-smooth andnot for distributions [1]. According to Boltzmann’s original interpretation, however, both the Boltzmann equationand Boltzmann H-theorem should only hold when the so-called Boltzmann-Grad limit is invoked, i.e. based on thelimit operator L BG ≡ lim N → + ∞ Nσ ∼ O (1) (see Ref. [6]).In striking departure from such a picture: • The axiomatic ab initio-theory based on the Master kinetic equation and the present PMI/DKE theory areapplicable to an arbitrary finite Boltzmann-Sinai CDS. This means that they hold for hard-sphere systemshaving a finite number of particles and with finite diameter and mass, i.e., without the need of invoking validityof asymptotic conditions. • The main departure with respect to Boltzmann kinetic theory arises because, as earlier discovered (see in par-ticular the related discussion reported in Ref.[6]), the Boltzmann-Shannon entropy associated with an arbitrarystochastic 1 − body PDF ρ ( N )1 ( t ) = ρ ( N )1 ( x , t ) solution of the Master kinetic equation is identically conserved.Thus both PMI and DKE are essentially unrelated to the Boltzmann-Shannon entropy. • In the case of the Master kinetic equation the physical mechanism responsible for the occurrence of both PMIand DKE is unrelated with the Boltzmann-Shannon entropy. In fact, as shown here, it arises because of theproperties of the MKI functional I M ( ρ ( N )1 ( t ) , b ) when it is expressed in terms of an arbitrary stochastic PDF ρ ( N )1 ( t ) = ρ ( N )1 ( x , t ) solution of the Master kinetic equation. The only requirement is that the initial PDF ρ ( N )1 o ( x ) is prescribed so that the corresponding MKI functional I M ( ρ ( N )1 o ( x ) , b ) exists. • As shown here the MKI functional is a suitably-weighted phase-space moment of ρ ( N )1 ( x , t ) which can beinterpreted as an information measure for the same PDF, namely belongs to the interval [0 , , and exhibits amonotonic-decreasing time-dependence, i.e., the property of macroscopic irreversibility. • In addition both I M ( ρ ( N )1 ( t ) , b ) and its time derivative ∂∂t I M ( ρ ( N )1 ( t ) , b ) vanish identically if and only if the1 − body PDF coincides with a Maxwellian kinetic equilibrium of the type (5). This warrants in turn also theoccurrence of the DKE-phenomenon for ρ ( N )1 ( x , t ), i.e., that for t − t o → + ∞ the same PDF must decay to aMaxwellian kinetic equilibrium of this type. • Finally, it is interesting to point out the peculiar behavior of the MKI functional I M ( ρ ( N )1 ( t ) , b ) and its timederivative ∂∂t I M ( ρ ( N )1 ( t ) , b ) when the Boltzmann-Grad limit is considered. In particular the 1 − and 2 − bodyoccupation coefficients k ( N )1 ( r , t ) and k ( N )2 ( r , r , t ) which appear in the Master kinetic equation (see AppendixB, Eqs.(78) and (79)) become respectively ( L BG k ( N )1 ( r , t ) = 1 L BG k ( N )2 ( r , r , t ) = 1 . (61)As a consequence the limit functionals L BG I M ( ρ ( N )1 ( t ) , b ) and L BG ∂∂t I M ( ρ ( N )1 ( t ) , b ) , are necessarily identicallyvanishing. This means that the present theory applies properly when the exact Master kinetic equation is con-sidered and not to its asymptotic approximation obtained in the Boltzmann-Grad limit, namely the Boltzmannkinetic equation (see Refs.[3, 4]).7An interesting issue, in the context of the PMI/DKE theory for the Master kinetic equation, is the role of MCBCin giving rise to the phenomena of macroscopic irreversibility and decay to kinetic equilibrium. Let us analyze forthis purpose the two cases represented by unary and binary hard-sphere elastic collisions.First, let us recall the customary treatment of collision boundary conditions for unary collision events (also referredto as the so-called mirror reflection CBC; see for example Cercignani [8, 13]). This refers to the occurrence ata collision time t i of a single unary elastic collision for particle 1 at the boundary ∂ Ω . Let us denote by n theinward normal to the stationary rigid boundary ∂ Ω at the point of contact with the same particle and respectively x ( − )1 ( t ) = (cid:16) r ( t ) , v ( − )1 ( t ) (cid:17) and x (+)1 ( t ) = (cid:16) r ( t ) , v (+)1 ( t ) (cid:17) the incoming and outgoing particle states while v (+)1 is determined by the elastic collision law for unary collisions, namely v (+)1 = v ( − )1 − n n · v ( − )1 . (62)Then, the PDF-conserving CBC for the 1 − body PDF requires that the following identity holds ρ ( N )1 ( x (+)1 ( t ) , t i ) = ρ ( N ) ( x ( − )1 ( t i ) , t i ) , (63)with ρ ( N )1 ( x (+)1 ( t ) , t i ) ≡ ρ ( N )(+)1 ( x (+)1 ( t ) , t i ) and ρ ( N ) ( x ( − )1 ( t i ) , t i ) ≡ ρ ( N )( − ) ( x ( − )1 ( t i ) , t i ) denoting the outgoing andincoming 1 − body PDF respectively. This identifies the PDF-conserving CBC usually adopted in Boltzmann kinetictheory [9] (Grad [12]; see also Refs.