Macroscopic irreversibility and decay to kinetic equilibrium for classical hard-sphere systems

TL;DR

This paper investigates the conditions under which macroscopic irreversibility and decay to kinetic equilibrium occur in classical hard-sphere systems, using an axiomatic approach and a novel functional called Master kinetic information.

Contribution

It introduces the Master kinetic information functional and demonstrates its role in characterizing irreversibility and decay to equilibrium within an axiomatic kinetic framework.

Findings

Macroscopic irreversibility is characterized by the Master kinetic information functional.

Decay to kinetic equilibrium is established for suitable solutions of the Master kinetic equation.

The Master kinetic information is unrelated to Boltzmann-Shannon entropy and Fisher information.

Abstract

In this paper the conditions are investigated 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. The problem is set in the framework of the axiomatic "ab initio" approach to classical statistical mechanics recently developed [Tessarotto \textit{et al}., 2013-2017] and the related establishment of an exact kinetic equation realized by 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 here identified with the \textit{Master kinetic information}. The goal is to show that, provided the same PDF is realized in terms of an arbitrary suitably-smooth particular solution of the Master kinetic equation the two properties indicated above are indeed realized and that the same functional is unrelated either with the Boltzmann-Shannon entropy and the Fisher information.

Equations

$${\rho }_{1M}^{(N)}({\mathbf{v}}_1)=\frac{n_o}{{\pi }^{3/2}{\left(2T_o/m\right)}^{3/2}}\exp \left\{-\frac{m{\left({\mathbf{v}}_1-{\mathbf{V}}_o\right)}^2}{2T_o}\right\},$$

$$M_{X_E}({\rho }_1(t))\equiv \underset{{\Gamma }_1}{\int }d{\mathbf{x}}_1{\rho }_1({\mathbf{x}}_1,t)X_E({\mathbf{x}}_1,t)=-\underset{{\Gamma }_1}{\int }d{\mathbf{x}}_1{\rho }_1({\mathbf{x}}_1,t)\ln \frac{{\rho }_1({\mathbf{x}}_1,t)}{A_1}\equiv S({\rho }_1(t)).$$

$$\frac{\partial }{\partial t}S({\rho }_1(t))\ge 0$$

$$\frac{\partial }{\partial t}S({\rho }_1(t))=0\iff {\rho }_1({\mathbf{x}}_1,t)={\rho }_{1M}^{(N)}({\mathbf{v}}_1)$$

$$\frac{\partial S({\rho }_1^{(N)}(t))}{\partial t}\equiv 0$$

$$0\le I_M({\rho }_1^{(N)}(t))\le 1$$

$$\frac{\partial }{\partial t}{I}_M({\rho }_1^{(N)}(t))\le 0$$

$$0\le I_M({\rho }_1^{(N)}(t))\le I_M\left({\rho }_{1o}^{(N)}({\mathbf{x}}_1)\right)\le 1.$$

$$I_M\left({\rho }_{1o}^{(N)}({\mathbf{x}}_1)\right)=1.$$

$$\left\{\begin{array}[]{c}I_{M}(\rho_{1}^{(N)}(t))=0\\ \frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t))=0\end{array}\Leftrightarrow\rho_{1}^{(N)}(\mathbf{x}_{1},t)=\rho_{1M}^{(N)}(\mathbf{v}_{1}),\right.$$

$$\left\{\begin{array}[]{c}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})\equiv\frac{K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})}{K_{Mo}},\\ K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})=-\int\limits_{\Gamma_{1(1)}}d\mathbf{x}_{1}\overline{\Theta}_{1}^{(\partial\Omega)}(\overline{\mathbf{r}})M(\mathbf{v}_{1},\mathbf{b})\frac{\rho_{1}^{(N)}(\mathbf{x}_{1},t)}{\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)}\frac{\partial^{2}\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)}{\partial\mathbf{r}_{1}\cdot\partial\mathbf{r}_{1}},\\ K_{Mo}=\sup\left\{1,K_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b})\right\}.\end{array}\right.$$

$${\hat{\rho }}_1^{(N)}({\mathbf{x}}_1,t)\equiv \frac{{\rho }_1^{(N)}({\mathbf{x}}_1,t)}{k_1^{(N)}({\mathbf{r}}_1,t)},$$

$$M({\mathbf{v}}_1,\mathbf{b})\equiv {\left({\mathbf{v}}_1\cdot \mathbf{b}\right)}^2,$$

$$M({\mathbf{v}}_1,{\mathbf{v}}_2,\mathbf{b})=\frac{1}{2}\left[M({\mathbf{v}}_1,\mathbf{b})+M({\mathbf{v}}_2,\mathbf{b})\right]$$

$$I_{M}(\rho_{1o}^{(N)},\mathbf{b})=\left\{\begin{array}[]{c}1,\\ K_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b}).\end{array}\right.$$

$$K_M({\rho }_{1o}^{(N)}({\mathbf{x}}_1),\mathbf{b})>0.$$

$$K_M({\rho }_1^{(N)}(t),\mathbf{b})\ge 0.$$

$$I_M({\rho }_1^{(N)}(t),\mathbf{b})\ge 0$$

$$K_M({\rho }_1^{(N)}(t),\mathbf{b})=0\iff {\rho }_1^{(N)}({\mathbf{x}}_1,t)\equiv {\rho }_{1M}^{(N)}({\mathbf{v}}_1).$$

$$K_M({\rho }_1^{(N)}(t),\mathbf{b})\equiv -\underset{{\Gamma }_{1(1)}}{\int }d{\mathbf{x}}_1{\overline{\Theta }}_1^{(\partial \Omega )}(\overline{\mathbf{r}})M({\mathbf{v}}_1,\mathbf{b})k_1^{(N)}({\mathbf{r}}_1,t)\frac{{\partial }^2{\hat{\rho }}_1^{(N)}({\mathbf{x}}_1,t)}{\partial {\mathbf{r}}_1\cdot \partial {\mathbf{r}}_1}.$$

$$K_M({\rho }_1^{(N)}(t),\mathbf{b})\equiv \underset{{\Gamma }_{1(1)}}{\int }d{\mathbf{x}}_1{\overline{\Theta }}_1^{(\partial \Omega )}(\overline{\mathbf{r}})M({\mathbf{v}}_1,\mathbf{b})\frac{\partial k_1^{(N)}({\mathbf{r}}_1,t)}{\partial {\mathbf{r}}_1}\cdot \frac{\partial {\hat{\rho }}_1^{(N)}({\mathbf{x}}_1,t)}{\partial {\mathbf{r}}_1}=$$

$$K_M({\rho }_1^{(N)}(t),\mathbf{b})\equiv K_M^{(1)}({\rho }_1^{(N)}({\mathbf{x}}_1,t),\mathbf{b})+\Delta K_M^{(1)}({\rho }_1^{(N)}({\mathbf{x}}_1,t),\mathbf{b}).$$

$$\begin{array}[]{c}K_{M}^{(1)}(\rho_{1}^{(N)}(\mathbf{x}_{1},t),\mathbf{b})=\left(N-1\right)\int\limits_{\Gamma_{1(1)}}d\mathbf{x}_{1}\int\limits_{\Gamma_{1(2)}}d\mathbf{x}_{2}\overline{\Theta}_{1}^{(\partial\Omega)}(\overline{\mathbf{r}})\overline{\Theta}_{2}^{(\partial\Omega)}(\overline{\mathbf{r}})\times\\ \frac{\partial\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)}{\partial\mathbf{r}_{1}}\cdot\frac{\partial}{\partial\mathbf{r}_{2}}\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{2},t)k_{2}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},t)M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})\overline{\Theta}\left(\left|\mathbf{r}_{2}-\mathbf{r}_{1}\right|-\sigma\right).\end{array}$$

$$\Delta K_M^{(1)}({\rho }_1^{(N)}({\mathbf{x}}_1,t),\mathbf{b})\equiv \left(N-1\right)\underset{{\Gamma }_{1(1)}}{\int }d{\mathbf{x}}_1{\overline{\Theta }}_1^{(\partial \Omega )}(\overline{\mathbf{r}})\frac{\partial {\hat{\rho }}_1^{(N)}({\mathbf{x}}_1,t)}{\partial {\mathbf{r}}_1}\cdot \underset{{\Gamma }_{1(2)}}{\int }d{\mathbf{x}}_2\times$$

$${\overline{\Theta }}_2^{(\partial \Omega )}(\overline{\mathbf{r}})M({\mathbf{v}}_1,\mathbf{b})\overline{\Theta }\left(\left|{\mathbf{r}}_2-{\mathbf{r}}_1\right|-\sigma \right){\hat{\rho }}_1^{(N)}({\mathbf{x}}_2,t)\frac{\partial }{\partial {\mathbf{r}}_2}{k}_2^{(N)}({\mathbf{r}}_1,{\mathbf{r}}_2,t),$$

$$K_M({\rho }_1^{(N)}(t),\mathbf{b})=\underset{{\Gamma }_{1(1)}}{\int }d{\mathbf{x}}_1\underset{{\Gamma }_{1(2)}}{\int }d{\mathbf{x}}_2{\overline{\Theta }}_1^{(\partial \Omega )}(\overline{\mathbf{r}}){\overline{\Theta }}_2^{(\partial \Omega )}(\overline{\mathbf{r}})M({\mathbf{v}}_1,{\mathbf{v}}_2,\mathbf{b})\times$$

