Low-Density Asymptotic Behavior of Observables of Hard Sphere Fluids

The paper deals with a rigorous description of the kinetic evolution of a hard sphere system in the low-density (Boltzmann– Grad) scaling limit within the framework of marginal observables governed by the dual BBGKY (Bogolyubov–Born– Green–Kirkwood–Yvon) hierarchy. For initial states specified by means of a one-particle distribution function, the link between the Boltzmann–Grad asymptotic behavior of a nonperturbative solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables and a solution of the Boltzmann kinetic equation for hard sphere fluids is established. One of the advantages of such an approach to the derivation of the Boltzmann equation is an opportunity to describe the process of the propagation of initial correlations in scaling limits.

As is well known, many-particle systems are described by means of two objects: observables and states.A functional of the mean value of observables defines a duality between observables and states and as a consequence there exist two approaches to the description of the evolution within the framework of the evolution of observables and states, respectively [25].Traditionally the evolution of many-particle systems is described within the framework of the evolution of states governed by the BBGKY hierarchy for marginal distribution functions.An equivalent approach to the description of the evolution of many-particle systems is given in terms of marginal observables governed by the dual BBGKY hierarchy [26].
The objective of the paper is to develop an approach to the description of the kinetic evolution of a hard sphere system within the framework of the evolution of observables.For this purpose in Section 2 we consider the microscopic description of the evolution of a hard sphere system within the framework of marginal observables governed by the dual BBGKY hierarchy.Then in Section 3 the origin of the dual kinetic evolution is stated; namely, a low-density (Boltzmann-Grad) limit of a nonperturbative solution of the Cauchy problem of the dual BBGKY hierarchy is established.In Sections 4 and 5 for initial states specified by means of a one-particle distribution function the link between the dual Boltzmann hierarchy for the limit marginal observables and the Boltzmann kinetic equation and the process of the propagation of initial chaos is established.In Sections 6 and 7 obtained results extended on hard spheres fluids, namely, 2 Advances in Mathematical Physics for initial states specified by means of a one-particle distribution function and initial correlation functions, characterized condensed states.Finally, in Section 8 we conclude with some perspectives for future research.