[2, 3]). The obvious physical implication of Eq.(63) is that ρ ( N )1 ( x (+)1 ( t ) , t i )(and ρ ( N ) ( x ( − )1 ( t i ) , t i )) should be necessarily an even function of the velocity component n · v ( − )1 . Indeed as shownin Refs.[2, 3] the PDF-conserving CBC (63) should be replaced with a suitable CBC identified with the MCBCcondition (see also Appendix C). When realized in terms of its causal form (predicting the outgoing PDF in terms ofthe incoming one) the MCBC for unary collisions is just: ρ ( N )(+)1 ( x (+)1 ( t ) , t i ) = ρ ( N )( − ) ( x (+)1 ( t i ) , t i ) , (64)with ρ ( N )( − ) ( x (+)1 ( t i ) , t i ) denoting the incoming 1 − body PDF evaluated in terms of the outgoing state x (+)1 ( t i ) . Assuming left-continuity (see related discussion in Ref.[2]), this can then be identified with ρ ( N )( − ) ( x (+)1 ( t i ) , t i ) ≡ ρ ( N ) ( x (+)1 ( t i ) , t i ) , thus yielding ρ ( N )(+)1 ( x (+)1 ( t ) , t i ) = ρ ( N ) ( x (+)1 ( t i ) , t i ) . (65)Eq.(65) provides the physical prescription for the collision boundary condition, which is referred to as MCBC, holdingfor the 1 − body PDF at arbitrary unary collision events. It is immediate to realize that the function ρ ( N ) ( x (+)1 ( t i ) , t i )need not generally be even with respect to the velocity component n · v ( − )1 . In addition Eq.(65), just as (63),also permits the existence of the customary collisional invariants which in the case of unary collisions are X =1 , (cid:12)(cid:12)(cid:12) n · v ( − )1 (cid:12)(cid:12)(cid:12) , v · (cid:2) − n n (cid:3) , v . As a consequence, one can show that Eq.(65) warrants at the same time also thevalidity of the so-called no-slip boundary conditions for the fluid velocity field V ( r , t ) carried by the 1 − body PDF ρ ( N )1 ( x , t ).The treatment of MCBC holding for the 2 − body PDF in case of binary collision events is analogous and is recalledfor convenience in Eq.(92) of Appendix C.Let us briefly analyze the qualitative physical implications of Eqs.(65) and (92) as far as the DKE theory is concerned.First, we notice that unary collisions cannot produce in a proper sense a velocity-isotropization effect since, as shownby Eq.(65), in such a case MCBC gives rise only to a change in the velocity distribution occurring during a unarycollision due to a single component of the particle velocity, namely n · v ( − )1 . As a consequence, this explains whyunary collisions do not affect the rate of change of the MKI functional (see THM.2). Second, Eq.(92) shows - on thecontrary - that binary collisions actually do affect by means of MCBC a velocity-spreading for the 1 − and 2 − bodyPDF. In particular, since the spreading effect occurs in principle for all components of particle-velocities affectingboth particles 1 and 2, this explains why binary collisions are actually responsible for the irreversible time-evolutionof the MKI functional (see THMs 2 and 3).In turn, as implied by THM.3, DKE arises because of the phenomenon of macroscopic irreversibility (THM.2). Thelatter arises due specifically to the possible occurrence of a velocity-space anisotropy which characterizes the 1 − bodyPDF when the same PDF differs locally from kinetic equilibrium. In turn, this requires also that the 1 − body PDFbelongs to the functional class of admissible stochastic PDFs n ρ ( N )1 ( x , t ) o . In difference to Boltzmann kinetic theory,8however, the key physical role is actually ascribed to the MKI functional I M ( ρ ( N )1 ( t )) rather than the Boltzmann-Shannon entropy S ( ρ ( N )1 ( t )). In fact, recalled above, the same functional remains constant in time once the Masterkinetic equation is adopted. Rather, as shown by THM.2, it is actually the Master kinetic information I M ( ρ ( N )1 ( t ))which exhibits the characteristic signatures of macroscopic irreversibility.The key differences arising between the two theories, i.e., the Boltzmann equation-DKE and the Master equation-DKE, are of course related to the different and peculiar intrinsic properties of the Boltzmann and Master kineticequations. In particular, as discussed at length elsewhere (see Refs.