$$F({\mathbf{r}}_1,{\mathbf{r}}_2,t)\frac{\partial {\hat{\rho }}_1^{(N)}({\mathbf{x}}_1,t)}{\partial {\mathbf{r}}_1}\cdot \frac{\partial }{\partial {\mathbf{r}}_2}{\hat{\rho }}_1^{(N)}({\mathbf{x}}_2,t)\overline{\Theta }\left(\left|{\mathbf{r}}_2-{\mathbf{r}}_1\right|-\sigma \right),$$

$$\Delta M({\mathbf{v}}_1,{\mathbf{v}}_2,\mathbf{b})={\mathbf{b}\cdot \mathbf{n}}_{12}\left|{\mathbf{n}}_{12}\cdot {\mathbf{v}}_{12}^{(+)}\right|{\mathbf{v}}_{12}^{(+)}\cdot \mathbf{b}-{\left({\mathbf{b}\cdot \mathbf{n}}_{12}\right)}^2{\left({\mathbf{n}}_{12}\cdot {\mathbf{v}}_{12}^{(+)}\right)}^2,$$

$$\frac{\partial }{\partial t}{K}_M({\rho }_1^{(N)}(t),\mathbf{b})=-(N-1){\sigma }^2\underset{U_{1(1)}}{\int }d{\mathbf{v}}_1\underset{U_{1(2)}}{\int }d{\mathbf{v}}_2\underset{\Omega }{\int }d{\mathbf{r}}_1{\int }^{(-)}d{\mathbf{\Sigma }}_{21}\left|{\mathbf{v}}_{12}^{(+)}\cdot {\mathbf{n}}_{12}\right|$$

$${\left({\mathbf{b}\cdot \mathbf{n}}_{12}\right)}^2{\left({\mathbf{n}}_{12}\cdot {\mathbf{v}}_{12}^{(+)}\right)}^2\frac{\partial {\hat{\rho }}_1^{(N)}({\mathbf{r}}_1,{\mathbf{v}}_1^{(+)},t)}{\partial {\mathbf{r}}_1}\cdot \frac{\partial }{\partial {\mathbf{r}}_2}{\hat{\rho }}_1^{(N)}({\mathbf{r}}_2={\mathbf{r}}_1+\sigma {\mathbf{n}}_{21},{\mathbf{v}}_2^{(+)}t)$$

$$k_2^{(N)}({\mathbf{r}}_1,{\mathbf{r}}_2={\mathbf{r}}_1+\sigma {\mathbf{n}}_{21},t)\le 0.$$

$$\frac{\partial }{\partial t}{I}_M({\rho }_1^{(N)}(t),\mathbf{b})\le 0$$

Full text

kinetic theory, classical statistical mechanics, Boltzmann equation, H-theorem

I 1 - Introduction

In this investigation the problem is posed of the proof-of-principle for two phenomena which characterize the statistical description of $N-$body hard-sphere systems and laying at the very foundations of classical statistical mechanics (CSM) and kinetic theory alike. The issue, more precisely, is 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).

In the following the case is considered of the so-called Boltzmann-Sinai classical dynamical system (CDS) noi1 which advances in time the microscopic state $\mathbf{x}\equiv\left\{\mathbf{x}_{1},...,\mathbf{x}_{N}\right\}$ of a set of $N$ extended like particles represented by smooth hard spheres noi7 of diameter $\sigma>0$, with $\mathbf{x}_{i}\equiv\left(\mathbf{r}_{i},\mathbf{v}_{i}\right)$, $\mathbf{r}_{i}$ and $\mathbf{v}_{i}$ denoting Newtonian center of mass state, position and velocity of the $i-$th particle. The same particles are assumed: A) subject to instantaneous (unary, binary and multiple) elastic collisions which leave unchanged the particles angular momenta and B) immersed in a bounded domain $\Omega$ of the Euclidean space $\mathbb{R}^{3}$ of finite measure.

For definiteness, the treatment is set in the framework of the ”ab initio” axiomatic approach to CSM recently-developed in Refs.noi1 ; noi2 ; noi3 ; noi4 ; noi5 ; noi6 ; noi7 (see also Refs.noi8 ; noi9 ; noi10 ) and the consequent establishment of an exact, i.e., non-asymptotic, kinetic equation noi3 , denoted as Master kinetic equation.  The new approach radically departs from standard approaches to be found in the literature such as the Boltzmann and Enskog kinetic equations Boltzmann1972 ; Enskog ; CHAPMA-COWLING which apply only in an asymptotic sense for large $N-.$body hard sphere systems, i.e., in which the number of particles $N$ is considered $\gg 1$.  In fact, the remarkable distinguishing feature of the new equation is that, unlike the aforementioned kinetic equations, it holds in the case of the finite Boltzmann-Sinai CDS (shortly referred to as $S_{N}-$CDS), namely for arbitrary hard-sphere systems having a finite number $N$ of particles** **and in which each particle is allowed to have, in addition, a *finite-size, namely is characterized by a finite diameter $\sigma>0,$ and a *finite-mass $m>0.$ ** **

The goal of the paper is to pose in such a context the problem of the existence of both PMI and DKE holding in the case of finite hard-sphere systems. The conjecture is that - just as the ergodicity property of the $S_{N}-$CDS Sinai1970 ; Sinai1989 - the possible occurrence of such phenomena in actual physical, i.e. necessarily finite, systems, might/should not depend on the number $N$ of constituent particles of the system. In particular we intend to show that these properties actually emerge as necessary implications of the Master kinetic equation itself. Incidentally, in doing so, the finiteness requirement on the $S_{N}-$CDS completely rules out for further possible consideration either the Boltzmann or the Enskog kinetic equations, these equations being manifestly inapplicable to the treatment of systems of this type.

Specifically, in the following the case $N>2$ is considered everywhere, which is by far the most physically-relevant one. In this occurrence, in fact, non-trivial $2-$body occupation coefficients arise (see related notations which are applicable for $N>2$  recalled in Appendices A and B below). For completeness the case $N=2$ is nevertheless briefly discussed in Appendix D.

I.1 1A - Motivations and background

Both properties indicated in the title concern the statistical behavior of an ensemble $S_{N}$ of like particles which are advanced in time by a suitable $N-$body classical dynamical system, here identified with the $S_{N}-$CDS. Specifically they 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) $\rho_{1}^{(N)}(t)\equiv\rho_{1}^{(N)}(\mathbf{x}_{1},t).$ The latter is required to belong to a suitable functional class $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t)\right\}$ of 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 non-unique, functional which should be globally defined in the future (i.e., for all times $t\geq t_{o}$ being $t_{o}$ a suitable initial time) bounded and non-negative 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 $1-$body PDF $\rho_{1}^{(N)}(t)\equiv\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ which, accordingly, should be globally defined and decay for $t\rightarrow+\infty$ to a stationary and spatially-uniform Maxwellian PDF

[TABLE]

where $\left\{n_{o}>0,T_{o}>0,\mathbf{V}_{o}\right\}$* *are suitable constant fluid fields.

Both PMI and DKE correspond to physical phenomena which might/should possibly arise in disparate classical $N-$body systems. The clue 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 ikt1 ; ikt2 ; ikt4 ; ikt7 ; ikt8 to collisional/collisionless and non-relativistic/relativistic gases and plasmas Cr2010 ; Cr2011 ; Cr2011a ; Crema2012 ; Crema2013 ; Crema2013a ; Crema2013b ; Crema2014 . However, the most notable example is provided by dilute gases characterized by a large number of particles ($N\equiv\frac{1}{\varepsilon}\gg 1$) and a small (i.e., infinitesimal) diameter $\sigma\sim O(\varepsilon^{1/2})$ of the hard-spheres. In fact, the property of macroscopic irreversibility indicated above is related to the Carnot’s second Law of Classical Thermodynamics and the historical attempt of its first-principle-proof performed originally by Ludwig Boltzmann in 1872 Boltzmann1972 . Indeed both phenomena lie at the very root of Boltzmann and Grad kinetic theories Boltzmann1972 ; Grad , although a different characterization of the concept of PMI is actually involved. 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 moment

[TABLE]

Here $X_{E}(\mathbf{x}_{1},t)\equiv-\ln\frac{\rho_{1}(\mathbf{x}_{1},t)}{A_{1}},$ $\rho_{1}(\mathbf{x}_{1},t)$ and $A_{1}$ 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 so-called entropic inequality

[TABLE]

while, furthermore, the entropic equality condition

[TABLE]

should hold. The latter equation implies therefore that, if $\rho_{1}(t)$ and $S(\rho_{1}(t))$ exist globally, then necessarily $lim_{t\rightarrow+\infty}\rho_{1}(\mathbf{x}_{1},t)=\rho_{1M}^{(N)}(\mathbf{v}_{1})$. However, both Boltzmann and Grad theories are actually specialized to the treatment of the so-called *Boltzmann-Grad limit *obtained introducing, first, the dilute-gas ordering $\sigma\sim O(\varepsilon^{1/2})$ with $\varepsilon\equiv\frac{1}{N}\ll 1$ and, then, taking the continuum limit $\varepsilon\rightarrow 0$ (for a review of the topic see again Ref.noi10 ).

