Incollection,
On solution of BBGKY hierarchy for quantum many-particle systems
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
We construct a new representation for a solution of the initial
value problem to the quantum BBGKY hierarchy of equations 1
$$
ddt F(t)=e^a(-N)e^-aF(t),
$$
$$
F(t)\mid_t=0=F(0),
$$
where an analog of the annihilation operator:
$(af)_n(1,łdots,n)=Tr_n+1f_n+1(1,$ $łdots,n,n+1),$
is defined and bounded in the space of sequences of trace operators
$L_\alpha^1(F)$~ (where $F$ is the
Fock space), the von Neumann operator
$N=\bigoplusłimits_n=1^ınfty N_n$~
($h=2\pi\hbar$ is a Planck constant):
$(Nf)_n=N_nf_n
\equiv1i\hbar\big(f_nH_n-H_nf_n\big),$
is defined in the domain $D(N)\subset
L^1_\alpha(F)$ described in 2~($H_n$ is
a self-adjoin $n$-particle Hamilton operator 2).
We prove the criterion of the cumulant representation for a solution
of the initial value problem to the quantum BBGKY hierarchy.
Constructed representation for a solution has the form of an
expansion over particle clusters
$$
F_s(t,Y)=\sumłimits_n=0^ınfty1n!Tr_s+1,łdots,s+n\times
$$
$$
U_1+n(t;Y,XY)F_s+n(0;X),
$$
whose evolution are described by the corresponding order cumulant
(semi-invariant) of the evolution operators of finitely
many-particle quantum systems
$$
\Big(U_1+|XY|(t)F_|X|(0)\Big)(Y,XY)=
$$
$$
= \sumłimits_P:\,(Y,XY)=\bigcupłimits_iX_i
(-1)^|P|-1(|P|-1)!\times
$$
$$
\times\prodłimits_i=1^|P|
U_|X_i|(-t;X_i)
F_|X|(0;X)\prodłimits_j=1^|P|U^-1_|X_j|(-t;X_j),
$$
where $U_n(-t)= e^-i\hbartH_n,$
$U_n^-1(-t)= e^i\hbartH_n$ and
$\sumłimits_P$ is the sum of all possible partitions
$P$ of the set $\Y,XY\=(1\cup
s,s+1,łdots,s+n)$
into $|P|$ nonempty mutually nonintersecting subsets
$X_i\subset\Y,XY\$, the symbol $1\cupłdotss$
implies, that $s$ particles evolve as a cluster, the same way as
particles $s+1,łdots,s+n,$, i.e., in this case the number of
elements of the set $\Y,XY\$ is equal to
$|Y|+|XY|=1+n$.
We prove 3 the existence and uniqueness theorem for initial data
from the space of sequences of trace operators
$L_\alpha^1(F)$. If
$F(0)ınL_\alpha^1(F)$, then for $\alpha>e$
there exists a unique global ($-ınfty<t<ınfty$) solution to the
initial value problem to the quantum BBGKY hierarchy given by the
above formula. This solution is a strong solution for $F(0)ın
D(N)\subset
L^1_\alpha(F)$ and a weak one for arbitrary
initial data from the space $L^1_\alpha(F)$.
We note, that the parameter $\alpha$ can be interpreted as a
quantity inverse to the density of the system.
We also discuss the problem of the construction of a solution in the
space of sequences of bounded operators describing states of
infinitely many particle systems, in particular, the sequence of
$n$-particle equilibrium operators belongs to this space.
For initial data from such space each term of the solution expansion
contains the divergent traces. The stated cumulant nature of the
solution expansion guarantee the compensation of divergent traces in
every its term.
On the basis of cluster expansions for the evolution operators of
finitely many particle quantum systems we give the classification of
the possible solution representations of the quantum BBGKY hierarchy
in the case of Maxwell-Boltzmann statistics.\\
1) C.~Cercignani, V.I.~Gerasimenko and D.Ya.~Petrina, ıt
Many-particle dynamics and kinetic equations. Dordrecht: Kluwer,
1997.\\
2) D.Ya.~Petrina, Mathematical Foundations of Quantum
Statistical Mechanics. Continuous Systems. Amsterdam: Kluwer, 1995.\\
3) V.I.~Gerasimenko, V.O.~Shtyk, Initial value problem of
quantum BBGKY hierarchy of many-particle systems. Ukrainian Math.
J., v.9, 2006, P.1175-1191.
