##
**Many-particle dynamics and kinetic equations.**
*(English)*
Zbl 0933.82001

Mathematics and its Applications (Dordrecht). 420. Dordrecht: Kluwer Academic Publishers. viii, 244 p. (1997).

This book is devoted to the investigation of the dynamics of infinite systems interacting via pair short range potential.

Chapter 1 presents the Hamiltonian dynamics for finite classical systems of hard-core particles interacting via smooth short range pair potential. For such systems the Liouville equation is considered and an existence theorem for solutions with initial data from the space of summable functions is proved. A preliminary information about the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is given. In Chapter 2 it is shown that for such systems the evolution operators of the BBGKY hierarchy are bounded, strongly continuous in the space \(L^1\) of sequences of summable functions. These operators form a group in \(L^1\), the infinitesimal generator of which is defined on an everywhere dense subset of \(L^1\). Then the BBGKY hierarchy is considered as an abstract evolution equation in \(L^1\) with the infinitesimal generator in the right hand side. The existence of solutions for a class of interaction potentials and initial data in \(L^1\) is proved.

Chapter 3 studies infinite systems of hard spheres. Distribution functions describing the states of such systems are in the space \(L^\infty\) of sequences of functions which are bounded with respect to the configuration variables and decreasing with respect to the momentum ones. Solutions of the initial value problem for the BBGKY hierarchy in the space \(L^\infty\) are represented by iteration series. A local existence theorem for arbitrary initial data form \(L^\infty\) and convergence in thermodynamic limit for these solutions are proved. Then for initial data corresponding to locally perturbed equilibrium states these solutions are continued to an arbitrary large time interval and existence of the thermodynamic limit is obtained. In the one-dimensional case a global existence theorem for the BBGKY hierarchy is proved for initial data not necessarily close to an equilibrium state.

Chapter 4 considers the Boltzmann-Grad limit method of derivation of the Boltzmann equation from the BBGKY hierarchy for a hard sphere particle system in dimension three. The Boltzmann-Grad limit means that the diameter \(a\) of the particles tends to zero and the activity \(z\) tends to infinity so that \(a^2z\) remains fixed. It is shown that for the Cauchy problem of the Boltzmann hierarchy there exists a unique solution and that the iteration series of the BBGKY hierarchy in certain sense coverage term by term to the corresponding series of the Boltzmann hierarchy in a finite time interval. The existence of the Boltzmann-Grad limit of the equilibrium states for the case of canonical and grand canonical ensembles are proved.

The last Chapter 5 is devoted to the derivation of kinetic equations from the BBGKY hierarchy. Two different approaches are discussed. One is Bogolyubov’s method of derivation of the Boltzmann equation directly from the BBGKY hierarchy based on the construction of the hierarchy’s solution by iteration series. Another approach which is based on the representation of the hierarchy’s solutions by functional series was developed by Cohen and Green. In the final section of Chapter 5 it is shown that when the initial data possess a certain factorization property the Cauchy problem for the BBGKY hierarchy is identical to the corresponding initial value problem for a generalized kinetic equation.

The book contains a large list of references. It may be useful for graduate students and researchers interested in theoretical and mathematical physics.

Chapter 1 presents the Hamiltonian dynamics for finite classical systems of hard-core particles interacting via smooth short range pair potential. For such systems the Liouville equation is considered and an existence theorem for solutions with initial data from the space of summable functions is proved. A preliminary information about the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is given. In Chapter 2 it is shown that for such systems the evolution operators of the BBGKY hierarchy are bounded, strongly continuous in the space \(L^1\) of sequences of summable functions. These operators form a group in \(L^1\), the infinitesimal generator of which is defined on an everywhere dense subset of \(L^1\). Then the BBGKY hierarchy is considered as an abstract evolution equation in \(L^1\) with the infinitesimal generator in the right hand side. The existence of solutions for a class of interaction potentials and initial data in \(L^1\) is proved.

Chapter 3 studies infinite systems of hard spheres. Distribution functions describing the states of such systems are in the space \(L^\infty\) of sequences of functions which are bounded with respect to the configuration variables and decreasing with respect to the momentum ones. Solutions of the initial value problem for the BBGKY hierarchy in the space \(L^\infty\) are represented by iteration series. A local existence theorem for arbitrary initial data form \(L^\infty\) and convergence in thermodynamic limit for these solutions are proved. Then for initial data corresponding to locally perturbed equilibrium states these solutions are continued to an arbitrary large time interval and existence of the thermodynamic limit is obtained. In the one-dimensional case a global existence theorem for the BBGKY hierarchy is proved for initial data not necessarily close to an equilibrium state.

Chapter 4 considers the Boltzmann-Grad limit method of derivation of the Boltzmann equation from the BBGKY hierarchy for a hard sphere particle system in dimension three. The Boltzmann-Grad limit means that the diameter \(a\) of the particles tends to zero and the activity \(z\) tends to infinity so that \(a^2z\) remains fixed. It is shown that for the Cauchy problem of the Boltzmann hierarchy there exists a unique solution and that the iteration series of the BBGKY hierarchy in certain sense coverage term by term to the corresponding series of the Boltzmann hierarchy in a finite time interval. The existence of the Boltzmann-Grad limit of the equilibrium states for the case of canonical and grand canonical ensembles are proved.

The last Chapter 5 is devoted to the derivation of kinetic equations from the BBGKY hierarchy. Two different approaches are discussed. One is Bogolyubov’s method of derivation of the Boltzmann equation directly from the BBGKY hierarchy based on the construction of the hierarchy’s solution by iteration series. Another approach which is based on the representation of the hierarchy’s solutions by functional series was developed by Cohen and Green. In the final section of Chapter 5 it is shown that when the initial data possess a certain factorization property the Cauchy problem for the BBGKY hierarchy is identical to the corresponding initial value problem for a generalized kinetic equation.

The book contains a large list of references. It may be useful for graduate students and researchers interested in theoretical and mathematical physics.

Reviewer: S.Poghosyan (Erevan)

### MSC:

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

82C40 | Kinetic theory of gases in time-dependent statistical mechanics |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |

45K05 | Integro-partial differential equations |