Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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Quantum
Bio-Informatics
IV
eds. L.
Accardi,
W .
Freudenb
er
g
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co. (pp. 321- 337)
BOGOLIUBOV
TYPE
EQUATIONS
VIA
INFINITE-DIMENSIONAL
EQUATIONS
FOR
MEASURES
V.V.
KOZLOV
AND
O.G.
SMOLYANOV
Steklov Mathematical Institute,
Russian
Academy
of
Sciences
E-mail
kozlov@pran.
ru
Introduction
Lomonosov
Moscow
State
University
E-mail: smolyanov@yandex.ru
We
introduce
a new
method,
of
developing some systems
of
equations de-
scribing evolution
of
both
quantum
and
classical systems
of
statistical
me-
chanics, which does
not
use
the
so called
thermodynamical
limit
[7]
. A
particular
case
of
one
of
the
systems
is
the
famous Bogoliubov
system
of
equations (this
system
is also called
the
Born-Bogoliubov-Green-Kirkwood-
Yvon chain
of
equations).
Each
of
the
systems
of
equation
either
follows from
or
is equivalent
to
an
equation
with
respect
to
functions
of
real variable
taking
values in
the
space
of
some measures
on
the
infinite-dimensional
phase
space. In
the
classical case
the
latter
equation
is
the
infinite-dimensional Liouville
equation, i.e.
the
adjoint
equation
to
the
equation
for first integrals
of
the
(system of)
Hamilton
equation(s). In
the
quantum
case
the
Liouville equa-
tion
is
substituted
by
an
equation,
with
respect
to
functions
taking
values
in
the
space
of
measures
on
the
phase
space, which is a generalization of
the
equation
describing
the
evolution of
the
Wigner
function.
One
needs
to
consider,
in
infinite-dimensional case, equations
with
respect
to
functions
taking
values
in
some spaces
of
measures, because
there
does
not
exist a
Lebesgue measure
on
an
infinite-dimensional space, i.e. a measure which
if
translation
invariant,
(I-additive,
(I-finite
and
locally finite
(=finite on
some balls)
and
hence
it
is impossible
to
consider,
instead
of measures,
their
densities. In
particular,
we need
to
define
the
notion
of
the
Wigner
321
322
measure.
Actually
the
systems
of
equations
that
we consider below
are
sys-
tems
of
equations
describing
evolution
of
densities
of
measures,
on
finite-
dimensional
spaces, which
are
either
finite-dimensional
projections
of
mea-
sures,
on
some infinite-dimensional spaces, whose
evolution
is governed
by
some infinite-dimensional
equations
or
the
results
of
a
procedure
of
the
so called
desintegration
[13]
of
similar
measures.
The
classical Bogoliubov
system
of
equation
is a
particular
case of one
of
the
latter
systems.
One
can
say
that,
instead
of
the
thermodynamical
limit, we use
an
infinite-dimensional
phase
space.
This
allows
to
develop
not
only
the
sys-
tems
of
equations
with
respect
to
densities of
particles
and
of
the
collec-
tions
of
particles
but
also some
systems
of
equations
with
respect
to
finite-
dimensional
densities
of
probabilities.
The
paper
is
organized
as follows.
First
we consider
the
classical case
when
the
state
is
described
by
a
nonnegative
measure
on
the
phase
space.
After
that
we define
the
notion
of
a
Wigner
measure, briefly
formulate
some
properties
of
that
object
and
introduce
the
equation
describing
the
evolution
of
the
Wigner
measure.
After
that
having
noticed
the
similarity
between
the
Liouville
equation
for
nonnegative
measures
and
the
equation
for
the
Wigner
measure
we
formulate
the
quantum
analogs of
those
results
of
the
paper
that
are
related
to
the
classical case. Finally, we discuss a
generalization
of
a
Poincare
model
of
irreversible
but
symmetrical
with
respect
to
time
evolution.
Below we
want
to
clarify
main
ideas
and
in
some
places we
omit
some
technical
assumptions.
1.
Symplectic
locally
convex
spaces
and
Hamilton's
equations.
For
any
locally convex
space
(LCS) E we
denote
by
E*
the
vector
space
of
all
continuous
linear
functionals
on
E
equipped
with
a
topology
compatible
with
the
duality
between
E*
and
E. We
assume
that
the
scalar
field is
the
field of all real
numbers
and
that
all LCS
are
Hausdorff
ones. For
any
LCS
E,
G we
denote
by
L(E,
G)
the
vector
space
of
all
linear
continuous
mappings
from E
into
G;
for
any
n E N we
denote
by
Bn(E)
the
vector
space
of
all
n-linear
functionals
on
the
Cartesian
product
of n copies
of
E.
