In opposite, fulfilment of the mentioned requirement guarantees observance of the virial relations (see Ap- pendix) and consequently belonging of resulting equa- tions to the class of BBGKY hierarchies (since virial re- lations do follow already from most general properties of many-particle dynamics and Liouville equations [1–3]). In view of these facts, it seems impossible to accept the old idea that under the Boltzmann-Grad limit (BGL) the “hard-sphere BBGKY hierarchy” is equivalent to so- called “hard-sphere Boltzmann hierarchy” (see e.g. [24, 28, 29, 31]). For our system it looks as With time, as was explained in Sec.3, also quite similar pre-collision correlations do arise, at u$_{j}$ · ρ$_{j}$ < 0 and again at | u$_{j}$ × ρ$_{j}$ | / | u$_{j}$ | < r$_{0}$ , since statistical picture acquires more and more time-reversal symmetry. At the same time, at | u$_{j}$ × ρ$_{j}$ | / | u$_{j}$ | > r$_{0}$ there are al- most no reasons for correlations. Hence, in directions perpendicular to u$_{j}$ all CFs become more and more sharp functions of ρ$_{j}$ . At that. the integral value of correlation per one atom goes to zero ∝ πr 2 $_{0}$λ = 1 /n . But this has no significance since correlations concentrate just where they are most effective. This is confirmed by the virial relations (7). The latter show also that really significant correlational characteristics are n $^{k}$C$_{k}$ which turn under BGL into singular generalized functions. where there are no forbidden regions, that is ρ$_{j}$ can take arbitrary values from the whole R $^{3}$, and the boundary conditions (19) at | ρ$_{j}$ | → r$_{0}$ → 0 are thrown, that is op- erators L 0 j are replaced by trivial translation operators, − ( v$_{j}$ − V ) · ∂/∂ρ$_{j}$ . We have seen that the Boltzmann-Grad limit does not lead to any essential simplifications (and in this sense it does not exist). In particular, it does not lead to the mythologic Botzmannian kinetics with Boltzmann equa- tion, Boltzmann-Lorentz equation, etc.), except non- interesting spatially homogeneous case [39]. Physical and statistical reasons of this pleasant fact already were over and over again explained in [6–8] and earlier [12–17] and later [1, 3, 4, 19, 22, 40, 41]. The fact is pleasant because it shows once again that real many-particle dynamical chaos does not degenerates into miserable stochastics. This means that in corresponding equations for CFs any C$_{k}$ is now connected to C$_{k}$$_{+1}$ only, and not to C$_{k}$$_{−}$$_{1}$ . Thus, these equations form two-diagonal hier- archy qualitatively different from the original one ! As the result of such frivolity, these equations allow for factored solution with “propagation of chaos”, when F$_{k}$ = F$_{0}$ ∏ k j =1 G$_{m}$ ( p$_{j}$ ) , C$_{k}$ = 0 at k > 0 . and F$_{0}$ = C$_{0}$ undergoes the “Boltzmann-Lorentz equation”. Evidently, this “Boltzmann hierarchy” contraries to the above emphasized requirement: it uses the boundary conditions to write “collision integrals” but neglects the same conditions in higher equations what determine the integrands (as if the latter took particles from a “parallel world”) ! As the consequence, the Boltzmann hierarchy does not satisfy the virial relations. Its wrong is clear already from observation that it results when one first damages Eqs.20 by replacing L 0 j with − ( v$_{j}$ − V ) · ∂/∂ρ$_{j}$ and only after that goes to r$_{0}$ = 0 [37]. Therefore, it is not surprising that the famous Lanford’s attempt [24] to prove the mentioned idea was in fact unsuccessful [8, 38]. Possible formal cause of this nonsuccess is very simple: the smaller is r$_{0}$ the less smooth functions of ρ$_{j}$ are all the CFs [3, 4], since conditions (19) ensure continuity and smoothness of the probability measures F$_{k}$ along phase trajectories only but not in perpendicular dimensions. Indeed, let u$_{j}$ · ρ$_{j}$ > 0 , that is BP and an atom scat- ter one from another. If at that r$_{0}$ ≪ | ρ$_{j}$ | ≲ λ and additionally | u$_{j}$ × ρ$_{j}$ | / | u$_{j}$ | < r$_{0}$ (where × denotes vec- tor product), i.e. vectors u$_{j}$ and ρ$_{j}$ are nearly paral- lel, then the particles form an out -state which (almost certainly) arose from their recent collision (especially if | ρ$_{j}$ | ≪ λ ).According to (13), post-collision correlations of any order between such particles inevitably take place, and by order of magnitude all they are equal to C$_{0}$ = F$_{0}$ . to the virial relations (6), ∂C$_{k}$/∂n = ∫ k +1 C$_{k}$$_{+1}$ . Notice that this consideration is valid regardless of con- crete form of boundary conditions at | ρ$_{j}$ | = r$_{0}$ . There- fore, virial relations hold under replacement of the hard- sphere “mirror reflection” conditions (14) by some other rule. Correspondingly, equations (16) and (18) can be ex- tended to collision operators ̂ Γ with other “scattering laws” than (14). which are identical to the previous ones. Since, according to (5), initial conditions for C$_{k}$ and ˜ C$_{k}$ also are identical (all are zeros), we come to equalities C$_{k}$ = ˜ C$_{k}$ , that is to the virial relations (6), ∂C$_{k}$/∂n = k +1 C$_{k}$$_{+1}$ . At the same time, differentiation of Eqs.16 in respect to the density produces for C$_{k}$ = ∂C$_{k}$/∂n equations Integration of ( k +1)-th of Eqs.16 over p$_{k}$$_{+1}$ and ρ$_{k}$$_{+1}$ at | ρ$_{k}$$_{+1}$ | > r$_{0}$ yields for ˜ C$_{k}$ = ∫ k $_{+1}$C$_{k}$$_{+1}$ , in view of (5), equations ∂F$_{k}$ ∂t = − V · ∂F$_{k}$ ∂ R − k ∑ j =1 ( v$_{j}$ − V ) · ∂F$_{k}$ ∂ρ$_{j}$ + + 1 πλ ∫ d p ∫$_{(}$ $^{V}$−v ) · Ω > 0 d Ω (( V − v ) · Ω) × (21) × [ F$_{k}$$_{+1}$ ( ρ = 0 , P $^{∗}$, p $^{∗}$) − F$_{k}$$_{+1}$ ( ρ = 0 , P , p ) ] , ∂C$_{k}$ ∂t = − V · ∂C$_{k}$ ∂ R − k ∑ j =1 u$_{j}$ · ∂C$_{k}$ ∂ρ$_{j}$ − ̂ Γ C$_{k}$$_{+1}$ − n ̂ Γ C$_{k}$$_{+1}$ ∂ ˜ C$_{k}$ ∂t = − V · ∂ ˜ C$_{k}$ ∂ R − k ∑ j =1 u$_{j}$ · ∂ ˜ C$_{k}$ ∂ρ$_{j}$ − ̂ Γ C$_{k}$$_{+1}$ − n ̂ Γ ˜ C$_{k}$$_{+1}$ Appendix: Virial relations for hard-sphere system VI. CONCLUSION 5