[1, 3]), precisely because the Boltzmann equationis only an asymptotic approximation of the Master kinetic equation explains why a loss of information occurs inBoltzmann kinetic theory and consequently the related Boltzmann-Shannon entropy is not conserved.The present investigation shows that in the context of the Master kinetic equation, the macroscopic irreversibilityproperty, i.e., the monotonic time-decay behavior of the MKI functional, can be explained at a more fundamentallevel, i.e., based specifically on the time-variation of the b − directional total kinetic energy which occurs at arbitrarybinary collision events.The Master equation-DKE theorem (THM.3) given above provides a first-principle proof of the existence of thephenomenon of DKE occurring for the kinetic description of a finite number of extended hard-spheres, i.e., describedby means of the Master kinetic equation. More precisely, the DKE phenomenon affects the 1 − body PDFs belongingto the admissible functional class n ρ ( N )1 ( x , t ) o determined according to THM.1 and requiring also that the localcharacteristic scale-length L ρ ( t ) associated with ρ ( N )1 ( x , t ) is non-zero at all times. In this paper the problem of the property of microscopic irreversibility (PMI) and decay to kinetic equilibrium(DKE) of the 1 − body PDF has been addressed. In doing so original ideas and methods are adopted of the new abinitio-theory for hard-sphere systems recently developed in the context of Classical Statistical Mechanics [1, 2].These are not just small deviations from standard literature approaches. Such developments, in fact, have openedup a host of exciting new subjects of investigation and theoretical challenges in kinetic theory which arise thanks to,or in the context of, the ab initio approach to kinetic theory. Both are based in particular on the discovery of theMaster kinetic equation first reported in Ref. [3], equation which has been adopted also in the present paper.The ab initio-theory, and specifically the present paper, represent the attempt to reach a new foundational basisand axiomatic physical description of the classical statistical mechanics for hard-sphere systems. The topic which hasbeen pursued here - which represents also a challenging test of the ab initio theory itself - concerns the investigationof the physical origins of PMI and the related DKE phenomenon arising in finite N − body hard-sphere systems. Theseissues refer in particular to: • The proof of the non-negativity of Master kinetic information (THM.1, subsection 2A) together with the propertyof macroscopic irreversibility (PMI; THM.2, subsection 2B). • The establishment of THM.3 (subsection 2C) and the related proof of the property of decay to kinetic equilibrium(DKE). • The consistency of PMI and DKE with microscopic dynamics (Section 3). • The analysis of the main physical implications of DKE (Section 4).
The theory presented here departs in several respects from previous literature and notably from Boltzmann kinetictheory. The main differences actually arise because of the non-asymptotic character of the new theory, i.e., the factthat it applies to arbitrary dense or rarefied systems for which the finite number and size of the constituent particlesis accounted for [3]. In this paper basic consequences of the new theory have been investigated which concern thephenomenon of decay to global kinetic equilibrium.The present results are believed to be crucial, besides in mathematical research, for the physical applications of the abinitio-theory statistical theory, i.e., the Master kinetic equation. Indeed, regarding challenging future developments ofthe theory one should mention among others the following examples of possible (and mutually-related) routes worth tobe explored. One is related to the investigation of the possible effects due to arbitrarily prescribed, i.e., non-vanishing,initial (binary or multi-body) phase-space statistical correlations. As recalled above, in fact, the Master equation isappropriate only when suitably-prescribed configuration-space statistical correlations are taken into account. The9second goal concerns the investigation of the time-asymptotic properties of the same kinetic equation, for which thepresent paper may represent a useful basis. The third goal refers to the possible extension of the theory to mixturesformed by hard spheres of different masses and diameter which possibly undergo both elastic and anelastic collisions.Finally, the fourth one concerns the investigation of hydrodynamic regimes for which a key prerequisite is providedby the DKE theory established here.