Nevertheless, the possible realization of either PMI or DKE depends critically on the prescription of the functional class $\left\{\rho^{(N)}(\mathbf{x}_{1},t)\right\},$ so that their occurrence is actually non-mandatory. Indeed, both cannot occur - also for Boltzmann and Grad kinetic theories noi6 - if the $N-$body (microscopic) probability density function $\rho^{(N)}(\mathbf{x,}t)$ is identified with the deterministic $N-$body PDF noi1 , namely the $N-$body phase-space Dirac delta. This is defined as $\delta(\mathbf{x-x}(t))\equiv\prod\limits_{1=1,N}\delta(\mathbf{x}_{i}\mathbf{-x}_{i}(t)),$ with $\mathbf{x}\equiv\left\{\mathbf{x}_{1},...,\mathbf{x}_{N}\right\}$ denoting the state of the $N-$body system and $\mathbf{x}(t)\equiv\left\{\mathbf{x}_{1}(t),...,\mathbf{x}_{N}(t)\right\}$ is the image of an arbitrary initial state $\mathbf{x}(t_{o})\equiv\mathbf{x}_{o}$ generated by the same $N-$body CDS.  That such a PDF necessarily must realize an admissible particular solution of the $N-$body Liouville equation follows, in fact, as a straightforward consequence of the axioms of classical statistical mechanics noi1 .

Despite these premises, however, the case of a finite Boltzmann-Sinai CDS, which is characterized by a finite number of particles $N$ 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 that:

• •

First, Boltzmann and Grad kinetic theories are inapplicable to the finite Boltzmann-Sinai CDS.

• •

Second, as shown in Ref.noi6 the Boltzmann-Shannon entropy associated with an arbitrary particular solution $\rho^{(N)}(t)\equiv\rho^{(N)}(\mathbf{x}_{1},t)$ of the Master kinetic equation, i.e., the functional $S(\rho_{1}^{(N)}(t))\equiv M_{X_{E}}(\rho_{1}^{(N)}(t)),$ in contrast to $S(\rho_{1}(t))\equiv M_{X_{E}}(\rho_{1}(t)),$ is exactly conserved in the sense that identically

[TABLE]

must 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 an incompressible viscous Navier-Stokes granular fluid in terms of the Master kinetic equation is relevant and suggests that this may be indeed the case. In fact, as shown in Ref. noi8 , in such a case the decay of the fluid velocity field occurring in a bounded domain necessarily requires the existence of DKE too. However, besides the construction of the kinetic equation appropriate for such a case, a further unsolved issues lies in the determination of the functional class $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t)\right\}$ 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 $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t_{o})\right\}$. These should warrant that in the limit $t\rightarrow+\infty,$ $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ uniformly converges to the spatially-homogeneous and stationary Maxwellian PDF $\rho_{1M}(\mathbf{v}_{1})$ (1). Such a result, however, is highly non-trivial since it should rely on the establishment of a global existence theorem for the same $1-$body PDF $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ - namely holding in the whole time axis $I\equiv\mathbb{R}$, besides the same $1-$body phase space $\Gamma_{1}$ - for the involved kinetic 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 related to the asymptotic nature of the Boltzmann equation noi7 . 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 $H-$theorem inequality and decay asymptotically to kinetic equilibrium Cercignani1982 ; Villani .

I.2 1B - Goals of the investigation

In a series of papers noi1 ; noi2 ; noi3 ; noi4 ; noi5 ; noi6 ; noi7 a new kinetic equation has been established for hard sphere systems subject to elastic instantaneous collisions, denoted as Master kinetic equation (see Appendix A). Its remarkable feature is that unlike the Boltzmann and Enskog kinetic equations Boltzmann1972 ; Enskog the new kinetic equation and its corresponding Master collision operator are exact, i.e., they hold for an arbitrary finite $N-$body hard-sphere system $S_{N}$. In other words this means that in such a context $S_{N}$ is allowed to have in principle an arbitrary constant and finite number ($N$) of* *hard spheres, each one characterized by a *finite *diameter $\sigma>0$ and a finite mass $m>0$.

These peculiar features follow uniquely as a consequence of the new approach to classical statistical mechanics developed in Refs.noi1 ; noi2 ; noi3 and referred to as ”ab initio” axiomatic approach. As shown in the same references (for a review see also Ref.noi9 ), this is based on the adoption of appropriate extended functional setting and physics-based modified collision boundary conditions (MCBC; see Appendix B) noi1 ; noi2 which are prescribed in order to advance in time across arbitrary (unary, binary or multiple) collision events the $N-$body PDF. The related physical interpretation is intuitive. It can be viewed, in fact, as the jump condition for the $N-$body PDF along the phase-space Lagrangian trajectory $\left\{\mathbf{x}(t)\right\}$ for an ensemble of $N$ tracer particles noi0 ; noi9 following the same deterministic trajectory and undergoing a collision event at a suitable collision time.

Based on the discovery of the Master kinetic equation, a host of new developments have opened up. These concern in particular the investigation of the conceptual aspects and implications of the same equation which include (for an extended discussion see also Refs.noi4 ; noi5 ; noi6 ; noi7 ):

*The determination of the Master H-theorem: *as pointed out in Refs.noi5 ; noi6 based on the discovery of a family of generalized collisional invariants, the Master kinetic equation is found to admit a constant H-theorem in terms of the Boltzmann-Shannon entropy $S_{1}(\rho_{1}^{(N)}(t))$. 2. 2.

The derivation of the Boltzmann kinetic equation in terms of the Master kinetic equation. The Boltzmann equation can be recovered in an asymptotic sense when the so-called dilute-gas asymptotic ordering is introduced in the Master kinetic equation (see Refs.noi7 ; noi10 ). 3. 3.

*The global validity of the Master kinetic equation: *the Master kinetic equation has been shown to hold globally in time noi7 .

However, the question arises of the possible occurrence of both PMI and DKE for arbitrary finite-size and/or dense systems of hard spheres. The example-case recently pointed out noi8 , corresponding to the statistical description of an incompressible Navier-Stokes granular fluid, suggests that this may be indeed the case. The goal of the present paper is to propose a new approach, referred to as PMI/DKE theory, to the treatment of PMI and DKE for hard-sphere systems described by means of the Master kinetic equation. The core of the new theory is the first-principle proof of both microscopic irreversibility and DKE properties holding for the Master kinetic equation.

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). Second, in Section 2, based on the theory of the Master kinetic equation earlier developed noi3 and suitable integral and differential identities (see Appendices A and B), the properties of the MKI functional are investigated. These concern in particular the establishment of appropriate inequalities holding for 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 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.

II 2 - Axiomatic prescriptions for the MKI functional

In view of the considerations given above we now proceed constructing an explicit possible realization of the MKI functional in terms of suitable axiomatic prescriptions. **This **should be intended as a functional $I_{M}\left(\rho_{1}^{(N)}(t)\right)$ of the $1-$body PDF $\rho_{1}^{(N)}(t),$ with $\rho_{1}^{(N)}(t)\equiv\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ being identified with an arbitrary particular solution of the Master kinetic equation (see Eq.(54) in Appendix A). The same PDF is assumed globally defined, a property which in view of Ref.noi7 is warranted in particular if the initial PDF $\rho_{1o}^{(N)}(\mathbf{x}_{1})$ (prescribed by the initial problem (54) at the initial time $t_{o}\in I$) belongs to the functional class of stochastic $1-$body PDFs, i.e., strictly positive ordinary functions $\rho_{1}^{(N)}(t)\equiv\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ which are smoothly-differentiable. Hereon for definiteness the case $N>2$ is considered (see Appendix D for comments about the treatment of the case $N=2$).

More specifically, first (*MKI Prescription No.#0 *), the functional class $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t)\right\}$ is prescribed. This is identified with the subset of stochastic particular solutions of the Master kinetic equation which are images of the corresponding initial PDFs $\left\{\rho_{1o}^{(N)}(\mathbf{x}_{1})\right\}.$ In turn $\left\{\rho_{1o}^{(N)}(\mathbf{x}_{1})\right\}$ is the ensemble of all initial PDFs for which the functional $I_{M}\left(\rho_{1o}^{(N)}(\mathbf{x}_{1})\right)$ exists. In the remainder $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t)\right\}$ will be referred to as functional class of the admissible stochastic PDFs.

Second (*MKI Prescription No.#1 *), the functional $I_{M}(\rho_{1}^{(N)}(t))$ should be suitably prescribed so that, assuming that by construction the initial value $I_{M}\left(\rho_{1o}^{(N)}(\mathbf{x}_{1})\right)$ exists, then the same functional necessarily must exist globally in the future, i.e., for all $t\geq t_{o}$ where $t_{o}\in I$ is a suitable initial time. Third, we shall require (MKI Prescription No.#2) $I_{M}(\rho_{1}^{(N)}(t))$ to be real, non-negative and bounded in $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t)\right\}$ in the sense

[TABLE]

so that it can be interpreted as an information measure associated with the $1-$body PDF $\rho_{1}^{(N)}(t)\equiv\rho_{1}^{(N)}(\mathbf{x}_{1},t).$ For this reason the previous inequalities will be referred to as information-measure inequalities. Fourth, for consistency with the property of macroscopic irreversibility (PMI), (MKI Prescription No.#3) $I_{M}(\rho_{1}^{(N)}(t))$ 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:

[TABLE]

should identically apply $\forall t\geq t_{o},$ so that by construction

[TABLE]

which implies that $I_{M}(\rho_{1}^{(N)}(t))$ is also globally defined for all $t\in I\equiv\mathbb{R}$ with $t\gtrsim t_{o}.$ In addition, if $\left.\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t))\right|_{t=t_{o}}\neq 0$, without loss of generality its initial value $I_{M}\left(\rho_{1o}^{(N)}(\mathbf{x}_{1})\right)$ can always be set such that

[TABLE]

As a fifth condition, in order to warrant the existence of DKE we shall require (*MKI Prescription No.#4 *) the functional $I_{M}(\rho_{1}^{(N)}(t))$ to be prescribed in such a way that at an arbitrary time $t\in I,$ with $t\gtrsim t_{o},$ the vanishing of both $I_{M}(\rho_{1}^{(N)}(t))$ and its time derivative $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t))$ should occur if and only if the $1-$body PDF solution of the Master kinetic equation coincides with kinetic equilibrium. As a consequence, for the functional $I_{M}(\rho_{1}^{(N)}(t))$ the following propositions should be equivalent

[TABLE]

with $\rho_{1M}^{(N)}(\mathbf{v}_{1})$ being a kinetic equilibrium PDF of the form (1).

