Research Article | Open Access
Massimo Tessarotto, Claudio Cremaschini, "Macroscopic Irreversibility and Decay to Kinetic Equilibrium of the 1-Body PDF for Finite Hard-Sphere Systems", Advances in Mathematical Physics, vol. 2018, Article ID 1931308, 19 pages, 2018. https://doi.org/10.1155/2018/1931308
Macroscopic Irreversibility and Decay to Kinetic Equilibrium of the 1-Body PDF for Finite Hard-Sphere Systems
The conditions for the occurrence of the so-called macroscopic irreversibility property and the related phenomenon of decay to kinetic equilibrium which may characterize the 1-body probability density function (PDF) associated with hard-sphere systems are investigated. The problem is set in the framework of the axiomatic “ab initio” theory of classical statistical mechanics developed recently and the related establishment of an exact kinetic equation realized by the Master equation for the same kinetic PDF. As shown in the paper the task involves the introduction of a suitable functional of the 1-body PDF, identified here with the Master kinetic information. It is then proved that, provided the same PDF is prescribed in terms of suitably smooth, i.e., stochastic, solution of the Master kinetic equation, the two properties indicated above are indeed realized.
The axiomatic theory of classical statistical mechanics (CSM) recently proposed in a series of papers (see [1–4]) and referred to as ab initio theory of CSM provides a self-consistent pathway to the kinetic theory of hard-sphere systems, as well as in principle also point particles subject to finite-range interactions . Its theoretical basis and conditions of validity are indeed founded on a unique physical realization of the axioms which are set at the foundations of CSM [1–3], a fact which permits the treatment of phase-space and kinetic probability density functions (PDF) which are realized by either stochastic (i.e., ordinary) functions or distributions such as the body Dirac delta (or certainty function ). This feature is physically based being due to the prescription of the collision boundary conditions (CBC, ), i.e., the relationship occurring at collision events between incoming and outgoing multibody probability density functions PDF. The choice of the appropriate CBC indicated in , denoted as modified collision boundary condition (MCBC), is actually of crucial importance and departs from the customary realization/interpretation (of the same axioms) originally adopted in Boltzmann , Enskog [8, 9], and Grad  kinetic approaches (for a review of Grad’s kinetic theory based on CSM see also Cercignani [11, 12]). The same choice implies, in fact, a number of theoretical and physically relevant consequences. In particular, it follows that the new theory(i)unlike Enskog theory  applies also to finite body hard-sphere systems , namely, systems formed by like smooth hard-spheres of diameter and mass , in which the parameters (, ) remain all constant and finite . 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) immersed in a stationary bounded domain of the Euclidean space with finite canonical measure;(ii)has led to the discovery  of an exact kinetic equation holding globally in time  (i.e., for all ) for these systems and denoted as Master kinetic equation (recalled in Appendix A). In other words the Master equation is nonasymptotic in character with respect to the parameters the (finite) parameters (, ). In addition the same equation holds under suitable maximal entropy conditions for the statistical treatment of the so-called Boltzmann-Sinai classical dynamical system (CDS), which implies that initial (binary or multibody) 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 validity of the Boltzmann and Enskog kinetic equations and notably applies to arbitrary body PDFs which can be realized either in terms of stochastic functions or distributions;(iii)is time-reversal invariant , namely, the Master kinetic equation is time-reversal () symmetric. In other words, the same equation is invariant with respect to the transformation Thus, representing the absolute time as , with being a prescribed (arbitrary) initial time, it follows that the transformation leaves invariant the initial time and the instantaneous position of an arbitrary particle, while reversing the signature (i.e., versus) of its velocity Accordingly, thanks to symmetry the two initial-value problems associated with the Master kinetic equation in the two cases are related in such a way that, respectively, the initial body PDF at time , , and the corresponding time-evolved PDF are carried into the transformed body PDFs and , respectively, prescribed according to the law(iv)conserves the corresponding Boltzmann-Shannon (BS) statistical entropy . This is identified with the phase-space moment with being an arbitrary stochastic PDF solution of the Master kinetic equation and an arbitrary positive constant such that the initial PDF is such that the corresponding BS functional is defined. As a consequence it follows that an arbitrary smooth solution of the Master kinetic equation satisfies the constant H-theorem for all (see again related discussion in ).
