Incollection,
On the solutions of the nonlinear Liouville hierarchy
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
We investigate the initial-value problem of the nonlinear
Liouville hierarchy. The nonlinear Liouville hierarchy that describes the evolution of
correlation functions, arises in many problems of statistical
mechanics concerning many-particle systems.
However, today it is still
insufficiently studied from the
mathematical point of view.
It is well known, that all possible states of
a classical system of a finite number of particles are described by
the functions interpreted as probability density functions. These
functions are solutions of the initial-value problem of the
Liouville hierarchy
--- the first-order partial
differential equations, whose characteristic equations are Hamilton
equations. If the state of a system is presented in terms of a
cluster expansion in new (correlation) functions one evidently
obtains an equivalent description of this state. Now evolution of
the correlation functions is determined by the nonlinear Liouville
hierarchy
--- certain nonlinear first-order partial differential
equations.
We construct an explicit solution of such nonlinear Liouville hierarchy
and present as an expansion in terms of
particle clusters whose evolution is described by a cumulant
(semi-invariant) of the evolution operators. The
interaction potential of the general form is considered, which makes
it possible to describe the general structure of the generator of
the nonlinear Liouville hierarchy. The existence of a strong
solution of the Cauchy problem with initial data from the space of
translation-invariant with respect to the configuration variables functions is proved.
The nonlinear Liouville hierarchy
is basic in the substantiation of the derivation of the nonlinear BBGKY hierarchy whose solutions describe the correlation dynamics of
infinite systems of particles. Moreover, the Bogolyubov's correlation-weakening principle for the system of hard spheres is proved.
The correlation functions may be employed to directly calculate the
specific characteristics of the system, i.e., fluctuations, defined
as the average values of the square deviations of an observable from
its average value, as well as macroscopic values which are not
averages of observables.