The implication of *MKI Prescriptions #0-#4 *is that, provided a realization of the MKI can be found in the functional class of the initial conditions indicated above $\left\{\rho_{1o}^{(N)}(\mathbf{x}_{1})\right\}$ the existence of both PMI and DKE for the Master kinetic equation would actually be established.

In the sequel the goal is to show that the MKI functional can be identified with the functional

[TABLE]

Here $\rho_{1}^{(N)}(t)\equiv\rho_{1}^{(N)}(\mathbf{x}_{1},t),$ $\rho_{1o}^{(N)}(\mathbf{x}_{1})$ and $\widehat{\rho}_{1}^{(N)}(t)\equiv\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)$ are respectively the $1-$body PDF solution of the initial problem associated with the Master kinetic equation (see Eq.(54) in Appendix A), with $\rho_{1o}^{(N)}(\mathbf{x}_{1})$ being the initial PDF, and the *renormalized *$1-$body PDF

[TABLE]

with $k_{1}^{(N)}(\mathbf{r}_{1},t)$ being the $1-$body occupation coefficient whose definition is recalled in Appendix B (see Eq.(63)). As a consequence in the previous equation $\frac{\rho_{1}^{(N)}(\mathbf{x}_{1},t)}{\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)}=k_{1}^{(N)}(\mathbf{r}_{1},t)$. Furthermore $\overline{\Theta}_{1}^{(\partial\Omega)}(\overline{\mathbf{r}})$ is the boundary theta function given by Eq.(59) (see Appendix A) and finally $\frac{1}{2}M(\mathbf{v}_{1},\mathbf{b})$ denotes the directional kinetic energy along $\mathbf{b}$ carried by particle $1\mathbf{,}$ namely the dynamical variable

[TABLE]

with $\mathbf{b}$ denoting an arbitrary constant unit vector. Hence

[TABLE]

identifies the corresponding total directional kinetic energy carried by particles $1$ and $2$. As a consequence it follows that if $K_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b})\geq 1$ or $0\leq K_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b})<1$, then consistent with (8) by construction respectively one should obtain

[TABLE]

II.1 2A - Proof of the non-negativity of the MKI information measure

The strategy adopted for the proof of the *No.#1 *and No.#2 MKI Prescriptions is to prove initially the validity of the information-measure left inequality in Eq.(6), namely that $I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ cannot have negative values for arbitrary $t\geq t_{o}$. To reach such a result the following proposition is established.

THM. 1 - Non-negativity of $I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$

*Let us assume that $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ is an arbitrary stochastic particular solution of the Master kinetic equation (54) such that the integral *$K_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b})$ is non-vanishing. Then it follows necessarily that:

• •

Proposition P1${}_{1}:$

[TABLE]

• •

Proposition P1${}_{2}:$ *for all $t\in I$ with *$t>t_{o}$

[TABLE]

• •

Proposition P1${}_{3}:$ then necessarily the inequality

[TABLE]

must hold too.

• •

Proposition P1${}_{4}:$ *Finally the following necessary and sufficient condition holds at a given time *$t\in I$ with $t\geq t_{o}:$

[TABLE]

*Proof - *One first notices that $K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ can be equivalently written in the form

[TABLE]

Hence integrating by parts, noting that the gradient term $\frac{\partial\overline{\Theta}_{1}^{(\partial\Omega)}(\overline{\mathbf{r}})}{\partial\mathbf{r}_{1}}$ gives a vanishing contribution to the phase-space integral and upon invoking Eq.(70) reported in Appendix B it follows

[TABLE]

Now noting that $\mathbf{n}_{12}\delta\left(\left|\mathbf{r}_{2}-\mathbf{r}_{1}\right|-\sigma\right)=-\frac{\partial}{\partial\mathbf{r}_{2}}\overline{\Theta}\left(\left|\mathbf{r}_{2}-\mathbf{r}_{1}\right|-\sigma\right)$ and ignoring again a vanishing contribution carried by $\frac{\partial}{\partial\mathbf{r}_{2}}\overline{\Theta}_{2}^{(\partial\Omega)}(\overline{\mathbf{r}})$, the rhs of previous equation can once more be integrated by parts yielding

[TABLE]

Hence in terms of the total directional kinetic energy carried by particles $1$ and $2,$ namely $M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})$ (see Eq.(14)) the functional $K_{M}^{(1)}(\rho_{1}^{(N)}(t),\mathbf{b})$ takes the form

[TABLE]

Instead the second term $\Delta K_{M}^{(1)}(\rho_{1}^{(N)}(\mathbf{x}_{1},t),\mathbf{b})$ reads

[TABLE]

with the gradient $\frac{\partial}{\partial\mathbf{r}_{2}}k_{2}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},t)$ being given by the differential identity (75) reported in Appendix B. The procedure is analogous to the one followed above for the calculation of $\frac{\partial k_{1}^{(N)}(\mathbf{r}_{1},t)}{\partial\mathbf{r}_{1}}$ and can be iterated at arbitrary order $s=1,N-1$ (see Eq.(83) in Appendix B). As a result it follows that $K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ can be represented in terms of a sum of the form $K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})\equiv\sum\limits_{j=1,N-1}K_{M}^{(j)}(\rho_{1}^{(N)}(\mathbf{x}_{1},t),\mathbf{b})$ in which each functional $K_{M}^{(j)}(\rho_{1}^{(N)}(\mathbf{x}_{1},t),\mathbf{b})$ is non-negative and symmetric. Therefore $K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ takes the form of a non-negative and symmetric functional of the type

[TABLE]

with $M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})$ being the total directional kinetic energy (14) and $F(\mathbf{r}_{1},\mathbf{r}_{2},t)$ a suitable real scalar kernel which is symmetric in the variables $\mathbf{r}_{1}$ and $\mathbf{r}_{2}.$ This proves validity of the inequality (18) (Proposition P11). As a result, invoking Eq.(15) it follows that the inequalities (16) and (17), and hence Propositions P12 and P13 manifestly hold too. Finally, one notices that $K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})=0$ if and only if identically $\frac{\partial}{\partial\mathbf{r}_{1}}\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)\equiv 0$. Since $\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)$ is by construction a solution of the Master kinetic equation this requires necessarily that (19) must hold too (Proposition P1${}_{4})$. Q.E.D.

II.2 2B - Proof of PMI for the Master kinetic equation

The next step is to prove the monotonic time-decreasing behavior of the MKI functional, which involves *No.#3 *and *No.#4 MKI Prescriptions *and consequently also the validity of No.#2 MKI Prescription. The first two refer respectively to the validity of the time derivative inequality (7) and the conditions of existence of kinetic equilibrium (10), while the latter one concerns the right-hand inequality $I_{M}(\rho_{1}^{(N)}(t))\leq 1$. In order to prove these properties let us preliminarily determine the variation of the total directional kinetic energy $M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})$ (see Eq.(14)) across a binary collision, namely the quantity $\Delta M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})\equiv M(\mathbf{v}_{1}^{(+)},\mathbf{v}_{2}^{(+)},\mathbf{b})-M(\mathbf{v}_{1}\mathbf{v}_{2},\mathbf{b})$. One obtains

[TABLE]

the rhs being expressed in terms of the outgoing particle velocities $(\mathbf{v}_{1}^{(+)},\mathbf{v}_{2}^{(+)})$. Then the following proposition holds.

**THM. 2 - Property of macroscopic irreversibility **(Master equation PMI theorem)

*Let us assume that $\rho_{1}^{(N)}(\mathbf{r}_{1},\mathbf{v}_{1},t)$ is an arbitrary stochastic particular solution of the Master kinetic equation (54) such that the integral *$K_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b})$ is non-vanishing. Then it follows that

• •

Proposition P2**1: *one finds that for all *$t\geq t_{o}:$

[TABLE]

• •

Proposition P2**2: the inequality

[TABLE]

*holds identically for all *$t\geq t_{o}$ so that necessarily

[TABLE]

• •

Proposition P2**3: *one finds that a given time *$t\in I$ with $t\geq t_{o}:$

[TABLE]

*Proof - *Upon time-differentiation of the functional $K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ and invoking the first form of the Master kinetic equation (see Eq.(51) in Appendix A) one obtains

[TABLE]

namely

[TABLE]

Hence, thanks to the differential identity (84) it follows:

[TABLE]

Performing an integration by parts and upon invoking the first differential identity (86) (reported in Appendix B) this delivers:

[TABLE]

[TABLE]

where $\frac{\partial}{\partial\mathbf{r}_{1}}\left[\delta(\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|-\sigma)\right]=-\frac{\partial}{\partial\mathbf{r}_{2}}\left[\delta(\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|-\sigma)\right]$. Hence performing a further integration by parts and using the second differential identity (86) (Appendix B) the previous equation yields

[TABLE]

where the symmetry property with respect to the exchange of states $\left(\mathbf{x}_{1},\mathbf{x}_{2}\right)$ has been invoked. In the previous equation the integration on the Dirac delta can be performed at once letting

