Kinetic equations, their algebraic structure and invariant reductions
Introduction
It is well known that the classical Bogolubov-Boltzmann kinetic equations under the condition of manyparticle correlations [1-12] at weak short range interaction potentials describe long waves in a dense gas medium. The same equation, called the Vlasov one, as it was shown by N. Bogolubov [5], describes also exact microscopic solutions of the infinite Bogolubov chain [4] for the manyparticle distribution functions, which was widely studied making use of both classical approaches in [2,6,11,13-23] and in [24-32], making use of the generating Bogolubov functional method and the related quantum current algebra representations.
A.A. Vlasov proposed his kinetic equation [33] for electron-ion plasma, based on general physical reasonings, that in contrast to the short range interaction forces between neutral gas atoms, interaction forces between charged particles slowly decrease with distance, and therefore the motion of each such particle is determined not only by its pair-wise interaction with either particle, yet also by the interaction with the whole ensemble of charged particles. In this case the Bogolubov equation for distribution functions in a domain
where
is the temporal evolution parameter,
denotes the canonical Poisson bracket [6,33,34] on the product
and
is an interparticle interaction potential, - reduces to the Vlasov equation if to put in (1.1)
that is to assume that the two-particle correlation function [2,3,11,23] vanishes:
for all
and
Then one easily obtains from (1.1) that
for all
and
Remark here that the equation (1.4) is reversible under the time reflection
thus it is obvious that it can not describe thermodynamically stable limiting states of the particle system in contrast to the classical Bogolubov-Boltzmann kinetic equations [1,2,4,6,11,24,27], being a priori time nonreversible owing to the choice of boundary conditions in the correlation weakening form. This means that in spite of the Hamiltonicity of the Bogolubov chain for the distribution functions, the Bogolubov-Boltzmann equation a priori is not reversible. It is also evident that the condition (1.3) does not break the Hamiltonicity - the equation (1.4) is Hamiltonian with respect to the following Lie-Poisson-Vlasov bracket:
where
respectively
are smooth functionals on the functional manifold
consisting of functions fast decreasing at the boundary
of the domain
. The statement above easily ensues from the following proposition.
Proposition 1.1 Let
denote a set of many-particle distribution functions. Then the classical Bogolubov-Poisson bracket [4,18,24,25] on the functional space
reduces invariantly on the subspace
to the Lie-Poisson-Vlasov bracket (1.5).
Concerning the general case when we are work with an innite Bogolubov chain of kinetic equations on the many-particle distribution functions and forced to break it at some place, numbered by some natural number N ∈ N; the usual approaches always give rise to the resulting inconsistency [3,5] of the chain and, as a result, to the nonphysical solutions. The most successful approach to obtaining the Boltzmann kinetic equation for the one-particle distribution function was suggested still many years ago by N. Bogolubov [1,2], based on the e⁄ective application of the so called weak correlation condition. So far, to the regret, this approach, being conjugated with the complex problem of solving functional equations, also gives rise to the inconsistency of the higher order kinetic equations. Nonetheless, being inspired by former studies [6, 16, 11] of these problems, based on the geometrical interpretation of the Bogolubov kinetic equations chain, we devised a new functional analytic approach to constructing its compatible reduction a priori free of any unphysical consequences. We also succeeded in constructing a reduced set of kinetic equations, based on a suitably devised Dirac type invariant reduction scheme of the corresponding many-particle Lie-Poisson phase space. The approach to solving this problem and its di⁄erent consequences will be analyzed in more detail in sections to follow below.
The Lie-Poisson-Vlasov bracket: Lie-algebraic approach
The bracket expression (1.5) allows a slightly different Lie-algebraic interpretation, based on considering the functional space
as a Poissonian manifold, related with the canonical symplectic structure on the diffeomorphism group
of the domain
first described [35,36] still in 1887 by Sophus Lie. Namely, the following classical theorem holds.