Based on the ab initio theory of CSM, in this paper the problem is posed of the existence of two phenomena which are expected to characterize the statistical description of finite body hard-sphere systems and therefore should lay at the very foundation of CSM and kinetic theory. These are related to the physical conditions for the possible occurrence of the so-called property of macroscopic irreversibility (PMI) and the consequent one represented by the decay to kinetic equilibrium (DKE) which characterize the body (kinetic) PDF in these body systems, i.e., when body-factorized initial conditions are considered for the body Liouville equation . The conjecture is that—in some sense in analogy with the ubiquitous character of the ergodicity property which characterizes hard-sphere systems and hence the CDS [14, 15]—the occurrence of such phenomena should be independent of the number of constituent particles of the system and therefore apply to actual physical systems for which the parameters , are obviously all finite.
1.1. Motivations and Background
Both properties indicated above concern the statistical behavior of an ensemble of like particles which are advanced in time by a suitable body classical dynamical system, identified here with the CDS. Specifically they arise in the context of the kinetic description of the same CDS, i.e., in terms of the corresponding body (kinetic) probability density function (PDF) The latter is required to belong to the functional class of suitably smooth and strictly positive ordinary functions which are particular solutions of the relevant kinetic equation.
In fact, PMI should be realized by means of a suitable, but still possibly nonunique, functional which should be globally defined in the future (i.e., for all times , being the initial time) bounded and nonnegative, and therefore 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 and monotonic, i.e., in particular decreasing, time-dependence.
Regarding, instead, the second property of DKE this concerns the asymptotic behavior of the body PDF which, accordingly, should be globally defined and decay for to a stationary and spatially uniform Maxwellian PDFwhere are constant fluid fields.
Both PMI and DKE correspond to physical phenomena which are actually expected to arise in disparate classical body systems. The clue for their realization is represented by the ubiquitous occurrence of kinetic equilibria and consequently, in principle, also of the corresponding possible manifestation of macroscopic irreversibility and decay processes. Examples of the former ones are in principle easy to be found, ranging from neutral fluids  to collisional/collisionless and nonrelativistic/relativistic gases and plasmas [17–19].
However, the most notable example is perhaps provided by dilute hard-sphere systems (“gases”) characterized by a large number of particles () and a small (i.e., infinitesimal) diameter of the same hard-spheres, for which the Boltzmann equation applies. Indeed the Boltzmann equation is actually specialized to the treatment of dilute hard-sphere systems in the Boltzmann-Grad limit discussed in the Lanford theorem [20–22] (for a detailed discussion of the topics in the context of the ab initio theory see also ). In such a case the body PDF can be formally obtained by introducing the Boltzmann-Grad limit operator  whereby the limit function is denotedand 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 of Classical Thermodynamics. More precisely, it is related to the first-principle-proof originally attempted by Ludwig Boltzmann in 1872 . Actually it is generally agreed that both phenomena lie at the very heart of Boltzmann and Grad kinetic theories [7, 10] and the related original construction of the Boltzmann kinetic equation (1872). In particular, the goal set by Boltzmann himself in his 1872 paper was the proof of Carnot’s Law providing at the same time also a possible identification of thermodynamic entropy. This was achieved in terms of what is nowadays known as Boltzmann-Shannon (BS) statistical entropy, which is identified with the phase-space momentHere , , and denote, respectively, the BS entropy density, an arbitrary particular solution 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  the same functional should satisfy the entropic inequalitywhile, furthermore, the entropic equality conditionshould hold. The latter equation implies therefore that, provided and exist globally , then necessarily , with denoting the stationary and spatially uniform Maxwellian PDF (6).
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., affecting only the BS entropy through the Boltzmann H-theorem indicated above, or microscopic in the sense that the same Boltzmann equation should be considered as irreversible (“microscopic irreversibility”). Thus, in principle, in the second case the further issue emerges of the possible physical origin of microscopic irreversibility in special reference to the Lanford’s derivation of the Boltzmann equation and subsequent related comments discussed respectively by Uffink and Valente and Ardourel in [24, 25] (see also Drory ).
However, as shown in , the Boltzmann equation is actually symmetric. Such a conclusion is of basic importance since it overcomes the so-called Loschmidt paradox, i.e., the objection raised by Loschmidt in 1876  regarding the original Boltzmann formulation of his namesake kinetic equation and H-theorem . In fact, Loschmidt claimed that the Boltzmann H-theorem inequality should change sign under time-reversal and thus violate the microscopic time-reversibility of the underlying hard-sphere classical dynamical system. In his long-pondered reply given in 1896  Boltzmann himself introduced what was later referred to as the modified form of the Boltzmann H-theorem .