[TABLE]

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.(89) in Appendix C) the integral $\int^{(+)}d\mathbf{\Sigma}_{21}$ can be transformed to a corresponding integration on $\int^{(-)}d\mathbf{\Sigma}_{21}.$ Thus the contributions in the two phase-space integrals only differ because of the variation $\Delta M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})$ of the total directional kinetic energy of particles $1$ and $2.$ This implies that

[TABLE]

where the solid-angle integration is performed on the incoming particles whereas $\Delta M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})$ is evaluated in terms of the outgoing particles $(+)$ and therefore must be identified with the second equation on the rhs of Eq.(26). Consider now the dependences in terms of the outgoing particle velocities $\mathbf{v}_{1}^{(+)}$ and $\mathbf{v}_{2}^{(+)}$ in the previous phase-space integral. The velocity dependences contained in the factors $\left|\mathbf{v}_{12}\cdot\mathbf{n}_{12}\right|$ and $\frac{\partial\widehat{\rho}_{1}^{(N)}(\mathbf{r}_{1},\mathbf{v}_{1}^{(+)},t)}{\partial\mathbf{r}_{1}}\cdot\frac{\partial\widehat{\rho}_{1}^{(N)}(\mathbf{r}_{2},\mathbf{v}_{2}^{(+)}t)}{\partial\mathbf{r}_{2}}$ are manifestly symmetric with respect to the variables $\mathbf{v}_{1}^{(+)}$ and $\mathbf{v}_{2}^{(+)}.$ On the other hand, as a whole, the same integral should remain unaffected with respect to the exchange of the outgoing particle velocities $\mathbf{v}_{1}^{(+)}\Leftrightarrow\mathbf{v}_{2}^{(+)}.$ This means that the only term in $\Delta M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})$ which can give a non-vanishing contribution is $-\left(\mathbf{b}\cdot\mathbf{n}_{12}\right)^{2}\left(\mathbf{n}_{12}\cdot\mathbf{v}_{12}^{(+)}\right)^{2}.$ As a consequence the previous integral reduces to

[TABLE]

(38)and hence is necessarily negative or null, the second case occurring only if $\frac{\partial\widehat{\rho}_{1}^{(N)}(\mathbf{r}_{1},\mathbf{v}_{1}^{(+)},t)}{\partial\mathbf{r}_{1}}\equiv 0.$ The proof of Proposition P22 is straightforward since $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})\equiv\frac{1}{K_{Mo}}\frac{\partial}{\partial t}K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ so that the inequality (38) implies Eq.(28) and (29) too.** **Finally, since $\widehat{\rho}_{1}^{(N)}(\mathbf{r}_{1},\mathbf{v}_{1},t)$ is a solution of the Master kinetic equation $\frac{\partial}{\partial\mathbf{r}_{1}}\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)\equiv 0$ occurs if and only if $\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)$ coincides with a Maxwellian kinetic equilibrium of the type (1). This proves also Proposition P2${}_{3}.$ Q.E.D.

II.3 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 $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ necessarily must decay asymptotically for $t-t_{o}\rightarrow+\infty$ to kinetic equilibrium, i.e., that the limit function $\lim_{t-t_{o}\rightarrow+\infty}\rho_{1}^{(N)}(\mathbf{x}_{1},t)\equiv\rho_{1\infty}^{(N)}(\mathbf{x}_{1})$ exists and it necessarily coincides with a Maxwellian kinetic equilibrium of the type (1). In this regard the following proposition holds.

THM. 3 - Asymptotic behavior of $I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ (Master equation-DKE theorem)

Let us assume that the initial condition $\rho_{1o}^{(N)}(\mathbf{x}_{1})\in\left\{\rho_{1o}^{(N)}(\mathbf{x}_{1})\right\}$ is* such that the corresponding functional* $K_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b})$ is non-vanishing, i.e., in view of THM.1 necessarily $>0$. Then is follows that the corresponding time-evolved solution of the Master kinetic equation $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ *in the limit *$t-t_{o}\rightarrow+\infty$ necessarily must decay to kinetic equilibrium, i.e.,

[TABLE]

*Proof - *In order to reach the thesis it is sufficient to prove that necessarily

[TABLE]

In fact, let us assume ”ad absurdum” that $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})\leq-k^{2}$ with $k^{2}>0$ a real constant. Then THM.2 (proposition P22) requires that

[TABLE]

a result which contradicts THM.1. This proves the validity of Eq.(40). Furthermore, by construction $\frac{\partial}{\partial t}I_{M}\equiv\frac{1}{K_{Mo}}\frac{\partial}{\partial t}K_{M}$ and furthermore $\frac{\partial}{\partial t}K_{M}$ is identified with the functional $W_{M}(\rho_{1}^{(N)}(t),\mathbf{b})\leq 0$ which is determined by Eq.(38). At this point one notices that, thanks to continuity of the functional $W_{M}(\rho_{1}^{(N)}(t),\mathbf{b}),$ necessarily the identity

[TABLE]

holds where, thanks to global existence of the $1-$body PDF (see Ref.noi7 ), the limit function

[TABLE]

necessarily exists. As a consequence Eq.(40) requires also the equation

[TABLE]

to hold. Upon invoking proposition P24 of THM.2 this implies that necessarily $\rho_{1\infty}^{(N)}(\mathbf{x}_{1})=\rho_{1M}^{(N)}(\mathbf{v}_{1})$ so the thesis (39) is proved. Incidentally, thanks to THM. 1, this requires also that

[TABLE]

Q.E.D.

II.4 2D - Remarks

A few remarks are worth being pointed out regarding the results presented above.

Remark #1: The choice of the MKI functional considered here (see Eq.(11)) is just one of the infinite particular admissible realizations which meet the complete set of MKI-prescriptions indicated above. In particular the choice of the velocity moment $M(\mathbf{v}_{1},\mathbf{b})$ considered here (see Eq.(13)) remains in principle arbitrary, since $\left|\mathbf{v}_{1}\cdot\mathbf{b}\right|^{2}$ can be equivalently replaced, for example, by any factor of the form $\left|\mathbf{v}_{1}\cdot\mathbf{b}\right|^{2n},$ with $n\geq 1.$ Furthermore it is obvious that $M(\mathbf{v}_{1},\mathbf{b})$ can be replaced by any function of the form $M(\mathbf{v}_{1},\mathbf{b})+\Delta M(\mathbf{v}_{1},\mathbf{b})$, being $\Delta M(\mathbf{v}_{1},\mathbf{b})$ prescribed in such a way that its contribution to $\frac{\partial}{\partial t}I_{M}$ vanishes identically so that the validity of the inequality (28) in THM. 2 is preserved. This implies in turn that the prescription of the MKI functional $I_{M}(\rho_{1}^{(N)}(t))$ remains in principle non-unique. 2. 2.

Remark #2: A possible issue is related to the requirement that the renormalized $1-$body PDF, as the $1-$body PDF itself, are strictly positive at all times and are non-vanishing. Here it is sufficient to state that an elementary consequence of the theory of the Master kinetic equation developed in Ref.noi3 is that, provided the corresponding initial $N-$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 positive too at all times and everywhere in the $1-$body phase-space. 3. 3.

Remark #3: It must be stressed that the signature of the time derivative $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ actually depends crucially on the adoption of the causal form of MCBC (i.e., see Eq.(87) or (89) in Appendix C) rather than the anti-causal one (given instead by Eq.(88)). The first choice of course is mandatory in view of the causality principle. Indeed, it is immediate to prove that $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ changes signature if the anti-causal MCBC Eq.(88) is invoked. 4. 4.

Remark #4: THM.2 warrants that Macroscopic irreversibility, namely the inequality $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})\leq 0$ occurs specifically because of: a) the time-variation of the $\mathbf{b}-$directional total kinetic energy which occurs at arbitrary binary collision events; b) the occurrence of a velocity-space anisotropy in the $1-$body PDF, i.e., the fact that the same PDF may not coincide with a local Maxwellian PDF. 5. 5.

*Remark #5: *The existence of the limit function $\lim_{t\rightarrow+\infty}\rho_{1}^{(N)}(\mathbf{x}_{1},t)=\rho_{1\infty}^{(N)}(\mathbf{x}_{1})$ follows uniquely as a consequence of the global existence theorem holding for the Master kinetic equation noi7 . 6. 6.

*Remark #6: *Last but not least, the fact that the same limit function may coincide or not with the Maxwellian kinetic equilibrium (1) depends specifically on the functional setting prescribed for the same PDF $\rho_{1}^{(N)}(\mathbf{x}_{1},t).$ More precisely DKE can only occur provided $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ is a suitably-smooth stochastic PDF such that the MKI functional exists for the corresponding initial PDF at time $t_{o},$ i.e., $\rho_{1}^{(N)}(\mathbf{x}_{1},t_{o})=\rho_{1o}^{(N)}(\mathbf{x}_{1}).$

THMs 1-3 represent the main results of the PMI/DKE theory developed here. In particular, they show that the notion of macroscopic irreversibility and that of decay to kinetic equilibrium are intimately connected. The crucial issues which remain to be addressed are whether these phenomena are actually consistent with the fundamental symmetry properties of the underlying Boltzmann-Sinai CDS and to analyze the physical origin and implications of the present theory.

A detailed discussion on these topics is given below in the following two sections.

III 3 - Consistency of MPI/DKE theory with microscopic dynamics

The problem to be posed in the present section concerns the investigation of consistency between the occurrence of the MPI/DKE phenomenon and the time-evolution of the underlying time-reversible, conservative and energy conserving $N-$body Boltzmann-Sinai classical dynamical system $S_{N}-$CDS.