The Dual BBGKY Hierarchy with Hard Sphere Collisions
As is well known, the evolution of many-particle systems can be described within the framework of a sequence of marginal (-particle) distribution functions as well as in terms of a sequence of marginal observables.In this section we construct a nonperturbative solution of the Cauchy problem of a hierarchy of evolution equations for marginal observables of a hard sphere system.We consider identical particles of a unit mass with a diameter  > 0, interacting as hard spheres with elastic collisions.Every particle is characterized by its phase coordinates For configurations of such a system the following inequalities are satisfied: for at least one pair (, ) :  ̸ =  ∈ (1, . . ., )} is the set of forbidden configurations in the configuration space of  > 1 hard spheres.Let C  be the space of sequences  = ( 0 ,  1 , . . .,   , . ..) of bounded continuous functions on R 3 × (R 3 \ W  ) which are symmetric with respect to permutations of the arguments  1 , . . .,   , equal to zero on the set of forbidden configurations W  and equipped with the norm: If  ≥ 0, the evolution of marginal observables () = ( 0 ,  1 (,  1 ), . . .,   (,  1 , . . .,   ), . ..) ∈ C  of a system of a nonfixed number of hard spheres is described by the Cauchy problem of the weak formulation of the following hierarchy of evolution equations [26]: where the coefficient  > 0 is a scaling parameter (the ratio of the diameter  > 0 to the mean free path of hard spheres) and on the set C ,0 ⊂ C  of the continuously differentiable functions with compact supports the operators L() and L int ( 1 ,  2 ) in a dimensionless form are defined by the formulas respectively.In (3) the symbol ⟨⋅, ⋅⟩ denotes a scalar product,  is the Dirac measure, , and the momenta  *  1 ,  *  2 are defined by the equalities We refer to recurrence evolution equations (1) as the dual BBGKY hierarchy for hard spheres in a dimensionless form.
If  ≤ 0, a generator of the dual BBGKY hierarchy for hard spheres is defined by the expression of corresponding form [10].
To construct a solution of recurrence evolution equations (1) on the space C  ≡ C(R 3 × (R 3 \ W  )) we introduce the group of operators   () that describes dynamics of  hard spheres.It is defined by means of the phase trajectories of a hard sphere system almost everywhere on the phase space R 3 × (R 3 \ W  ), namely, beyond of the set M 0  of the zero Lebesgue measure, as follows: where   () ≡   (,  1 , . . .,   ) is a phase trajectory of th particle constructed in [7,10], and the set M 0  consists of the phase space points which are specified by initial data, generating multiple collisions of hard spheres in the evolutionary process, that is, collisions of more than two particles, more than one two-particle collision at the same instant, and infinite number of collisions on a finite time interval.
On the space C  one-parameter mapping ( 5) is an isometric * -weak continuous group of operators; that is, it is a  * 0 -group [27].
The infinitesimal generator L  of a group of operators ( 5) is defined in the sense of a * -weak convergence of the space C  and it has the structure , and the operators L() and L int ( 1 ,  2 ) are defined by formulas (3).
A nonperturbative solution of the Cauchy problems (1) and ( 2) is determined by the following expansions: The generating operators of expansions (6) is the (1 + )thorder cumulant of groups of operators (5) defined by the following expansion: where  ≡ (1, . . ., ),  ≡ ( The simplest examples of expansions (6) for marginal observables have the following form: On the space C  for the Cauchy problems (1) and ( 2) the following statement is true [28].
In the case of -ary-type marginal observables solution expansion (6) has the form where the generating operator of this expansion is the (1 +  − )th-order cumulant of groups of operators (7), and, if 1 ≤  < , we have  ()  () = 0. We remark also that expansion (6) can be also represented in the form of the perturbation (iteration) series [26] as a result of applying of analogs of the Duhamel equation to cumulants (7) of groups of operators (5). Let ) be the space of integrable functions that are symmetric with respect to permutations of the arguments  1 , . . .,   , equal to zero on the set of forbidden configurations W  and equipped with the norm: where initial state of finitely many hard spheres is described by means of a sequence of the marginal distribution functions Owing to estimate (9), functional (12) exists under the condition that  <  −1 .

The Kinetic Evolution of Hard Sphere Observables
We consider the problem of the rigorous description of the kinetic evolution of hard spheres within the framework of marginal observables by giving of a low-density (Boltzmann-Grad) asymptotic behavior of the Cauchy problem of the dual BBGKY hierarchy (1), (2).
and then for arbitrary finite time interval there exists the Boltzmann-Grad limit of marginal observables (6) in the sense of a * -weak convergence of the space which is determined by the expansions where the operator L 0 int ( 1 ,  2 ) is the collision operator of point particles, namely, Before proving this statement we give some comments.We consider the Boltzmann-Grad limit of a special case of marginal observables, namely, the additive-type marginal observables.If for the initial additive-type marginal observable    1 the following condition is satisfied: then, according to statement (15), for additive-type marginal observables (10) we derive where the limit marginal observable  (1)   () is determined as a special case of expansion (16): We make several examples of expansions (20) of the limit additive-type marginal observable: If for the initial -ary-type marginal observable    the following condition is satisfied: then, according to statement (15), for -ary-type marginal observables (11) we derive where the limit marginal observable  ()  () is determined as a special case of expansion (16): If  0 ∈ C  , then the sequence () = ( 0 ,  1 (), . . .,   (), . ..) of limit marginal observables ( 16) is a generalized global solution of the Cauchy problem of the dual Boltzmann hierarchy with hard sphere collisions where the operator L 0 int is defined by (17).This fact is proved similar to the case of an iteration series of the dual BBGKY hierarchy [26].
It should be noted that equations set (25) has the structure of recurrence evolution equations.Indeed, we make a few examples of the dual Boltzmann hierarchy with hard sphere collisions (25): Thus, in the Boltzmann-Grad scaling limit the kinetic evolution of hard spheres is described in terms of limit marginal observables (16) governed by the dual Boltzmann hierarchy with hard sphere collisions (25).Similar approach to the description of the mean field asymptotic behavior of quantum many-particle systems was developed in paper [36].
We outline the sketch of the proof of the limit theorem.For the group of operators (5) the analog of the Duhamel equation is valid [27] ⋅  −1 (, (1, . . ., ) \  1 )  1 (,  1 )   , where the operator L int (, ) is defined by formula (3).Then for the (1 + )th-order cumulant of groups of operators (5) the analog of the Duhamel equation holds For arbitrary finite time interval * -weak continuous group of operators (5) has the following Boltzmann-Grad scaling limit in the sense of a * -weak convergence of the space Taking into account assumption ( 14) and an analog of the Duhamel equation (29), then in view of formula (31), for cumulants of asymptotically perturbed groups of operators we have As a result of equality (32) we establish the validity of statement (15) for nonperturbative solution (6) of the Cauchy problem of the dual BBGKY hierarchies (1) and (2).