The key implication is therefore that, in contrast to Boltzmann’s own statement and the traditional subsequent mainstream literature interpretation (see, for example, by Cercignani, Lebowitz in [30, 31] and more recently the review given by Gallavotti ), 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 for the Boltzmann equation itself. On the contrary, consistent with the physical interpretation of the Loschmidt paradox provided in , this must be regarded only as property of macroscopic irreversibility (or PMI) of the body PDF solution of the Boltzmann equation. In other words, the Boltzmann inequality (10) necessarily holds independent of the orientation of the time axis (arrow of time) and therefore cannot represent a true (i.e., microscopic) property which 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 critical way on the prescription of the functional class , so that their occurrence is actually nonmandatory. Indeed, both cannot occur—in principle also for Boltzmann and Grad kinetic theories—if the body probability density function is identified with the deterministic body PDF , namely, the body phase-space Dirac delta. This is defined as , with denoting the state of the body system and is the image of an arbitrary initial state generated by the same body CDS. That such a PDF necessarily must realize an admissible particular solution of the body Liouville equation follows, in fact, as a straightforward consequence of the axioms of classical statistical mechanics .
Despite these premises, however, the case of a finite Boltzmann-Sinai CDS, which is characterized by a finite number of particles and/or a finite-size of the hard-spheres and/or a dense or locally dense system, is more subtle and—as explained below—even unprecedented since it has actually remained unsolved to date. The reasons are as 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 of the Master kinetic equation, i.e., the functional , in contrast to , is exactly conserved in the sense that identicallymust hold. As a consequence the validity itself of Boltzmann H-theorem breaks down in the case of the Master kinetic equation. Third, an additional motivation is provided by the conjecture that both PMI and DKE might occur only if the Boltzmann-Grad limit is actually performed, i.e., only in validity of Boltzmann equation and H-theorem.
Hence the question which arises is whether in the case of a finite Boltzmann-Sinai CDS the phenomenon of DKE may still arise. Strong indications seem to be hinting at such a possibility. In this regard the example-case which refers to the statistical description of a Navier-Stokes fluid described by the incompressible Navier-Stokes equations (INSE) in terms of the Master kinetic equation is relevant and suggests that this may be indeed the case. In fact, thanks also to comparisons with the mean-field inverse kinetic approach to INSE , in such a case the decay of the fluid velocity field occurring in a bounded domain necessarily demands the existence of DKE. In other words, in the limit the body PDF must decay uniformly to the stationary and spatially uniform Maxwellian PDF (6).
However, besides the construction of the kinetic equation appropriate for such a case, a further unsolved issue lies in the determination of the functional class for which both PMI and DKE should/might be realized. In particular, the possible occurrence of both PMI and DKE should correspond to suitably smooth, but nonetheless still arbitrary, initial conditions . These should warrant that in the limit , uniformly converges to the spatially homogeneous and stationary Maxwellian PDF (6). Such a result, however, is highly nontrivial since it should rely on the establishment of a global existence theorem for the same body PDF —namely, holding in the whole time axis , besides the same body phase space —for the involved kinetic equation which is associated with the 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 related to the asymptotic nature of the Boltzmann equation . In fact for the same equation it is not known in satisfactory generality whether smooth enough solutions of the same equation exist which satisfy the theorem inequality and decay asymptotically to kinetic equilibrium [23, 30].
1.2. 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 body hard-sphere systems and therefore should lay at the very foundation of classical statistical mechanics and kinetic theory alike. These are related to the physical conditions for the possible occurrence of both PMI and the consequent one represented by the possible occurrence of DKE which should characterize the kinetic PDF in these systems. These phenomena are well known to occur in the case of dilute hard-sphere systems, i.e., in the Boltzmann-Grad limit. In particular, for an exhaustive treatment of the related issues which arise in the context of the ab initio theory we refer to discussions reported in . Nevertheless, as indicated above, their existence in the case of finite hard-sphere systems is partly motivated by a previous investigation dealing with the kinetic description incompressible Navier-Stokes granular fluids .