First issue: consistency with the microscopic reversibility principle - This is related to the famous objection raised 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 manifestation of a decay of the $1-$body PDF to kinetic equilibrium, i.e., the uniform Maxwellian PDF of the form (1), as predicted by the above Master equation-DKE Theorem. That a satisfactory answer to this question is actually possible follows from elementary considerations which are based on the axiomatic “ab initio” statistical description realized by the Master kinetic equation. In this regard it is worth recalling the discussion reported above concerning the role of MCBC regarding the functional $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t)).$ In particular, it is obvious that the signature depends on whether the causal (or anti-causal) form of MCBC is invoked (see Appendix C). Such a choice is not arbitrary since, for consistency with the causality principle, it must depend on the microscopic arrow of time, i.e., the orientation of the time axis chosen for the reference frame. Based on these premises, consistency between the occurrence of macroscopic irreversibility associated with the DKE phenomenon and the principle of microscopic reversibility can immediately be established. Indeed, it is sufficient to notice that when a time-reversal or a velocity-reversal is performed on the $S_{N}$ $-$CDS the form of the collision boundary conditions (i.e., in the present case the MCBC provided by Eq.(87) in Appendix C) must be changed, replacing them with the corresponding anti-causal ones, i.e., Eq.(88). This manifestly implies that MKI functional decreases in both cases, i.e., after performing the time-reversal, so that no contradiction can possibly arise in this case between THM.3 and the microscopic reversibility principle. 2. 2.

Second issue: consistency with Poincare’ recurrence theorem (PRT) - Similar considerations concern the consistency with PRT as well as the conservation of total (kinetic) energy for the $S_{N}$ $-$CDS (see also the related Zermelo’s objection in the Introduction). In fact, first, as shown in Ref.noi5 by construction the Master collision operator admits the customary Boltzmann collisional invariants, including total kinetic energy of colliding particles. Hence, total energy conservation is again warranted for $S_{N}$ $-$CDS. Second, regarding PRT, it concerns the Lagrangian phase-space trajectories of the $S_{N}$ $-$CDS, i.e., the fact that almost all of these trajectories return arbitrarily 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 strongly both on the same initial condition and the notion of distance to be established on the same phase-space. Nevertheless, such a ”recurrence effect” influences only the Lagrangian time evolution of the $N-$body PDF which occurs along the same Lagrangian $N-$body phase-space trajectories. Instead, the same recurrence effect has manifestly no influence on the time evolution of the Eulerian $1-$body PDF which is advanced in time in terms of the Eulerian kinetic equation represented by the Master kinetic equation. Therefore the mutual consistency of DKE and PRT remains obvious.

Hence, in the framework of the axiomatic “ab initio” statistical theory based on the Master kinetic equation the full consistency is warranted with the microscopic dynamics of the underlying Boltzmann-Sinai CDS.

IV 4 - Physical implications

Let us now investigate the physical interpretation and main implications emerging from the PMI/DKE theory developed 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 as well as the occurrence of the DKE-phenomenon are both determined by the Boltzmann H-theorem. As recalled above, this is expressed in terms of the production rate for the Boltzmann-Shannon entropy $\frac{\partial}{\partial t}S(\rho_{1}(t)),$ with $S(\rho_{1}(t))$ being interpreted as a measure of the ignorance associated with a solution of the Boltzmann equation. In fact the customary interpretation is that they arise specifically because of the validity of the entropic inequality (3), i.e., the monotonic increase of $S(\rho_{1}(t))$, and the corresponding entropic equality (4) stating a necessary and sufficient condition for kinetic equilibrium. Such a theorem is actually intimately related with the equation itself. In fact both the theorem and the equation generally hold only for stochastic PDFs $\rho_{1}(t)=\rho_{1}(\mathbf{x}_{1},t)$ which are suitably-smooth and not for distributions noi1 . According to Boltzmann’s original interpretation, however, both the Boltzmann equation and Boltzmann H-theorem should only hold when the so-called Boltzmann-Grad limit is invoked, i.e. based on the limit operator $L_{BG}\equiv\lim_{\begin{subarray}{c}N\rightarrow+\infty\\ N\sigma^{2}\sim O(1)\end{subarray}}$ (see Ref. noi3 ; noi7 ; noi10 ).

In striking departure from such a picture:

• •

The axiomatic ”ab initio” theory based on the Master kinetic equation and the present PMI/DKE theory are applicable to an arbitrary finite Boltzmann-Sinai CDS. This means that they hold for hard-sphere systems having a finite number of particles and with finite diameter and mass, i.e., without the need of invoking validity of asymptotic conditions.

• •

The main departure with respect to Boltzmann kinetic theory arises because, as earlier discovered noi6 , the Boltzmann-Shannon entropy associated with an arbitrary stochastic $1-$body PDF $\rho_{1}^{(N)}(t)=\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ 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 PMI and DKE is unrelated with the Boltzmann-Shannon entropy. In fact, as shown here, it arises because of the properties of the MKI functional $I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ when it is expressed in terms of an arbitrary stochastic PDF $\rho_{1}^{(N)}(t)=\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ solution of the Master kinetic equation. The only requirement is that the initial PDF $\rho_{1o}^{\left(N\right)}(\mathbf{x}_{1})$ is prescribed so that the corresponding MKI functional $I_{M}(\rho_{1o}^{(N)}(\mathbf{x}_{1}),\mathbf{b})$ exists.

• •

As shown here the MKI functional is a suitably-weighted phase-space moment of $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$ which can be interpreted as an information measure for the same PDF, namely belongs to the interval $\left[0,1\right],$ and exhibits a monotonic-decreasing time-dependence, i.e., the property of macroscopic irreversibility.

• •

In addition both $I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ and its time derivative $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ vanish identically if and only if the $1-$body PDF coincides with a Maxwellian kinetic equilibrium of the type (1). This warrants in turn also the occurrence of the DKE-phenomenon for $\rho_{1}^{(N)}(\mathbf{x}_{1},t)$, i.e., that for $t-t_{o}\rightarrow+\infty$ the same PDF must decay to a Maxwellian kinetic equilibrium of this type.

• •

Finally, it is interesting to point out the peculiar behavior of the MKI functional $I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ and its time derivative $\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ when the Boltzmann-Grad limit is considered. In particular the $1-$ and $2-$body occupation coefficients $k_{1}^{(N)}(\mathbf{r}_{1},t)$ and $k_{2}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},t)$ which appear in the Master kinetic equation (see Appendix B, Eqs.(63) and (VIII)) become respectively

[TABLE]

As a consequence the limit functionals $L_{BG}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ and $L_{BG}\frac{\partial}{\partial t}I_{M}(\rho_{1}^{(N)}(t),\mathbf{b}),$ are necessarily identically vanishing. This means that the present theory applies properly when the exact Master kinetic equation is considered and not to its asymptotic approximation obtained in the Boltzmann-Grad limit, namely the Boltzmann kinetic equation (see Refs.noi3 ; noi7 ).

An interesting issue, in the context of the PMI/DKE theory for the Master kinetic equation, is the role of MCBC in giving rise to the phenomena of macroscopic irreversibility and decay to kinetic equilibrium. Let us analyze for this 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 referred to as the so-called mirror reflection CBC; see for example Cercignani Cercignani1969a ; Cercignani1975 ). This refers to the occurrence at a collision time $t_{i}$ of a single unary elastic collision for particle $1$ at the boundary $\partial\Omega.$** Let us denote by $\mathbf{n}_{1}$ **the inward normal to the stationary rigid boundary $\partial\Omega$ at the point of contact with the same particle and respectively $\mathbf{x}_{1}^{(-)}(t_{1})=\left(\mathbf{r}_{1}(t_{1}),\mathbf{v}_{1}^{(-)}(t_{1})\right)$ and $\mathbf{x}_{1}^{(+)}(t_{1})=\left(\mathbf{r}_{1}(t_{1}),\mathbf{v}_{1}^{(+)}(t_{1})\right)$ the incoming and outgoing particle states while $\mathbf{v}_{1}^{(+)}$ is determined by the elastic collision law for unary collisions, namely

[TABLE]

Then, the PDF-conserving CBC for the $1-$body PDF requires that the following identity holds

[TABLE]

with $\rho_{1}^{\left(N\right)}(\mathbf{x}_{1}^{(+)}(t_{1}),t_{i})\equiv\rho_{1}^{\left(N\right)(+)}(\mathbf{x}_{1}^{(+)}(t_{1}),t_{i})$ and $\rho^{\left(N\right)}(\mathbf{x}_{1}^{(-)}(t_{i}),t_{i})\equiv\rho^{\left(N\right)(-)}(\mathbf{x}_{1}^{(-)}(t_{i}),t_{i})$ denoting the outgoing and incoming $1-$body PDF respectively. This identifies the PDF-conserving CBC usually adopted in Boltzmann kinetic theory Boltzmann1972 (Grad Grad ; see also related discussions in Refs.noi2 ; noi3 ; noi4 ). The obvious physical implication of Eq.(48) is that $\rho_{1}^{\left(N\right)}(\mathbf{x}_{1}^{(+)}(t_{1}),t_{i})$ (and $\rho^{\left(N\right)}(\mathbf{x}_{1}^{(-)}(t_{i}),t_{i}))$ should be necessarily an even function of the velocity component $\mathbf{n}_{1}\cdot\mathbf{v}_{1}^{(-)}.$ Indeed as shown in Refs.noi2 ; noi3 the PDF-conserving CBC (48) should be replaced with a suitable CBC identified with the MCBC condition (see also Appendix C). When realized in terms of its causal form (predicting the outgoing PDF in terms of the incoming one) the MCBC for unary collisions is just:

