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r$_{0}$ of the potential Φ( ρ ) , corresponding free-path length of BP, λ = ( πr 2 $_{0}$n ) − $^{1}$, and integrated correlations 

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Thus, W$_{0}$ ( t, R ; λ ) = ∫ F$_{0}$ ( t, R , P ; 1 /πr 2 $_{0}$λ ) d P is proba- bility distribution of the BP’s path. Then Eqs.6 yield 

These exact relations hold at any values of r$_{0}$ and n , hence, under BGL ( r$_{0}$ → 0 , n → ∞ , λ = const) too. They show that anyway by order of magnitude W$_{k}$ ( t, R ; λ ) ∼ W$_{0}$ ( t, R ; λ ) . 

## III. PRE-COLLISION CORRELATIONS AND FAILURE OF MOLECULAR CHAOS 

Is the Boltzmann’s molecular chaos (“Stoßahlansatz”) compatible with virial relations (6), (7)? May be, cor- relations there are purely post-collision and therefore do not contradict Boltzmann s ansatz proclaiming absence of pre-collision correlations? 

Unfortunately, this is vain hope, and the answer is no. To make sure of this, let us agree that post-collision ( out - ) and pre-collision ( in -) states of BP and an atom satisfy ( ρ · u ) > 0 or ( ρ · u ) < 0 , respectively, - where ρ = r − R , u = v − V , with V = P /M and v$_{j}$ = p$_{j}$ /m being velocities, - and consider the first three of equations (4), 

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Here, we introduced the Liouville operator 

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and made use of the relative atoms’ coordinates ρ$_{j}$ . 

Evidently, the last term in Eq.9 represents generation of pair correlation by BP-atom interaction, and the result is “post-collision correlation” since it passes to finishing “ out -state” of the particles. If we neglected three-particle correlation represented by CF C$_{2}$ then solution to Eq.9 would be just this post-collision correlation: C$_{1}$ = C out 1 . Substitution of this C$_{1}$ to Eq.8 would yield a closed “Boltzmann-Lorentz” equation for the BP’s distribution C$_{0}$ = F$_{0}$ , thus realizing the dream of “molecular chaos”. 

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However, the last term in Eq.10, for C$_{2}$ , quite simi- larly (even under neglect of C$_{3}$ ) produces three-particle correlations out of the two-particle one, C out 1 . Resulting C$_{2}$ contributes, via integral in Eq.9, back to evolution of C$_{1}$ , and produces, in particular, pre-collision pair cor- relations between particles going to meet one another. In such way, C$_{1}$ acquires pre-collision component C in 1 , thus turning the dream into a rough approximation. This is illustrated schematically by Fig.1. 

For further, first notice that Eqs.4, 8-10 can be written in equivalent form 

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Since these equations, as well as virial relations (6) and (7), are valid at any smooth interaction potential, we can extend them to the limit of “hard-sphere” interaction, for instance, defined by Φ( r ) = ( r$_{0}$/ | r | ) p Φ$_{0}$ with p → ∞ and Φ$_{0}$ = const ∼ T . At that, clearly, the BP’s free path length must tend to a constant, λ → const . Therefore at any stage of this limit transition all CFs C$_{k}$ are equally smooth functions of time (except the very beginning of evolution), momenta and coordinates, excluding “colli- sion regions” | ρ$_{j}$ | < r$_{0}$ + ϵ , where ϵ → 0 along with char- acteristic thickness h of the potential wall δ = r$_{0}$/p → 0 (at that, satisfying ϵ/δ → ∞ ). 

In these collision regions, in opposite, C$_{k}$ ( k > 0 ) become more and more sharp functions of ρ$_{j}$ , so that 

## IV. HARD-SPHERE LIMIT 