The Derivation of the Boltzmann Kinetic Equation
We shall establish the link between the constructed asymptotic behavior of marginal observables of a hard sphere system (Theorem 2) and the description of kinetic evolution of states by means of a one-particle marginal distribution function governed by the Boltzmann kinetic equation.
In case of the absence of correlations between particles at initial time, that is, for initial states satisfying a chaos condition [10], the sequence of initial marginal distribution functions for a system of hard spheres has the form where X R 3 \W  is the Heaviside step function of the allowed configurations.This assumption about initial state is intrinsic for the kinetic theory, because in this case all possible states of gases are described by means of a one-particle distribution function.
If () ∈ C  and | 0 1 (  )| ≤  exp(−( 2  /2)), then the Boltzmann-Grad limit of mean value functional (12) exists under the condition that [7]: , and it is determined by the following series expansion: In consequence of the following equality for the limit additive-type marginal observables (20) (below it is proved in more general case) ( (1) where function  (1)   () is given by expansion (20) and the distribution function  1 (,  1 ) is given by the series Advances in Mathematical Physics 7 where the following operator was introduced: ( and the group of operators  * 1 () is a group of adjoint operators to operators (5) in the sense of mean value functional (12).
The distribution function  1 () is a solution of the Cauchy problem of the Boltzmann kinetic equation Thus, we establish that hierarchy (25) for additive-type marginal observables and initial state (34) describes the evolution of hard sphere systems just as the Boltzmann kinetic equation (39).
We differentiate over the time variable expression (37) in the sense of the pointwise convergence of the space where the operator L 0, * int (  ,  + 2) is defined by formula (38).
Using the product formula for the one-particle marginal distribution function  1 (,   ) defined by series expansion (37) in case of initial data (34) where the group property of one-parameter mapping ( 5) is applied, we express the second summand in the righthand side of equality (41) in terms of ∏ 2 =1  1 (, ), and, consequently, we get (39).
We remark that in a one-dimensional space the collision integral of the Boltzmann equation with elastic hard sphere collisions identically equals zero.In a one-dimensional space the Boltzmann-Grad limit is not trivial in case of hard sphere dynamics with inelastic collisions [38].In paper [38] for one-dimensional granular gas the process of the creation of correlations in the Boltzmann-Grad limit was also described.

On Propagation of Initial Chaos in a Low-Density Limit
If the initial states of hard spheres are specified by a sequence of marginal distribution functions (33), then the property of the propagation of initial chaos holds in the Boltzmann-Grad limit.It is a result of the validity of the following equality for the limit -ary marginal observables (24); that is,  () (0) = (0, . . .,  0  ( 1 , . . .,   ), 0, . ..), where for finite time interval the limit one-particle marginal distribution function  1 () is defined by series expansion (37) and therefore it is governed by the Cauchy problem of the Boltzmann kinetic equations ( 39) and (40).

Advances in Mathematical Physics
In fact, taking into account the validity of the following equality for expansion (16) of the function  ()   () and product formula (42), for the limit one-particle marginal distribution function defined by series expansion (37), we finally verify the validity of equality (43).Thus, in the Boltzmann-Grad scaling limit an equivalent approach to the description of the kinetic evolution of hard spheres within the framework of the Cauchy problem of the Boltzmann kinetic equations (39) and ( 40) is given by the Cauchy problem of the dual Boltzmann hierarchy with hard sphere collisions (25) and (26) for the additivetype marginal observables.In case of the nonadditive-type marginal observables a solution of the dual Boltzmann hierarchy with hard sphere collisions (25) is equivalent to the property of a propagation of initial chaos in the sense of equality (43).