Theorem 1.2 The Lie-Poisson bracket at point
on the coadjoint space
is equal to the expression
for any smooth right-invariant functionals
Proof. By classical definition [33-37] of the Poisson bracket of smooth functions
on the symplectic space
it is easy to calculate that
where
Since the expressions
and
owing the right-invariance of the vector fields
the Poisson bracket (1.7) transforms into
for all
and any
The Poisson bracket (1.8) is easily generalized to
for any smooth functionals
finishing the proof.
Concerning our special problem of describing evolution equations for one-particle distribution functions, we will consider the one particle cotangent space
over a domain
and the canonical Poisson bracket
on
for which, by definition, for any
where
Denote now by
the related functional Lie algebra and
its adjoint space with respect to the standard bilinear symmetric form
on the product
where
The constructed Lie algebra with respect to the bilinear symmetric form (1.11) proves to be metrized, that is
and
for any
and
is a smooth functional on
its gradient
at point
is naturally defined via the limiting expression
for arbitrary element
Define now the Poisson structure
by means of the standard Lie-Poisson [9,33,34-36,38,39] expression:
for arbitrary functionals
It is evident that the expression (1.14) identically coincides with the Poisson bracket (1.5).
Consider a functional
and the related coadjoint action of the element
at a fixed element
where
is the corresponding evolution parameter. It is easy observe that
is a Hamiltonian equation with the functional
taken as its Hamiltonian, being simultaneously equivalent to the following canonical Hamiltonian flow:
if to choose as a Hamiltonian the following functional
where
is a two-particle interaction potential,
It is easy to observe here that the Hamiltonian (1.18) is obtained from the corresponding classical Bogolubov Hamiltonian expression
where
denotes an infinite vector from the space
of multiparticle distribution functions, and if to impose on it the constraint (1.2). Thus we have stated the following proposition.
Proposition 1.3 The Boltzmann-Vlasov kinetic equation (1.4) is a Hamiltonian system on the functional manifold
with respect to the canonical Lie-Poisson structure (1.14) with Hamiltonian (1.18). As a consequence, the flow (1.4) is time reversible.
Boltzmann-Vlasov kinetic equations and microscopic exact solutions
Proposition 1.1, stated above, claims that the Boltzmann-Vlasov equation (1.4) is a suitable reduction of the whole Bogolubov chain upon the invariant functional subspace
Moreover, this invariance in no way should be compatible a priori [5,19,21,24,25,27] with the other kinetic equations from the Bogolubov chain, and can be even contradictory. Nonetheless, as it was stated [5] by N. Bogolubov, namely owing to this invariance of the subspace
the Boltzmann-Vlasov equation (1.4) in the case of the Boltzmann-Enskog hard sphere approximation of the inter-particle potential possesses exact microscopical solutions which are compatible with the whole hierarchy of the Bogolubob kinetic equations. The latter is, obviously, equivalent to its Hamiltonicity on the manifold with respect to the Lie-Poisson bracket (1.14). The Boltzmann-Enskog kinetic equation [3,5,11,12,23] equals.
where
a particle diameter,
a unit vector,
and, by definition,
for all
satisfying the condition
. The equation (1.20) easily reduces to the Vlasov-Enskog equation
for all
owing to its Hamiltonicity on the space
If in addition there exists a nontrivial interparticle potential, the equation above is naturally generalized to the kinetic equation
which remains to be Hamiltonian on
and possesses, in particular, the following exact singular solution:
where
- phase space coordinates in
of
interacting particles in the domain
Specified above the Hamiltonicity problem and the existence of exact solutions to the Botzmann-Vlasov kinetic equation (1.22) is deeply related to that of describing correlation functions [2,11,23], suitably breaking the infinite Bogolubov chain [2,4,11,24,30,31] of manyparticle distribution functions. Namely, if to introduce manyparticle correlation functions [2,11,23] for related Bogolubov distribution functions as
where
then the Vlasov equation (1.22) is obtained from the Bogolubov hierarchy at
and
for all
.
