Macroscopic Irreversibility and Decay to Kinetic Equilibrium of the 1-Body PDF for Finite Hard-Sphere Systems

Massimo Tessarotto,Claudio Cremaschini

doi:10.1155/2018/1931308

Abstract

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.

Highlights

The axiomatic theory of classical statistical mechanics (CSM) recently proposed in a series of papers 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 point particles subject to finite-range interactions [5]
Its theoretical basis and conditions of validity are 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 functions or distributions such as the N−body Dirac delta
This feature is physically based being due to the prescription of the collision boundary conditions (CBC, [2]), i.e., the relationship occurring at collision events between incoming and outgoing multibody probability density functions PDF

Summary

Introduction

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 point particles subject to finite-range interactions [5]. The crucial new results that we intend to display in this paper concern the proof-of-principle of two phenomena which are expected to characterize the statistical description of finite N−body hard-sphere systems and 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.

Axiomatic Prescription of the MKI Functional

Physical Implications

Conclusions

Realizations of the Master Kinetic Equation

Integral and Differential Identities for the Occupation Coefficients

Causal and Anticausal Forms of Collisional Boundary Conditions