Therefore, main goal of the paper is to show that these properties actually emerge as necessary implications of the ab initio theory of CSM. Incidentally, in doing so, the Master kinetic equation must be necessarily adopted. In fact, the finiteness requirement on the CDS rules out for further possible consideration either the Boltzmann or the Enskog kinetic equations, these equations being inapplicable to the treatment of systems of this type . Specifically, in the following the case is considered everywhere, which is by far the most physically relevant one. In this occurrence, in fact, nontrivial body occupation coefficients arise (see related notations which are applicable for recalled in Appendices A and B below). For completeness the case is nevertheless briefly discussed in Appendix D.
For this purpose, first, in Section 2, the MKI functional is explicitly determined. We display in particular its construction method (see No.#1- #4 MKI Prescriptions). Based on the theory of the Master kinetic equation earlier developed  and suitable integral and differential identities (see Appendices A, B, and C), the properties of the MKI functional are investigated. These concern in particular the establishment of appropriate inequalities holding for the same functional (Theorem 1, Section 2.1), the signature of the time derivative of the same functional (Theorem 2, Section 2.2) and the property of DKE holding for a suitable class of body PDFs (Theorem 3, Section 2.3). In the subsequent Sections 3 and 4, the issue of the consistency of the phenomena of PMI and DKE with microscopic dynamics is posed together with the physical interpretation and implications of the theory. The goal is to investigate the relationship of the DKE theory developed here with the microscopic reversibility principle and the Poincaré recurrence theorem. Finally in Section 5 the conclusions of the paper are drawn and possible applications/developments of the theory are pointed out.
2. Axiomatic Prescription of the MKI Functional
In view of the considerations given above in this section the problem is posed of the explicit realization of the MKI functional in terms of suitable axiomatic prescriptions. The same functional, denoted , should depend on the body PDF , with being identified with a particular solution of the Master kinetic equation (see (A.4) in Appendix A holding for and Appendix D for the case ).
Unlike Boltzmann kinetic equation, the Master kinetic equation actually deals with the treatment of finite hard-sphere body systems, i.e., in which both the number of particles and their diameter remain finite . To achieve such a goal suitably prescribed physical collision boundary conditions (CBC) of the 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 , with denoting the corresponding incoming (−) and outgoing (+) states. In particular, upon invoking due to causality the assumption of left-continuity, i.e., the requirementthe incoming PDF is required to coincide with the same body PDF evaluated in terms of the incoming state and time [1, 3]. Hence, as recalled in Appendix C (see also ) from (C.1) it follows that the so-called causal form of the modified collision boundary condition (MCBC )is mandatory. A further important requirement concerns precisely setting also the related functional class of admissible solutions in such a way that, besides , also the same functional exists globally for arbitrary . For definiteness, we shall consider for this purpose the case of body PDFs which satisfy the initial conditionwith belonging to the functional class of stochastic body PDFs . For a generic belonging to the time axis this is the ensemble of body PDFs 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 exist which correspond either to arbitrary monomial functions of (or its components , for ) or to the entropy density , thus yielding the Boltzmann-Shannon (BS) entropy evaluated in terms of .
Concerning the choice of the setting the following remarks are in order. As a first remark, the previous requirements (A), (B), and (C) for , together with validity of MCBC (14), actually should warrant that the corresponding solution of the Master kinetic equation exists globally in the extended phase-space and that for all the same PDF belongs to the class of stochastic PDFs indicated above and also fulfills identically the constant H-theorem (12). Indeed, one can show  that global existence of solutions for the Master kinetic equation follows in elementary way from the body Liouville equation. Indeed, an arbitrary body PDF which is a particular solution of the Master kinetic equation realizes by construction also a particular factorized solution of the body Liouville equation, i.e., of the body PDF . The same PDF evolves uniquely in time along arbitrary phase-space Lagrangian trajectories, its Lagrangian time evolution being determined at arbitrary collision times by MCBC (14) .
As a second remark, the validity of assumptions (A) and (B) for and implies also suitable assumptions to apply all for the local characteristic scale length which characterize the same PDF More precisely, this is associated with the spatial variations of the body PDF prescribed aswhich necessarily assumed nonzero at all time . Hence is assumed to be bounded for all spanning the extended body phase space .
Finally, as a third remark (see also the further related discussion in Theorem 2 below), the previous requirements are expected to warrant also the global existence of the MKI functional , so that effectively realizes the functional class of admissible solutions indicated above.