ACKNOWLEDGMENTS
This work is dedicated to the dearest memory of Flavia, wife of M.T., recently passed away. The investigationwas developed in part within the research projects: A) the Albert Einstein Center for Gravitation and Astrophysics,Czech Science Foundation No. 14-37086G; B) the research projects of the Czech Science Foundation GA ˇCR grantNo. 14-07753P; C) the grant No. 02494/2013/RRC “
Kinetick´y pˇr´ıstup k proud˘en´ı tekutin ” (Kinetic approach to fluidflow) in the framework of the “Research and Development Support in Moravian-Silesian Region”, Czech Republic.Initial framework and motivations of the investigation were based on the research projects developed by the Con-sortium for Magnetofluid Dynamics (University of Trieste, Italy) and the MIUR (Italian Ministry for Universitiesand Research) PRIN Research Program “
Problemi Matematici delle Teorie Cinetiche e Applicazioni ” (MathematicalProblems of Kinetic Theories and Applications), University of Trieste, Italy. One of the authors (M.T.) is grateful tothe International Center for Theoretical Physics (Miramare, Trieste, Italy) for the hospitality during the preparationof the manuscript.
APPENDIX A: REALIZATIONS OF THE MASTER KINETIC EQUATION
For completeness we recall here the two equivalent forms of the Master kinetic equation [3]. In terms of therenormalized 1 − body PDF b ρ ( N )1 ( x , t ) (see Eq.(25) ) the first form of the same equation reads L b ρ ( N )1 ( x , t ) = 0 , (66)with L = ∂∂t + v · ∂∂ r denoting the 1 − body free-streaming operator. Hence it follows L ρ ( N )1 ( x , t ) = b ρ ( N )1 ( x , t ) L k ( N )1 ( r , t ) , (67)where explicit evaluation of the rhs the last equation (see also Eq.(88) below) yields b ρ ( N )1 ( x , t ) L k ( N )1 ( r , t ) = ( N − σ Z U d v Z d Σ v · n Θ ∗ ( r ) k ( N )2 ( r , r , t ) b ρ ( N )1 ( x , t ) b ρ ( N )1 ( x , t ) , (68)with Θ ∗ ( r ) ≡ Θ ( ∂ Ω) i ( r ) and k ( N )2 ( r , r , t ) being identified with the definitions given respectively by Eqs.(71) andEq.(78) in Appendix B. Then, consistent with Ref.[3] and upon invoking the causal form of MCBC (see Eq.(93) inAppendix C) the same equation can be written in the equivalent second form of the Master kinetic equation [3]. Thecorresponding initial-value problem, taking the form: ( L ρ ( N )1 ( x , t ) − C (cid:16) ρ ( N )1 | ρ ( N )1 (cid:17) = 0 ,ρ ( N )1 ( x , t o ) = ρ ( N )1 o ( x ) , (69)can be shown to admit a unique global solution [4]. Here the notation is standard [3]. Thus C (cid:16) ρ ( N )1 | ρ ( N )1 (cid:17) ≡ ( N − σ Z U d v Z ( − ) d Σ hb ρ ( N )1 ( r , v (+)1 , t ) b ρ ( N )1 ( r , v (+)2 , t ) − b ρ ( N )1 ( r , v , t ) b ρ ( N )1 ( r , v , t ) i ×| v · n | k ( N )2 ( r , r , t )Θ ∗ ( r ) (70)0identifies the Master collision operator, while ρ ( N )1 o ( x ) is the initial 1 − body PDF which belongs to the functionalclass n ρ ( N )1 o ( x ) o of stochastic, i.e., strictly-positive, smooth ordinary functions, 1 − body PDFs. Furthermore, thesolid-angle integral on the rhs of Eq.