As it was mentioned above, the constraint imposed on the infinite Bogolubov hierarchy is compatible with its Hamiltonicity. Yet in many practical cases this closedness procedure by means of imposing the conditions like
for all
at some fixed
gives rise to some serious dynamical problems related with its mathematical correctness. Namely, if to close this way the infinite Bogolubov chain of kinetic equations on manyparticle distribution functions, one easily checks that the imposed constraint (1.25) does not persists in time subject to the evolution of the distribution functions
This menas that these naively reduced kinetic equations are written down somehow incorrectly, as the reduced functional submanifold
should remain invariant in time. To dissolve this problem we are forced to consider the whole Bogolubov hierarchy of kinetic equationas on multiparticle distribution functions as a Hamiltonian system on the functional manifold
and correctly reduce it on the constructed above functional submanifoild
via the classical Dirac type [1, 3, 6, 19, 36] procedure. The kinetic equations obtained this way by means of the reduced Lie-Poisson-Bogolubov structure will evidently differ from those naively obtained by means of the direct substitution of the imposed constraint (1.25) into the Bogolubov chain of kinetic equations, and in due course will conserve the functional submanifold
invariant.
The invariant reduction of the Bogolubov distribution functions chain
Consider the constructed before Hamiltonian functional
(1.19)
and calculate the evolution of the distribution functions vector
under the simplest constraint (1.25) at
that is
for all
To perform this reduction on
we need [39-43] to constraint the
-extended Hamiltonian expression
for some smooth function
and next to determine it from the submanifold
invariance condition
for all
and
To calculate effectively the condition (1.29) let us first calculate the evolutions for distribution functions and
and
which can be rewritten equivalently as follows:
and
Having now substituted temporal derivatives (1.32) and (1.33) into the equality (1.29) in their explicit form, one obtains the following functional relationship:
which is satisfied iff
for all
Taking into account the result (1.35), one easily obtains from the equation (1.32) the invariantly reduced on the submanifold
kinetic equation on the one-particle distribution function:
which can be rewritten in the following compact form:
where we put, by definition,
The kinetic equation (1.36) naturally coincides exactly with that obtained before from the naively reduced evolution equation
on the submanifold
as it is globally invariant [18,24] with respect to the classical Lie-Poisson-Bogolubov structure on
The obtained result can be formulated as the following proposition.
Proposition 1.4 The first coorelation function Dirac type reduction on the functional submanifold
formed by relationships (1.27), reduces the corresponding Bogolubov chain of many-particle kinetic equations to the well known classical Vlasov kinetic equation.
Remark 1.5 It is worth to mention here that the well known classical Bogolubov approximation of the many-particle distribution functions as
with mapping
presenting smooth nonlinear functionals, independent of the temporal parameter
define a suitably different functional submanifold
upon which the reduced evolution flow
gives rise to a new Boltzmann type kinetic equation, being compatible with evolution equations for higher distribution functions, free of evolution inconsistencies and completely different from that derived before by Bogolubov [4].
The same way as above one can explicitly construct the system of invariantly reduced kinetic equations
on the submanifold
which already is not a priori globally invariant with respect to the Hamiltonian evolution flows on
and whose detail structure and analysis are postponed to another place. This cae
Conclusion
We studied a well known classical problem of constructing a compatible nite-particle reduction of the Bogolubov chain of many-particle distribution functions and analyzed a special class of the related dynamical systems of BoltzmannBogolubov and Boltzmann Vlasov type on innite dimensional functional manifolds, modeling kinetic processes in many-particle media. Based on the geometric approach, e⁄ectively devised to studying the corresponding many-particle Lie-Poisson functional phase space, we succeeded in dual analysis of the innite Bogolubov hierarchy of many-particle distribution functions and their Hamiltonian structure. Moreover, we proposed a new an e⁄ective approach to invariant Dirac type reduction of the Bogolubov hierarchy upon a suitably chosen invariant Poisson subspace endowed and deduced the related modied BoltzmannBogolubov kinetic equations on a nite set of multi-particle distribution functions.