A
symplectic
LCS is a
pair
(E,1)
where
E is
an
LCS, 1 is a linear
mapping
of
E*
into
E,
such
that
1*
=
-1.
A
Hamiltonian
system
is
the
triplet
(E, 1,
H)
where
(E,1)
is a
symplectic
LCS, called (as well
as
E)
323
the
phase
space
of
the
Hamiltonian
system,
and
'H
is a numerical function
on
E,
called
the
Hamiltonian
function.
The
Hamilton
equation
for
this
Hamiltonian
system
is
the
equation
f'(t)
= I'H'
f(t)
with
respect
to
a function f
of
a
real
argument
taking
values in
the
phase
space
E;
here 'H' is
the
(Gateaux)
derivative
of
the
function
'H.
The
equation
for first integrals for
the
Hamilton
equation,
or
the
Liou-
ville
equation
w.r.t. functions, is
the
equation:
8F
7§t(t) = .CI-t(F(t));
here F is a function
of
a real
argument
taking
values
in
some space
F(E)
of
functions
on
the
phase
space E
and
£1-(
is
the
linear
operator
on
F(E),
called
the
Liouville
operator,
defined
by
(£1-(
cp)
(x) =
{cp,
'H}(
x)
where
{-,
.}
is
the
Poisson bracket, which is defined for
any
two functions
cP
and
\Ii
on
E
by
{cp,
\Ii}(x) =
cp'
(x)
(I
(\Ii' (x))).
2.
Liouville's
equations
with
respect
to
measures.
The
Liouville
equation
with
respect
to
measures is
the
adjoint
to
the
equa-
tion
for first integrals for
the
Hamilton
equation.
Nevertheless
it
has
the
same
structure,
which follows from
an
infinite-dimensional version
of
the
Liouville
theorem
about
the
conservation of
the
phase
volume
(the
original
Liouville
theorem
has
no
sense for infinite-dimensional spaces because, as
it
has
already
been mentioned,
there
does
not
exists
"the
Lebesgue measure"
on
an
infinite-dimensional space.)
If
E is
an
LCS
then
a
set
AcE
is called cylindrical if
it
is
the
inverse
image
of
a Borel
subset
of
some finite-dimensional LCS G
under
a contin-
uous linear
mapping
f
of
E
onto
G (it is sufficient
to
assume
that
G is a
quotient
space
of E
and
that
f is
the
canonical
mappings
of
E
onto
G).
For
any
finite-dimensional
quotient
space G
of
E
we
denote
by
Ac
(E)
the
inverse image
of
the
o--algebra of Borel's
subsets
of
G
under
the
canonical
mapping
of
E
onto
G.
Let
9
be
a family
of
finite-dimensional
quotient
spaces
of
an
LCS E
satisfying
the
following condition: for
any
G
1
,
G
2
E 9
there
exists G
3
E 9
some
quotient
spaces
of
which
are
naturally
isomorphic
to
G
1
and
G
2
and
let
Ag(E)
=
UCEgAc(E)
(Ag(E)
may
be
the
algebra,
of
all cylindrical
subsets
of
E,
which is
denoted
by
A(E)).
We assume
that
Ag(E)
generates
324
the
Borel o--algebra of E. A
(9-
) cylindrical measure on E is a number-
valued function
v
on
Ag(E)
such
that,
for
any
G E 9
the
restriction
of
v
to
Ac(E)
is count ably additive.
The
set
of all cylindrical measures
on
(E,
Ag(E))
(resp.,
on
(E,
A(E)))
is
denoted
by
Mg(E)
(resp., by
M(E)).
A function
on
E is called G-cylindrical if
it
is measurable
with
respect
to
the
o--algebra
Ac(E);
a function is called
(9-
) cylindrical if
it
is G-cylindrical
for some G
E
g.
Any
bounded
cylindrical measure becomes count ably
additive
under
a
suitable
extension
of
the
space
[11]
(
the
assumption
of
bounded
ness is often included
in
the
definition
of
the
cylindrical measure).
Definition
2.1.