(70) is now evaluated on the subset in which v · n < , while r identifies r = r + σ n , while k ( N )1 ( r , t ) and k ( N )2 ( r , r , t ) coincide respectively with the 1 − and 2 − body occupation coefficients[3] and Θ ∗ ≡ Θ ∗ ( r i ) is prescribed by Θ ∗ ( r i ) ≡ Θ ( ∂ Ω) i ( r ) ≡ Θ (cid:16)(cid:12)(cid:12)(cid:12) r i − σ n i (cid:12)(cid:12)(cid:12) − σ (cid:17) , (71)with Θ( x ) being the strong Heaviside theta function Θ( x ) = (cid:26) y > y ≤ S N − ensemble strong theta-function Θ ( N ) . The latter is prescribed, according to Ref.[3], by requiringthat Θ ( N ) ( r ) = 1 (72)for all confguration vectors r ≡ { r , ..., r N } belonging to the collisionless subset of Ω ( N ) . This is identified with theopen subset of the N − body configuration domain Ω ( N ) ≡ Q i =1 ,N Ω in which each of the particles of S N is not in mutualcontact with any other particle of S N or with the boundary ∂ Ω of Ω . This means that at any configuration r , Θ ( N ) ( r )can be prescribed as Θ ( N ) ( r ) ≡ Y i =1 ,N Θ i ( r )Θ ( ∂ Ω) i ( r ) . (73)Here Θ ( ∂ Ω) i ( r ) identifies the i − th particle ”boundary” theta functionΘ ( ∂ Ω) i ( r ) ≡ Θ ( ∂ Ω) i ( r i ) = Θ (cid:16) | r i − r W i | − σ (cid:17) , (74)with r W i = r i − ρ n i and ρ n i being the inward vector normal to the boundary belonging to the center of the i − thparticle having a distance ρ/ i ( r ) is the ”binary-collision” theta function.A possible identification of Θ i ( r ) which warrants validity of Eq.(72) is given by the expressionΘ i ( r ) ≡ Y j =1 ,N ; i One notices that although the definitions (77) and (75) given in Appendix A for Θ i ( r ) coincide in the collisionlesssubset of Ω ( N ) , only the first one is applicable in the complementary collision subset. Based on these premises in thisappendix a number of integral and differential identities holding for the 1 − and 2 − body occupation coefficients aredisplayed.First, recalling Ref.[3], one notices that the realizations of the 1 − and s − body occupation coefficients k ( N )1 ( r i , t ) , k ( N )2 ( r , r , t ) , ..., k ( N ) s ( r , r , .. r s , t ) remain uniquely prescribed by the 1 − body PDF, being given by k ( N )1 ( r , t ) ≡ F Y j =2 ,N ρ ( N )1 ( x j , t ) k ( N )1 ( r j , t ) , (78) k ( N )2 ( r , r , t ) ≡ F Y j = s +1 ,N ρ ( N )1 ( x j , t ) k ( N )1 ( r j , t ) , (79) ...k ( N ) s ( r , r , .. r s , t ) ≡ F s Y j = s +1 ,N ρ ( N )1 ( x j , t ) k ( N )1 ( r j , t ) , (80)where F s denotes the integral operator F s ≡ Z Γ N d x Θ ( N ) ( r ) Y i =1 ,s δ ( x i − x i ) . (81)Therefore, since in the collisionless subset of Ω ( N ) the prescriptions (75) and (77) are equivalent, in the same subsetthe 1 − and 2 − body occupation coefficients, written in terms of Eq.(75), become explicitly k ( N )1 ( r , t ) = Z Γ d x ρ ( N )1 ( x , t ) k ( N )1 ( r , t ) Θ ( ∂ Ω)2 ( r )Θ ( | r − r | − σ ) × Z Γ d x ρ ( N )1 ( x , t ) k ( N )1 ( r , t ) Θ ( ∂ Ω)3 ( r ) Y j =1 , Θ ( | r − r j | − σ ) ....... Z Γ N ) d x N ρ ( N )1 ( x N , t ) k ( N )1 ( r N , t ) Θ ( ∂ Ω) N ( r ) Y j =1 ,N − Θ ( | r N − r j | − σ ) , (82)and k ( N )2 ( r , r , t ) = Z Γ d x ρ ( N )1 ( x , t ) k ( N )1 ( r , t ) Θ ( ∂ Ω)3 ( r ) Y j =1 , Θ ( | r − r j | − σ ) × Z Γ d x ρ ( N )1 ( x , t ) k ( N )1 ( r , t ) Θ ( ∂ Ω)4 ( r ) Y j =1 , Θ ( | r − r j | − σ ) ... (83) .... Z Γ N ) d x N ρ ( N )1 ( x N , t ) k ( N )1 ( r N , t ) Θ ( ∂ Ω) N ( r ) Y j =1 ,N − Θ ( | r N − r j | − σ ) . (84)Accordingly letting n jj = r uij / | r ij | with r ij = r i − r