content_image/396402.jpg content_image/396401.jpg content_image/396397.jpg content_image/396398.jpg content_image/396399.jpg content_image/396400.jpg
After their consideration in Sec.3 we will see for our- selves that collisions constantly give birth to various sta- tistical correlations, and not only post-collision ones but also pre-collision correlations. Then, in Sec.4, discuss and execute the hard-sphere limit of our BBGKY hierar- chy, taking in mind that it should stay compatible with the virial relations as they are independent on charac- ter of interactions (interaction potentials). The result is (i) modified BBGKY equations combined with (ii) fa- miliar boundary conditions to them at hyper-surfaces (in many-particle phase spaces) corresponding to collisions. The second ingredient allows to transform the first into usual “hard-sphere BBGKY hierarchy” but, at the same time, it forbids its reduction to the “Boltzmann hierar- chy” [24, 25, 28, 29, 31] under the Boltzmann-Grad limit. Thus, the correct theory does not present such marvel- lous simplifications as “propagation of chaos” and closed equation for one-particle distribution function (DF): as before, one has to solve an infinite hierarchy of equations! This situation will be discussed in Sec.5. 

## II. BBGKY HIERARCHY, ITS CUMULANT REPRESENTATION, AND VIRIAL RELATIONS 

We want to consider random walk R ( t ) of a Brown- ian particle (BP) in thermodynamically equilibrium ideal gas, assuming that at initial time moment t = 0 BP starts from certainly known position R (0) = 0 . The BBGKY equations for this problem can be either de- rived [22] directly from the Liouville equation, following Bogolyubov [5], or extracted [21] from general results of [1–3]. They reads as 

<img src='content_image/396402.jpg'>

where k = 0 , 1 , ... , H$_{k}$ is Hamiltonian of subsystem “ k atoms + BP ”, [ ..., ... ] means the Poisson brackets, Φ( r ) is potential of (short-range repulsive) interaction between BP and atoms, Φ $^{′}$( r ) = ∇ Φ( r ) , n is gas density (mean concentration of atoms), F$_{0}$ = F$_{0}$ ( t, R , P ; n ) is normal- ized DF of the BP’s coordinate R and momentum P , F$_{k}$ = F$_{k}$ ( t, R , r ( k $^{)}$, P , p ( k ) ; n ) are ( k + 1)-particle DFs for BP and k atoms [33], r$_{j}$ and p$_{j}$ denote coordinates and momenta of atoms, respectively, r ( k ) = { r$_{1}$ ... r$_{k}$ } , p ( k ) = { p$_{1}$ ... p$_{k}$ } , and ∫ k ... = ∫ ∫ ... d r$_{k}$ d p$_{k}$ . Initial conditions to these equations, corresponding to 

the gas equilibrium, at temperature T , are 

where M and m are masses of BP and atoms, re- spectively, G$_{m}$ ( p ) = (2 πT m ) − 3 / $^{2}$exp ( − p $^{2}$/ 2 T m ) is the Maxwell momentum distribution of a particle with mass m , and E ( r ) = exp [ − Φ( r ) /T ] . Besides, existence of the thermodynamical limit presumes the “cluster prop- erty” of DFs. that is vanishing of inter-particle correla- tions under large spatial separation of particles, so that 

<img src='content_image/396401.jpg'>

F$_{k}$ → F$_{k}$$_{−}$$_{1}$ G$_{m}$ ( p$_{s}$ ) at | r$_{s}$ − R | → ∞ , where 1 ≤ s ≤ k and F$_{k}$$_{−}$$_{1}$ does not include r$_{s}$ and p$_{s}$ . That are bound- ary conditions to Eq.1. 

In view of these conditions, in order to visually ex- tract inter-particle correlations, it is convenient to make the linear change of variables, from the DFs F$_{k}$ to new functions C$_{k}$ , [34] as follow: 

<img src='content_image/396397.jpg'>

and so on. with f ( r , p ) = E ( r ) G$_{m}$ ( p ) . 

Clearly, such defined C$_{k}$ can be named cumulant func- tions (CF) since represent irreducible correlations be- tween k gas atoms and total BP’s path R . In their terms the BBGKY hierarchy (1) takes the form [1, 2, 22] 

<img src='content_image/396398.jpg'>

where E $^{′}$( r ) = ∇ E ( r ) , ρ$_{k}$ ≡ r$_{k}$ − R , and P$_{kj}$ means replacement of pair of arguments x$_{j}$ = { r$_{j}$ , p$_{j}$ } (if it is present) by x$_{k}$ = { r$_{k}$ , p$_{k}$ } . The mentioned initial and boundary conditions simplify to 

<img src='content_image/396399.jpg'>

( 1 ≤ j ≤ k ). A careful enough scanning of these equa- tions results in observation [22] that exact relations 

take place. This is particular case of general “virial re- lations” found in [1, 2, 21] as exact properties of solu- tions to BBGKY equations describing molecular Brown- ian motion in fluids (they can be also deduced [3] from the generalized fluctuation-dissipation relations [35, 36]). 

These relations demonstrate existence and significance of all many-particle correlations. To see straight away that they keep significant in the Boltzmann-Grad limit (BGL), let us introduce characteristic interaction radius 

<img src='content_image/396400.jpg'>