[TABLE]

with $\rho^{\left(N\right)(-)}(\mathbf{x}_{1}^{(+)}(t_{i}),t_{i})$ denoting the incoming $1-$body PDF evaluated in terms of the outgoing state $\mathbf{x}_{1}^{(+)}(t_{i}).$ Assuming left-continuity (see related discussion in Ref.noi2 ). this can then be identified with $\rho^{\left(N\right)(-)}(\mathbf{x}_{1}^{(+)}(t_{i}),t_{i})\equiv\rho^{\left(N\right)}(\mathbf{x}_{1}^{(+)}(t_{i}),t_{i}),$ thus yielding

[TABLE]

Eq.(50) provides the physical prescription for the collision boundary condition, which is referred to as MCBC, holding for the $1-$body PDF at arbitrary unary collision events. It is immediate to realize that the function $\rho^{\left(N\right)}(\mathbf{x}_{1}^{(+)}(t_{i}),t_{i})$ need not generally be even with respect to the velocity component $\mathbf{n}_{1}\cdot\mathbf{v}_{1}^{(-)}.$ In addition Eq.(50), just as (48), also permits the existence of the customary collisional invariants which in the case of unary collisions are $X=1,\left|\mathbf{n}_{1}\cdot\mathbf{v}_{1}^{(-)}\right|,\mathbf{v}_{1}\cdot\left[\underline{\underline{\mathbf{1}}}-\mathbf{n}_{1}\mathbf{n}_{1}\right],$ $v_{1}^{2}.$ As a consequence, one can show that Eq.(50) warrants at the same time also the validity of the so-called no-slip boundary conditions for the fluid velocity field $\mathbf{V}(\mathbf{r}_{1},t)$ carried by the $1-$body PDF $\rho_{1}^{\left(N\right)}(\mathbf{x}_{1},t)$.

The treatment of MCBC holding for the $2-$body PDF in case of binary collision events is analogous and is recalled for convenience in Eq.(88) of Appendix C.

Let us briefly analyze the qualitative physical implications of Eqs.(50) and (88) 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 shown by Eq.(50), in such a case MCBC gives rise only to a change in the velocity distribution occurring during a unary collision due to a single component of the particle velocity, namely $\mathbf{n}_{1}\cdot\mathbf{v}_{1}^{(-)}$.** **As a consequence, this explains why unary collisions do not affect the rate of change of the MKI functional (see THM.2). Second, Eq.(88) shows - on the contrary - that binary collisions actually do affect by means of MCBC a velocity-spreading for the $1-$ and $2-$body PDF. In particular, since the spreading effect occurs in principle for all components of particle-velocities affecting both particles $1$ and $2$, this explains why binary collisions are actually responsible for the irreversible time-evolution of the MKI functional (see THM.s 2 and 3).

In turn, as implied by THM.3, DKE arises because of the phenomenon of macroscopic irreversibility (THM.2). The latter arises due specifically to the possible occurrence of a** **velocity-space anisotropy which characterizes the $1-$body PDF when the same PDF differs locally from kinetic equilibrium. In turn, this requires also that the $1-$body PDF belongs to the functional class of admissible stochastic PDFs $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t)\right\}$. In difference to Boltzmann kinetic theory, however, the key physical role is actually ascribed to the MKI functional $I_{M}(\rho_{1}^{(N)}(t))$ rather than the Boltzmann-Shannon entropy $S_{1}(\rho_{1}^{(N)}(t))$. In fact, as shown in Ref.noi6 the same functional remains constant in time once the Master kinetic equation is adopted. Rather, as shown by THM.2, it is actually the Master kinetic information $I_{M}(\rho_{1}^{(N)}(t))$ 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 kinetic equations. In particular, as discussed at length elsewhere (see Refs.noi1 ; noi3 ; noi6 ; noi7 ), precisely because the Boltzmann equation is only an asymptotic approximation of the Master kinetic equation explains why a loss of information occurs in Boltzmann 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 irreversibility property, i.e., the monotonic time-decay behavior of the MKI functional, can be explained at a more fundamental level, i.e., based specifically on the time-variation of the $\mathbf{b}-$directional total kinetic energy which occurs at arbitrary binary collision events.

The Master equation-DKE theorem (THM.3) given above provides a first-principle proof of the existence of the phenomenon of DKE occurring for the kinetic description of a finite number of extended hard-spheres, i.e., described by means of the Master kinetic equation. More precisely, the DKE phenomenon affects the $1-$body PDFs belonging to the admissible functional class $\left\{\rho_{1}^{(N)}(\mathbf{x}_{1},t)\right\}$ determined according to the MKI Prescription No.#0.

V 5 - Conclusions

In this paper the problem of the property of microscopic irreversibility (PMI) and decay to kinetic equilibrium (DKE) have been addressed. In doing so original ideas and implications are adopted of the new ”ab initio” approach for hard-sphere systems recently developed in the context of Classical Statistical Mechanics noi1 ; noi2 . These are not just small deviations from standard literature approaches. These developments, in fact, have opened up a host of exciting new problems and subjects of investigation in kinetic theory based on the Master kinetic equation for the so-called Boltzmann-Sinai classical dynamical system (CDS). In fact, the ”ab initio” approach, and the present paper in particular, represent an attempt at providing new foundational bases to the classical statistical mechanics of hard-sphere systems. The topic which has been pursued here - which represents also a challenging test for the validity of the new approach - concerns the investigation of the physical origins of PMI and the related DKE phenomenon arising *in finite *$N-$body hard-sphere systems. These issues refer in particular to:

• •

The proof of the non-negativity of Master kinetic information (THM.1, subsection 2A) together with the property of 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 kinetic theory. The main differences actually arise because of the non-asymptotic character of the new theory, i.e., the fact that it applies to arbitrary dense or rarefied systems for which the finite number and size of the constituent particles is accounted for noi3 . In this paper basic consequences of the new theory have been investigated which concern the phenomenon of decay to global kinetic equilibrium.

The present results are believed to be crucial, besides in mathematical research, for the physical applications of the ”ab initio” statistical theory, i.e., the Master kinetic equation. Indeed, regarding challenging future developments of the theory one should mention among others the following examples of possible (and mutually-related) routes worth to be explored. One is related to the investigation of the time-asymptotic properties of the same kinetic equation, for which the present paper may represent a useful basis. The second goal refers to the possible extension of the theory to mixtures formed by hard spheres of different masses and diameter which possibly undergo both elastic and anelastic collisions. The third one concerns the investigation of hydrodynamic regimes for which a key prerequisite is provided by the DKE theory here established.

VI Acknowledgments

Work 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 GAČR grant No. 14-07753P; C) the grant No. 02494/2013/RRC “kinetický přístup k proudĕní tekutin” (kinetic approach to fluid flow) 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 Consortium for Magnetofluid Dynamics (University of Trieste, Italy) and the MIUR (Italian Ministry for Universities and Research) PRIN Research Program “Problemi Matematici delle Teorie Cinetiche e Applicazioni”, University of Trieste, Italy. One of the authors (M.T.) is grateful to the International Center for Theoretical Physics (Miramare, Trieste, Italy) for the hospitality during the preparation of the manuscript.

VII Appendix A: Realizations of the Master kinetic equation

For completeness we recall here the two equivalent forms of the Master kinetic equation noi3 . In terms of the renormalized $1-$body PDF $\widehat{\rho}_{1}^{(N)}(\mathbf{x}_{1},t)$ (see Eq.(12) ) the *first form *of the same equation reads

[TABLE]

with $L_{1\left(1\right)}=\frac{\partial}{\partial t}+\mathbf{v}_{1}\cdot\frac{\partial}{\partial\mathbf{r}_{1}}$ denoting the $1-$body free-streaming operator. Hence it follows

[TABLE]

where explicit evaluation of the rhs the last equation (see also Eq.(84) below) yields

[TABLE]

with $\overline{\Theta}^{\ast}(\mathbf{r}_{2})\equiv\overline{\Theta}_{i}^{(\partial\Omega)}(\overline{\mathbf{r}})$ and $k_{2}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},t)$ being identified with the definitions given respectively by Eqs. (56) and Eq.(63) in Appendix B. Then consistent with Ref.noi3 and upon invoking the causal form of MCBC (see Eq.(89) in Appendix C) the same equation can be equivalent written in the equivalent second form of the Master kinetic equation noi3 . The corresponding initial-value problem, taking the form:

[TABLE]

can be shown to admit a unique global solution noi7 . Here the notation is standard noi3 . Thus

[TABLE]

identifies the Master collision operator while $\rho_{1o}^{(N)}(\mathbf{x}_{1})$ is the initial $1-$body PDF which belongs to the functional class $\left\{\rho_{1o}^{(N)}(\mathbf{x}_{1})\right\}$ of stochastic, i.e., strictly-positive, smooth ordinary functions, $1-$body PDFs. Furthermore, the solid-angle integral on the rhs of Eq.(55) is now evaluated on the subset in which $\mathbf{v}_{12}\cdot\mathbf{n}_{12}<0,$ while $\mathbf{r}_{2}$ identifies* *$\mathbf{r}_{2}=\mathbf{r}_{1}+\sigma\mathbf{n}_{21}$, while $k_{1}^{(N)}(\mathbf{r}_{1},t)$ and $k_{2}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},t)$ coincide respectively with the $1-$ and $2-$body occupation coefficients noi3 and $\overline{\Theta}^{\ast}\equiv\overline{\Theta}^{\ast}(\mathbf{r}_{i})$ is prescribed by

[TABLE]

with $\overline{\Theta}(x)$ being the strong Heaviside theta function \overline{\Theta}(x)=\left\{\begin{array}[]{lll}1&&y>0\\ 0&&y\leq 0\end{array}\right..

