∂C$_{k}$/∂ρ$_{j}$ ∼ δ − $^{1}$C$_{k}$ , similarly to Φ $^{′}$( ρ$_{j}$ ) ∼ δ − $^{1}$Φ( ρ$_{j}$ ) . Hence, the second and the fourth right-hand terms in Eq.(9), Eq.(10), etc., both are infinitely increasing ∝ δ − 1 in collision regions and, in view of absence of other such terms, should compensate one another. Equivalently, the expressions in square brackets in Eqs.11 should be- come eigenfunctions of operators L$_{j}$ corresponding to zero eigenvalues, that is same way as pre-collision momenta P in , p in in (14) are related to post-collision ones, P out , p $^{out}$, and the integration involves pre-collision states only. What is for the operators L 0 j , in respect to CFs they are defined as L 0 j = − ( v$_{j}$ − V ) · ∂/∂ρ$_{j}$ at | ρ$_{j}$ | > r$_{0}$ and by boundary conditions (13) at | ρ$_{j}$ | = r$_{0}$ . Thus, importantly, L 0 j are not mere translation operators: at collision surfaces | ρ$_{j}$ | = r$_{0}$ they act as creation oper- ators, creating ( k + 1)-order correlations from k -order ones. Therefore factually C$_{k}$ remain coupled with both C$_{k}$$_{+1}$ and C$_{k}$$_{−}$$_{1}$ , as in basic Eqs.4. In other words, due to conditions (13), hierarchy (17) keeps character- istic three-diagonal structure of equations for CFs ! Return from CFs back to DFs, according to the CFs definition (3), transforms these equations to asymptotically, inside the collision regions. This statement. in its turn, means that expression in the square brackets represents integral of motion, and its values at beginning and at end of the pair collision,are equal. Thus, visualizing only variables what change dur- ing collision, we can write where ρ$_{j}$ = r$_{0}$ Ω , with Ω being unit normal vector, and in -states and out -states correspond to Ω · u$_{j}$ < 0 and Ω · u$_{j}$ > 0 , respectively. and are connected via the limit collision dynamics (mirror reflection): and conditions (13) to along with (14). Their substitution into the “collision integral’ ̂ Γ F$_{k}$$_{+1}$ gives, similarly to (17), k On the other hand, integration of (12) over the collision region, - for instance, at j = k , - yields Substituting this equality into Eqs.11, for complement of the now forbidden regions | ρ$_{j}$ | ≤ r$_{0}$ we obtain Here again L 0 j = − ( v$_{j}$ − V ) · ∂/∂ρ$_{j}$ at | ρ$_{j}$ | > r$_{0}$ but at | ρ$_{j}$ | = r$_{0}$ action of the operator L 0 j onto DFs is defined by boundary conditions (19). Equations (16) as combined with boundary conditions (13)-(14) at collision surfaces | ρ$_{j}$ | = r$_{0}$ (and conditions (5) at infinity) or, equivalently, equations (18) combined with (19) and (14) represent direct analogue of so-called “hard-sphere BBGKY hierarchy” (see e.g. Eq.2.2 from [28] or Eq.4.1 from [29]). Importantly, the term “hard-sphere BBGKY hierar- chy” is adequate on the understanding only that when the boundary conditions are substituted into F$_{k}$$_{+1}$ inside the “collision integral” in equation for F$_{k}$ then simul- taneously and necessarily they are satisfied by F$_{k}$$_{+1}$ in the next equation for F$_{k}$$_{+1}$ itself (i.e. included into definition of the operators L 0 j as above). Otherwise one makes some “mathematical non-physics”, since resulting equations will be non-derivable from Liouville equation for an (infinitely) many hard sphere system ! with ρ = ρ$_{k}$$_{+1}$ , p = p$_{k}$$_{+1}$ . v = v$_{k}$$_{+1}$ = p$_{k}$$_{+1}$ /m , and momenta P ∗ , p ∗ being related to P , p in the where u$_{j}$ = v$_{j}$ − V . At boundaries of the forbidden regions. i.e. at | ρ$_{j}$ | = r$_{0}$ , these equations should be sup- plemented with boundary conditions (13)-(14). Combin- ing Eqs.16 with (15) and (13) we come to ∂C$_{k}$ ∂t = − V · ∂C$_{k}$ ∂ R + k ∑ j =1 L 0 j C$_{k}$ + + 1 πλ ∫ d p ∫$_{(}$ $^{V}$−v ) · Ω > 0 d Ω (( V − v ) · Ω) × (17) × [ G$_{m}$ ( p $^{∗}$) C$_{k}$ ( P $^{∗}$) − G$_{m}$ ( p ) C$_{k}$ ( P ) + + C$_{k}$$_{+1}$ ( ρ = r$_{0}$ Ω , P $^{∗}$, p $^{∗}$) − C$_{k}$$_{+1}$ ( ρ = r$_{0}$ Ω , P , p ) ] ∂C$_{k}$ ∂t = − V · ∂C$_{k}$ ∂ R − k ∑ j =1 u$_{j}$ · ∂C$_{k}$ ∂ρ$_{j}$ − n ̂ Γ C$_{k}$$_{+1}$ , (16) − ∂ ∂ P ∫$_{k}$ Φ $^{′}$( ρ$_{k}$ ) C$_{k}$ = − ̂ Γ C$_{k}$ ≡ (15) ≡ r 2 0 ∫ ∫ (( v$_{k}$ − V ) · Ω) C$_{k}$ ( ρ$_{k}$ = r$_{0}$ Ω) d p$_{k}$ d Ω P in + p in j = P out + p out j , Ω · ( v out j − V $^{out}$) = − Ω · ( v in j − V $^{in}$) , (14) ( 1 − Ω ⊗ Ω) ( v out j − V $^{out}$) = ( 1 − Ω ⊗ Ω) ( v in j − V $^{in}$) ∂F$_{k}$ ∂t = − V · ∂F$_{k}$ ∂ R + ∑ j =1 L 0 j F$_{k}$ + + 1 πλ ∫ d p ∫$_{(}$ $^{V}$−v ) · Ω > 0 d Ω (( V − v ) · Ω) × (20) × [ F$_{k}$$_{+1}$ ( ρ = r$_{0}$ Ω , P $^{∗}$, p $^{∗}$) − F$_{k}$$_{+1}$ ( ρ = r$_{0}$ Ω , P , p ) ] F$_{k}$ ( P $^{in}$, p in j ) = F$_{k}$ ( P $^{out}$, p out j ) at | ρ$_{j}$ | = r$_{0}$ (19) ∂F$_{k}$ ∂t = − V · ∂F$_{k}$ ∂ R − k ∑ j =1 u$_{j}$ · ∂F$_{k}$ ∂ρ$_{j}$ − n ̂ Γ F$_{k}$$_{+1}$ (18) C$_{k}$ ( P $^{in}$, p in j ) + G$_{m}$ ( p in j ) P$_{kj}$ C$_{k}$$_{−}$$_{1}$ ( P $^{in}$) = (13) = C$_{k}$ ( P $^{out}$, p out j ) + G$_{m}$ ( p out j ) P$_{kj}$ C$_{k}$$_{−}$$_{1}$ ( P $^{out}$) , L$_{j}$ [ C$_{k}$ + G$_{m}$ ( p$_{j}$ ) E ( ρ$_{j}$ ) P$_{kj}$ C$_{k}$$_{−}$$_{1}$ ] = 0 (12) 4 V. THE BOLTZMANN-GRAD LIMIT AND “MATHEMATICAL NON-PHYSICS”