Given these premises let us pose now the problem of the identification of the functional , based on the introduction of ’ad hoc’ physical requirements, to be referred to here as MKI Prescriptions No.#1-#4. The prescriptions are as follows:(i) MKI Prescription No.#1: the first one is that the functional should be determined in such a way that the existence of at a suitable initial time should warrant also that must necessarily exist globally in the future, i.e., for all As a consequence the functional class must be suitably prescribed.(ii) MKI Prescription No.#2: second, we shall require that is real, nonnegative, and bounded in the sense that This implies that can be interpreted as an information measure associated with the body PDF . For this reason the previous inequalities will be referred to as information-measure inequalities.(iii) MKI Prescription No.#3: third, for consistency with the property of macroscopic irreversibility, is prescribed in terms of a smoothly time-differentiable and monotonically time-decreasing functional in the sense that in the same time-subset the inequality should identically apply , so that This implies that is also globally defined for all with In addition, if , without loss of generality its initial value , can always be set such that(iv) MKI Prescription No.#4: fourth, in order to warrant the existence of DKE we shall require the functional to be prescribed in such a way that at an arbitrary time , with , the vanishing of both and its time derivative should occur if and only if the body PDF solution of the Master kinetic equation coincides with kinetic equilibrium. As a consequence, for the functional the following propositions should be equivalent: with being a kinetic equilibrium PDF of the form (6).
The immediate obvious implication of the previous prescriptions is that—provided a nontrivial realization of the MKI can be found in the functional class —the existence of both PMI and DKE for the Master kinetic equation is actually established. In the sequel the goal is to show, in particular, that the MKI functional can be identified by means of the prescriptionwhere and denote, respectively, a suitable (and possibly nonunique) moment-dependent phase-space functional and an appropriate normalization constant to be chosen in such a way to satisfy all the MKI prescriptions indicated above. In particular, as shown below, an admissible choice for and is provided bywhile denotes the directional kinetic energy (along the unit vector ) carried by particle , namely, the dynamical variablewith denoting a still arbitrary constant unit vector. Hence,identifies the corresponding total directional kinetic energy carried by particles and . Here the remaining notation is standard. Thus, , , and are, respectively, the body PDF solution of the initial problem associated with the Master kinetic equation (see (A.4) in Appendix A), the initial PDF, and the renormalized body PDFwhile furthermore is the body occupation coefficient recalled in Appendix B (see (B.1)). As a consequence in the previous equation it follows that . Furthermore, is the boundary theta function given by (A.10) (see Appendix A). Finally, regarding the initial value it follows that if, respectively, orthen correspondingly one obtains, consistent with (19), that the initial value of MKI functional is
2.1. Proof of the Nonnegativity of the MKI Information Measure
The strategy adopted for the proof of the MKI Prescriptions No.#1 and No.#2 is to show initially the validity of the information-measure left inequality in (17), namely, that cannot acquire negative values for arbitrary . The result is established by the following theorem.
Theorem 1 (nonnegativity of , , and ). Let us assume that is an arbitrary stochastic and suitably smoothly differentiable, particular solution of the Master kinetic equation (A.4) prescribed so that the integral (23) expressed in terms of the initial PDF, namely, , is nonvanishing. Then, it follows necessarily that(i)Proposition :(ii)Proposition : the corresponding time-evolved functional for all with is such that(iii)Proposition : for all with the functional fulfills the inequality(iv)Proposition : the following necessary and sufficient condition holds at a given time with :
Proof. One first notices that can be equivalently written in the form where in order that the same functional exists it is obvious that the renormalized body PDF must be of class . Integrating by parts and noting that the gradient term gives a vanishing contribution to the phase-space integral, this yields equivalently Therefore, upon invoking (B.9) reported in Appendix B, direct substitution deliversNext, invoking the identity and noting again that gives vanishing contribution, one can perform a further integration by parts with respect to . This permits to cast the rhs of previous equation in the formHere the two terms on the rhs of (36) are defined as follows: the first term is symmetric and nonnegative, so that it can be expressed so to carry the total directional kinetic energy of particles and (see (25)). Hence, it takes the form The second term reads insteadwhere is given by the differential identity (B.10) reported in Appendix B. Thus, upon invoking the identity , one notices that an integration by parts can be performed also with respect to This means that a procedure analogous to the one used for the calculation of can be invoked and iterated at all orders, i.e., up to the body occupation coefficient (see (B.11) in Appendix B). As a consequence the functional can be represented in terms of a finite sum of the form in which each term of the sum is nonnegative and symmetric. This implies therefore that the same functional can be cast in the formwith being the total directional kinetic energy (25) and a suitable real scalar kernel which is symmetric in the variables and Hence actually defines a nonnegative functional. This proves the validity of the inequality (31) (Proposition ).