j , one notices that in the collisionless subset of Ω ( N ) the following2differential identities hold for all s = 1 , N − ∂∂ r k ( N )1 ( r , t ) = ( N − Z Γ d x n δ ( | r − r | − σ ) × k ( N )2 ( r , r , t )Θ ( ∂ Ω)2 ( r ) b ρ ( N )1 ( x , t ) , (85) ∂∂ r k ( N )2 ( r , r , t ) = ( N − R Γ d x n δ ( | r − r | − σ ) × Q j =1 , j =1 Θ ( | r − r j | − σ ) Θ ( ∂ Ω)3 ( r ) k ( N )3 ( r , r , r , t ) b ρ ( N )1 ( x , t ) , ∂∂ r k ( N )2 ( r , r , t ) = ( N − R Γ d x n δ ( | r − r | − σ ) × Q j =1 , j =2 Θ ( | r − r j | − σ ) Θ ( ∂ Ω)3 ( r ) k ( N )3 ( r , r , r , t ) b ρ ( N )1 ( x , t ) , (86) ............... ∂∂ r k ( N ) s ( r , r , .., r s , t ) = ( N − s ) R Γ s +1) d x s +1 n s +1 δ ( | r s +1 − r | − σ ) × Q j =1 ,s ; j =1 Θ ( | r s +1 − r j | − σ ) Θ ( ∂ Ω)3 ( r ) k ( N ) s +1 ( r , r , ., r s +1 , t ) b ρ ( N )1 ( x s +1 , t ) ∂∂ r k ( N ) s ( r , r , .., r s , t ) = ( N − s ) R Γ d x s +1 n s +1 δ ( | r s +1 − r | − σ ) × Q j =1 ,s ; j =2 Θ ( | r s +1 − r j | − σ ) Θ ( ∂ Ω)3 ( r ) k ( N ) s +1 ( r , r , ., r s +1 , t ) b ρ ( N )1 ( x s +1 , t ) ..... ∂∂ r s k ( N ) s ( r , r , .., r s , t ) = ( N − s ) R Γ s +1) d x s +1 n ss +1 δ ( | r s +1 − r s | − σ ) × Q j =1 ,s ; j = s Θ ( | r s +1 − r j | − σ ) Θ ( ∂ Ω)3 ( r ) k ( N ) s +1 ( r , r , ., r s +1 , t ) b ρ ( N )1 ( x s +1 , t ) (87)As a consequence the following identities (the first one needed to evaluate the rhs of Eq.(67) in Appendix A) L k ( N )1 ( r , t ) = ( N − Z Γ d x v · n δ ( | r − r | − σ )Θ ∗ ( r ) k ( N )2 ( r , r , t ) b ρ ( N )1 ( x , t ) , (88) ∂ k ( N )1 ( r , t ) ∂ r · ∂ r = − ( N − Z Γ d x k ( N )2 ( r , r , t ) δ ( | r − r | − σ )Θ ( ∂ Ω)2 ( r ) n · ∂∂ r b ρ ( N )1 ( x , t ) , (89)hold too. However, the alternative realization of the factor Θ i ( r ) given by Eq.(77) (see Appendix A) has the virtueof excluding explicitly multiple collisions. The consequence is that when such a definition is adopted the differentialidentities ( δ ( | r − r | − σ ) ∂∂ r k ( N )2 ( r , r , t ) = 0 ,δ ( | r − r | − σ ) ∂∂ r k ( N )2 ( r , r , t ) = 0 , (90)both hold identically. The latter equations, in fact, manifestly hold also in the collision subset where δ ( | r − r | − σ ) =0 . APPENDIX C: CAUSAL AND ANTI-CAUSAL FORMS OF COLLISIONAL BOUNDARY CONDITIONS For definiteness, let us denote respectively the outgoing and incoming N − body PDFs ρ ( − )( N ) ( x ( − ) ( t i ) , t i ) and ρ (+)( N ) ( x (+) ( t i ) , t i ), with ρ ( ± )( N ) ( x ( ± ) ( t i ) , t i ) = lim t → t ( ± ) i ρ ( N ) ( x ( t ) , t ), where x ( − ) ( t i ) and x (+) ( t i ), with x ( ± ) ( t i ) =3lim t → t ( ± ) i x ( t ) , are the incoming and outgoing Lagrangian N − body states, their mutual relationship being againdetermined by the collision laws holding for the S N − CDS. Here it is understood that: • The S N − CDS is referred to a reference frame O ( r , τ ≡ t − t o ), having respectively spatial and time origins atthe point O which belongs to the Euclidean space R and at time t o ∈ I . • In addition, by assumption the time-axis is oriented. Such an orientation is referred to as microscopic arrow oftime .For an arbitrary N − body PDF ρ ( N ) ( x , t ) belonging to the extended functional setting and an arbitrary collisionevent occurring at time t i two