Regarding the specific identification of the occupation coefficients let us preliminarily recall the notion of $S_{N}-$* ensemble strong theta-function* $\overline{\Theta}^{(N)}.$ The latter is prescribed, according to Ref.noi3 , by requiring that

[TABLE]

for all configuration vectors $\overline{\mathbf{r}}\equiv\left\{\mathbf{r}_{1},...,\mathbf{r}_{N}\right\}$ belonging to the collisionless subset of $\Omega^{(N)}$. This is identified with the open subset of the $N-$body configuration domain $\Omega^{(N)}\equiv\prod\limits_{i=1,N}\Omega$ in which each of the particles of $S_{N}$ is not in mutual contact with any other particle of $S_{N}$ or with the boundary $\theta\Omega$ of $\Omega.$ this can be prescribed in terms of the n, i.e., in such a way that identically In agreement with Ref.noi3 this occurs for $N-$body PDFs which are represented by ordinary functions (i.e., are stochastic). This means that $\overline{\Theta}^{(N)}(\overline{\mathbf{r}})$ can be prescribed as

[TABLE]

Here $\overline{\Theta}_{i}^{(\partial\Omega)}(\overline{\mathbf{r}})$ identifies the $i-$th particle ”boundary” theta function

[TABLE]

with $\mathbf{r}_{Wi}=\mathbf{r}_{i}-\rho\mathbf{n}_{i}$ and $\rho\mathbf{n}_{i}$ the inward vector normal to the boundary belonging to the center of the $i-$th particle having a distance $\rho$ from the same boundary. Furthermore $\overline{\Theta}_{i}(\overline{\mathbf{r}})$ is the ”binary-collision” theta function. A possible identification of $\overline{\Theta}_{i}(\overline{\mathbf{r}})$ which warrants validity of Eq.(57) is manifestly given by the expression

[TABLE]

namely

[TABLE]

However an equivalent possible prescription of $\overline{\Theta}_{i}(\overline{\mathbf{r}})$ is also provided by the alternative realization obtained letting

[TABLE]

Indeed by construction in the subset of $\Omega^{(N)}$ in which for all $i=1,N$ the rhs of Eq.(60) is identically equal to unity the factor $\prod\limits_{\begin{subarray}{c}m,n=1,N\\ i<m<n\end{subarray}}\overline{\Theta}(\left|\mathbf{r}_{i}-\mathbf{r}_{m}\right|+\left|\mathbf{r}_{i}-\mathbf{r}_{n}\right|-2\sigma)\overline{\Theta}(\left|\mathbf{r}_{m}-\mathbf{r}_{n}\right|+\left|\mathbf{r}_{i}-\mathbf{r}_{m}\right|-2\sigma)$ is necessarily equal to unity too. Incidentally, we notice in fact that the latter factor carries the contributions due to triple collisions which are by construction ruled out in the domain of validity of Eq.(57).

VIII Appendix B - Integral and differential identities

One notices that although the definitions (62) and (60) given in Appendix A for $\overline{\Theta}_{i}(\overline{\mathbf{r}})$ coincide in the collisionless subset of $\Omega^{(N)},$ only the first one is applicable in the complementary collision subset. Based on these premises in this appendix a number of integral and differential identities holding for the $1-$ and $2-$body occupation coefficients are displayed.

First, recalling Ref.noi3 , one notices that the realizations of the $1-$ and $s-$body occupation coefficients $k_{1}^{(N)}(\mathbf{r}_{i},t),k_{2}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},t),$…, $k_{s}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},..\mathbf{r}_{s},t)$ remain uniquely prescribed by the $1-$body PDF, being given by

[TABLE]

where $F_{s}$ denotes the integral operator

[TABLE]

Therefore, since in the collisionless subset of $\Omega^{(N)}$ the prescriptions (60) and (62) are equivalent, in the same subset the $1-$ and $2-$body occupation coefficients, written in terms of Eq.(60), become explicitly

[TABLE]

and

[TABLE]

Accordingly letting $\mathbf{n}_{jj}=\mathbf{r}_{uij}/\left|\mathbf{r}_{ij}\right|$ with $\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$, one notices that in the collisionless subset of $\Omega^{(N)}$ the following differential identities hold for all $s=1,N-1$:

[TABLE]

As a consequence the following identities (the first one needed to evaluate the rhs of Eq.(52) in Appendix A)

[TABLE]

[TABLE]

hold too. However, the alternative realization of the factor $\overline{\Theta}_{i}(\overline{\mathbf{r}})$ given by Eq.(62) (see Appendix A) has the virtue of excluding explicitly explicitly multiple collisions. The consequence is that when such a definition is adopted the differential identities

[TABLE]

both hold identically. The latter equations, in fact, manifestly hold also in the collision subset where $\delta\left(\left|\mathbf{r}_{2}-\mathbf{r}_{1}\right|-\sigma\right)\neq 0.$

IX Appendix C: Causal and anti-causal forms of collisional boundary

conditions

For definiteness, let us denote respectively the outgoing and incoming $N-$body PDFs $\rho^{(-)\left(N\right)}(\mathbf{x}^{\left(-\right)}(t_{i}),t_{i})$ and $\rho^{(+)\left(N\right)}(\mathbf{x}^{\left(+\right)}(t_{i}),t_{i})$, with $\rho^{(\pm)\left(N\right)}(\mathbf{x}^{\left(\pm\right)}(t_{i}),t_{i})=\lim_{t\rightarrow t_{i}^{\left(\pm\right)}}\rho^{\left(N\right)}(\mathbf{x}(t),t)$, where $\mathbf{x}^{(-)}(t_{i})$ and $\mathbf{x}^{(+)}(t_{i})$, with $\mathbf{x}^{\left(\pm\right)}(t_{i})=\lim_{t\rightarrow t_{i}^{\left(\pm\right)}}\mathbf{x}(t),$ are the incoming and outgoing Lagrangian $N-$body states, their mutual relationship being again determined 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 $O\left(\mathbf{r},\tau\equiv t-t_{o}\right)$, having respectively spatial and time origins at the point $O$ which belongs to the Euclidean space $\mathbb{R}^{3}$ and at time $t_{o}\in I$.

• •

In addition, by assumption the time-axis is oriented. Such an orientation is referred to as microscopic arrow of time.

For an arbitrary $N-$body PDF $\rho^{\left(N\right)}(\mathbf{x},t)$ belonging to the extended functional setting and an arbitrary collision event occurring at time $t_{i}$ two possible realizations of the MCBC can in principle be given, both yielding a relationship between the PDFs $\rho^{(+)\left(N\right)}$ and $\rho^{(-)\left(N\right)}$. In the context of the ”ab initio” statistical approach based on the Master kinetic equation noi1 ; noi2 ; noi3 ; noi4 ; noi5 ; noi6 ; noi7 these are provided by the two possible realizations of the so-called modified CBC (MCBC). When expressed in Lagrangian form they are realized respectively either by the causal and* anti-causal MCBC*, namely

[TABLE]

or

[TABLE]

The corresponding Eulerian forms of the MCBC can easily be determined (see Ref.noi6 ). The one corresponding to Eq.(87) is, for example, provided by the condition

[TABLE]

where now $\mathbf{x}^{\left(+\right)}$ 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 principle in the reference frame $\left(\mathbf{r},\tau\equiv t-t_{o}\right)$ manifestly requires invoking Eq.(87). Indeed, Eq.(87) predicts the future (i.e., outgoing) PDF from the past (incoming) one. Therefore the choice (87) is the one which is manifestly consistent with the causality principle. On the other hand, if the arrow of time is changed, i.e. the time-reversal transformation with respect to the initial time (or time-origin) $t_{o},$ i.e., the map between the two reference frames

[TABLE]

with $\tau^{\prime}=-\tau$ is performed, it is obvious that for the transformed reference frame $O\left(\mathbf{r},\tau^{\prime}\right)$ the form of CBC consistent with causality principle becomes that given by Eq.(88). Analogous conclusions hold if a velocity-reversal is performed, implying the incoming states and corresponding PDF must be exchanged with corresponding outgoing ones and vice versa.

X 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 the special case $N=2.$ For this purpose, one notices that - thanks to Eq.(VIII) recalled in Appendix B (see also Ref.noi3 ) - in this case by construction $k_{2}^{(N)}(\mathbf{r}_{1},\mathbf{r}_{2},t)$ simply reduces to

[TABLE]

Accordingly, once the same prescription is invoked, both the Master kinetic equation (54) and the corresponding Master collision operator (55) remain formally unchanged. In a similar way it is important to remark that the expression of the functional $\frac{\partial}{\partial t}K_{M}(\rho_{1}^{(N)}(t),\mathbf{b})\equiv W_{M}(\rho_{1}^{(N)}(t),\mathbf{b})$ given by Eq. (38) is still correct also in such a case, being now given by

[TABLE]

It is then immediate to infer the validity of both the PMI theorem (THM.2) and the DKE property for the Master kinetic 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. In fact, binary collisions, as indicated above, are responsible for the MPI/DKE phenomenology and in such a case can only occur between particles $1$ and $2.$