The Boltzmann Equation for Hard Spheres Fluids
We consider initial states of a hard sphere system specified by the one-particle marginal distribution function ) in the presence of correlations, that is, initial states defined by the following sequence of marginal distribution functions: where the functions    ≡    ( 1 , . . .,   ) ∈   (R 3 × (R 3 \ W  )),  ≥ 2, are specified initial correlations.Since manyparticle systems in condensed states are characterized by correlations sequence (45) describes the initial state of the kinetic evolution of hard sphere fluids.
We assume that the Boltzmann-Grad limit of initial oneparticle marginal distribution function  0, 1 ∈  ∞  (R 3 × R 3 ) exists in the sense as above; that is, in the sense of a weak convergence the equality holds:  − lim →0 ( 2  0, 1 −  0 1 ) = 0, and in case of correlation functions let  − lim →0 (   −   ) = 0,  ≥ 2; then in the Boltzmann-Grad limit initial state (45) is defined by the following sequence of the limit marginal distribution functions: We consider relationships of the constructed Boltzmann-Grad asymptotic behavior of marginal observables with the nonlinear Boltzmann-type kinetic equation in case of initial states (46).
For the limit additive-type marginal observables (20) and initial states (46) the following equality is true: where the functions  (1)   () are represented by expansions (20) and the limit marginal distribution function  1 () is represented by the following series expansion: Series ( 48) is uniformly convergent for finite time interval under the condition as above (37).
The function  1 () represented by series (48) is a weak solution of the following Cauchy problem of the Boltzmann kinetic equation with initial correlations [39,40] This fact is proved similarly to the case of a perturbative solution of the BBGKY hierarchy for hard spheres represented by the iteration series [10,29].Thus, in case of initial states specified by one-particle marginal distribution function (46) we establish that the dual Boltzmann hierarchy with hard sphere collisions (25) for additive-type marginal observables describes the evolution of a hard sphere system just as the Boltzmann kinetic equation with initial correlations (49).

Propagation of Initial Correlations in a Low-Density Limit
The property of the propagation of initial correlations in a low-density limit is a consequence of the validity of the following equality for a mean value functional of the limit ary marginal observables: * 1 (, where the one-particle marginal distribution function  1 (,   ) is solution (48) of the Cauchy problem of the Boltzmann kinetic equation with initial correlations (49) and (50), and the inverse group to the group of operators  * 1 () we denote by ( * 1 ) −1 () =  * 1 (−) =  1 ().This fact is proved similarly to the proof of a property of a propagation of initial chaos (43).
We note that, according to equality (51), in the Boltzmann-Grad limit the marginal correlation functions defined as cluster expansions of marginal distribution functions, namely,   (, where the symbol ∑  means the sum over possible partitions  of the set of arguments ( 1 , . . .,   ) on || nonempty subsets   , and the one-particle marginal distribution function  1 () is a solution of the Cauchy problem of the Boltzmann kinetic equation with initial correlations (49) and (50).Thus, in case of the limit -ary marginal observables a solution of the dual Boltzmann hierarchy with hard sphere collisions (25) is equivalent to a property of the propagation of initial correlations for the -particle marginal distribution function in the sense of equality (51) or in other words the Boltzmann-Grad scaling dynamics does not create new correlations except initial correlations.

Conclusion
In the paper a new approach to the problem of the rigorous description of the kinetic evolution of a system of hard spheres with elastic collisions was developed.For this purpose we established the low-density (Boltzmann-Grad) asymptotic behavior of a solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables of hard spheres (1) and (2).The constructed scaling limit is governed by the set of recurrence evolution equations, namely, by the dual Boltzmann hierarchy with hard sphere collisions (25).
Furthermore, it was established that for initial states specified by a one-particle distribution function the evolution of additive-type marginal observables is equivalent to a solution of the Boltzmann kinetic equation (39) and the evolution of nonadditive-type marginal observables is equivalent to the property of the propagation of initial chaos for states (43).In other words the Boltzmann-Grad dynamics does not create correlations.