possible realizations of the MCBC can in principle be given, both yielding a relationshipbetween the PDFs ρ (+)( N ) and ρ ( − )( N ) . In the context of the ab initio statistical approach based on the Master kineticequation [1–3] these are provided by the two possible realizations of the so-called modified CBC (MCBC). Whenexpressed in Lagrangian form they are realized respectively either by the causal and anti-causal MCBC , namely ρ (+)( N ) ( x (+) ( t i ) , t i ) = ρ ( − )( N ) ( x (+) ( t i ) , t i ) , (91)or ρ ( − )( N ) ( x ( − ) ( t i ) , t i ) = ρ (+)( N ) ( x ( − ) ( t i ) , t i ) . (92)The corresponding Eulerian forms of the MCBC can easily be determined (see Ref.[2]). The one corresponding toEq.(91) is, for example, provided by the condition ρ (+)( N ) ( x (+) , t ) = ρ ( − )( N ) ( x (+) , t ) , (93)where now x (+) denotes again an arbitrary outgoing collision state.Once the time-axis is oriented, i.e. , the microscopic arrow of time is prescribed, the validity of the causality principlein the reference frame ( r , τ ≡ t − t o ) manifestly requires invoking Eq.(91). Indeed, Eq.(91) predicts the future ( i.e. ,outgoing) PDF from the past (incoming) one. Therefore the choice (91) is the one which is manifestly consistent withthe causality principle. On the other hand, if the arrow of time is changed, i.e. the time-reversal transformation withrespect to the initial time (or time-origin) t o , i.e. , the map between the two reference frames O ( r , τ ≡ t − t o ) → O ( r , τ ′ ) , (94)with τ ′ = − τ is performed, it is obvious that for the transformed reference frame O ( r , τ ′ ) the form of CBC consistentwith causality principle becomes that given by Eq.(92). Analogous conclusions hold if a velocity-reversal is performed,implying the incoming states and corresponding PDF must be exchanged with corresponding outgoing ones and viceversa. APPENDIX D: TREATMENT OF CASE N = 2 For completeness let us briefly comment on the particular realization of MPI/DKE theory which is achieved in thespecial case N = 2 . For this purpose, one notices that - thanks to Eq.(79) recalled in Appendix B (see also Ref.[3]) -in this case by construction k ( N )2 ( r , r , t ) simply reduces to k ( N )2 ( r , r , t ) ≡ . (95)Accordingly, once the same prescription is invoked, both the Master kinetic equation (69) and the correspondingMaster collision operator (70) remain formally unchanged. In a similar way it is important to remark that theexpression of the functional ∂∂t K M ( ρ ( N )1 ( t ) , b ) ≡ W M ( ρ ( N )1 ( t ) , b ) given by Eq. (51) is still correct also in such a case,being now given by ∂∂t K M ( ρ ( N )1 ( t ) , b ) ≡ W M ( ρ ( N )1 ( t ) , b ) = − ( N − σ Z U d v Z U d v × Z Ω d r Z ( − ) d Σ ∂ b ρ ( N )1 ( r , v (+)1 , t ) ∂ r · ∂ b ρ ( N )1 ( r = r + σ n , v (+)2 t ) ∂ r × (cid:12)(cid:12)(cid:12) v (+)12 · n (cid:12)(cid:12)(cid:12) ( b · n ) (cid:16) n · v (+)12 (cid:17) ≤ . (96)4It is then immediate to infer the validity of both the PMI theorem (THM.2) and the DKE property for the Masterkinetic equation (THM.3). As a consequence one concludes that MPI/DKE theory holds also in the special case N = 2. This conclusion is not unexpected. 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