r$_{0}$ of the potential Φ( ρ ) , corresponding free-path length of BP, λ = ( πr 2 $_{0}$n ) − $^{1}$, and integrated correlations 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 ) . 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), 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 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 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”. and made use of the relative atoms’ coordinates ρ$_{j}$ . Here, we introduced the Liouville operator ∂C$_{k}$ ∂t = − V · ∂C$_{k}$ ∂ R − n ∂ ∂ P ∫$_{k}$$_{+1}$ Φ $^{′}$( ρ$_{k}$$_{+1}$ ) C$_{k}$$_{+1}$ + + k ∑ j =1 L$_{j}$ [ C$_{k}$ + G$_{m}$ ( p$_{j}$ ) E ( ρ$_{j}$ ) P$_{kj}$ C$_{k}$$_{−}$$_{1}$ ] (11) L$_{j}$ = ( V − v$_{j}$ ) · ∂ ∂ρ$_{j}$ + Φ $^{′}$( ρ$_{j}$ ) · ( ∂ ∂ p$_{j}$ − ∂ ∂ P ) ∂C$_{2}$ ∂t = − V · ∂C$_{2}$ ∂ R + ( L$_{1}$ + L$_{2}$ ) C$_{2}$ − n ∂ ∂ P ∫$_{3}$ Φ $^{′}$( ρ$_{3}$ ) C$_{3}$ + + T (1 + P$_{21}$ ) G$_{m}$ ( p$_{2}$ ) E $^{′}$( ρ$_{2}$ ) [ P M T + ∂ ∂ P ] C$_{1}$ (10) ∂C$_{0}$ ∂t = − V · ∂C$_{0}$ ∂ R − n ∂ ∂ P ∫$_{1}$ Φ $^{′}$( ρ$_{1}$ ) C$_{1}$ , (8) ∂C$_{1}$ ∂t = − V · ∂C$_{1}$ ∂ R + L$_{1}$ C$_{1}$ − n ∂ ∂ P ∫$_{3}$ Φ $^{′}$( ρ$_{2}$ ) C$_{2}$ + + T G$_{m}$ ( p$_{1}$ ) E $^{′}$( ρ$_{1}$ ) [ P M T + ∂ ∂ P ] C$_{0}$ , (9) W$_{k}$ ( t, R ; λ ) = 1 λ k ( ∂ ∂λ − 1 ) k W$_{0}$ ( t, R ; λ ) (7) W$_{k}$ ( t, R ; λ ) ≡ n k ∫ [∫$_{1}$ ... ∫$_{k}$ C$_{k}$ ( t, R , r ( k $^{)}$, P , p ( k $^{)}$; n ) ] d P Time Coordinates C$_{1}$ out C$_{1}$ in C$_{2}$ FIG. 1. A simplest diagram transforming post-collision pair correlation, C out 1 , into pre-collision pair correlation, C in 1 , through three-particle correlation, C$_{2}$ . The “Brownian par- ticle” is represented by circles and entire arrows while gas atoms by dots, light lines and short arrows. III. PRE-COLLISION CORRELATIONS AND FAILURE OF MOLECULAR CHAOS IV. HARD-SPHERE LIMIT 3