In a similar way also the remaining propositions can be established. In fact, invoking (28) it follows that the inequalities (30) and (31)—and hence also Propositions and —manifestly hold too. Finally, regarding the proof of Proposition , one notices that if and only if identically . Since is by construction a solution of the Master kinetic equation it follows that this requires necessarily that must coincide with the local Maxwellian (see (6)) and hence (32) must hold too under the same realization (Proposition ).
The conclusion is therefore that the definition of the MKI functional (22) given above in terms of and (see (23)) is indeed consistent with the physical prerequisites represented by the MKI Prescriptions No.#1 and No.#2.
2.2. Proof of PMI for the Master Kinetic Equation
The next step is to prove that the functional defined above (see (28)) indeed exhibits a monotonic time-decreasing behavior which is consistent with the MKI Prescriptions No.#3 and No.#4, which are realized respectively by(i)the time derivative inequality (18) and the conditions of existence of kinetic equilibrium (21);(ii)the validity of the inequality .
In order to reach the proofs of these properties let us preliminarily determine the variation across a binary collision occurring between particles and of the total directional kinetic energy (see (25)), namely, the phase-space scalar function . One obtainsthe rhs being expressed in terms of the outgoing particle velocities only. Then, the following proposition holds.
Theorem 2 (property of macroscopic irreversibility (master equation PMI theorem)). Let us assume that is an arbitrary stochastic particular solution of the Master kinetic equation (A.4) with initial condition such that the integral exists and is nonvanishing. Then it follows that(i)Proposition : one finds that for all (ii)Proposition : the inequality holds globally (i.e., identically for all ) so that necessarily is globally defined too, being also prescribed so that(iii)Proposition : one finds that a given time with
Proof. Consider first the proof of Proposition which requires evaluation of the partial time derivative Upon invoking the first form of the Master kinetic equation (see (A.1) in Appendix A), explicit differentiation of delivers namely, upon integration by parts in the first integral on the rhs, Hence, thanks to the differential identity (B.12) it follows Performing an integration by parts with respect to and upon invoking the first differential identity (B.14) reported in Appendix B one obtains thereforewhere . Hence performing a further integration by parts with respect to and using the second differential identity on (B.14) (see Appendix B) the previous equation finally yieldswhere the symmetry property with respect to the exchange of states has been invoked. In the previous equation the integration on the Dirac delta can be performed at once letting 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 (C.3) in Appendix C) the integral on outgoing particles can be transformed to a corresponding integration on incoming ones, namely, Thus, the contributions in the two phase-space integrals only differ because of the variation of the total directional kinetic energy of particles and This implies thatwhere the solid-angle integration is performed on the incoming particles whereas is evaluated in terms of the outgoing particles and therefore must be identified with the second equation on the rhs of (40). Consider now the dependence in terms of the outgoing particle velocities and in the previous phase-space integral. The velocity dependence contained in the factors and is symmetric with respect to the variables and On the other hand, as a whole, the same integral should remain unaffected with respect to the exchange of the outgoing particle velocities This means that the only term in which gives a (possibly) nonvanishing contribution is As a consequence it is found thatand hence is necessarily negative or null, the second case occurring only if and consequently too.
The proof of Proposition follows in a similar way. In fact, first, one notices that thanks to the global validity of the body PDF  the body PDF necessarily belongs to the functional class of stochastic PDFs prescribed so that also the local characteristic scale length defined above (see (16)) is larger than zero and finite. As a consequence it follows that both the functional and (see (22)) are globally defined too. Consider in fact the representation of achieved in Theorem 1 and given by (35). Next, let us notice that thanks to (16) the characteristic scale length is necessarily strictly positive. Then, upon noting that and , with being the velocity moment , it follows that where the integral on the rhs is necessarily bounded. This happens because belongs to the functional class and therefore is bounded, while, at the same time, the phase-space moments indicated above necessarily exist. Furthermore, since