If
k is a vector field
on
E (i.e. a
mapping
from E into
E)
then
a
measure
v E
Mg(E)
is called differentiable along k if
there
exists a
function
f3'kO
on
E,
which is called a logarithmic derivative
of
v along k,
such
that
Ie
f(x)f3'k(x)v(dx)
= -
Ie
f'(x)hv(dx)
for
any
cylindrical function a differentiable along k
and
bounded
together
with
its
derivative along
k.
If
k(x)
= h E E for
any
x E E
then
we write
f3v
(h, .)
instead
of
f3'k
0
and
call
the
measure v differentiable along h; in
this
case
the
function
f3v
(h, .) is called
the
logarithmic derivative
of
v along
h.
The
measures f3'kOv
and
f3
V
(h,
·)v
are
called, respectively, derivatives
of
v along k
and
along h
and
are
denoted
by
v'
k
and
by
v'
h.
Actually
the
same
definition
can
be
applied
to
generalized measures like
the
Feynman
type
pseudo-measure.
Proposition
2.1.
If
the
linear
span
of
{k(x)
: x E
E}
is
contained
in
the
collection
of
all
vectors
from
E along
which
the
measure
v
is
differentiable
aO
ne
of
the
definitions
of
integral
with
r
espe
ct
to
a
bounded
cylindrical
measure
is
as
follows.
Consider
an
extension
of
the
initial
space
on
which
the
measure
is
countably
additive.
If
the
function
whose
integral
is
to
be
determined
admits
a
natural
extension
to
the
extended
space,
then
its
integral
with
respect
to
the
given
cylindrical
measure
is
defined
as
the
usual
int
eg
ral
of
the
extended
function
with
resp
ec
t
to
the
Lebesgue
ex
tension
of
the
measure
on
the
extended
space
gen
e
rated
by
the
given
cylindrical
mea-
sure.
If
the
given
cylindrical
measure
itself
is
count
ably
additive,
th
en
the
integral
with
res
pect
to
this
measure
is defined
as
the
integral
with
resp
ect
to
its
Lebesgue
extension.
For
cylindrical
functions,
integral
is
calculated
dir
ectly,
because
G-cylindri
ca
l
functions
are
measurable
with
respect
to
the
a-algebra
AG(E);
in
this
case we
do
not
ne
ed
to
assume
that
the
cylindrical
measure
is
bounded.
It
is also
worth
noticing
that
actu-
ally
both
integrals
with
respect
to
a
a-additive
measure
and
integrals
with
respect
to
a
cylindrical
measur
e
are
calculated
as
limits
of
proper
sequ
e
nces
of
some
integrals
over
finite-dimensional
spaces.
325
then the measure v is differentiable along k and
(3~
(
x)
=
(3v
( k ( x ) ,
x)
+
tr
k'
( x )
(see
[12}).
Remark
2.1.
If
the
dimension
of
a space E is finite,
then
the
algebra
of
sets
A(E)
coincides
with
the
cr-algebra
of
Borel
sets
and
M(E)
coincides
with
the
set
of
all count ably additive finite Borel measures. If, moreover,
the
measure v
has
a
smooth
density 9 E
Ll
(E),
with
respect
to
the
Lebesgue
measure
on
E,
then,
whatever
a vector
hE
E,
the
measure v is differentiable
along
h
and
the
derivative
of
v along h,
v'
h, has a density
f'
Oh
with
respect
to
the
Lebesgue measure;
the
logarithmic derivative of
the
measure
v along h coincides
with
the
logarithmic derivative
of
its density. However,
in
the
general case,
the
density
of
the
d
er
ivative
of
such a measure along
a vector field
k,
v'k,
does
not
coincide
with
the
derivative along
this
field
of
its
density,
and
the
logarithmic derivative
of
the
measure
along a vector
field does
not
coincide
with
the
logarithmic derivative
of
the
density
of
the
measure along
the
same
vector field.
Proposition
2.2.
If
a vector field k is Hamiltonian, then
(3~(x)
= (3V(k(x),
x)
(one does
not
assume that
k'
(x) is a trace class operator).
Corollary
2.1.
Under the assumptions
of
Remark
2.1, the derivative
of
a
measure along a Hamiltonian vector field has a density, which coincides with
the derivative
of
the density along this field, and its logarithmic derivative
coincides with the logarithmic derivative
of
the density.
Proposition
2.3.
If
F is a canonical transformation
of
the phase space,
v
E
Mg(E),
and
vF
is the image
of
a measure v with respect to the mapping
F-
1
,
then
d(vF)
(x) =
exp(
r
1
(3V(FUt,
x),
F(t,
x)dt),
dv
J
o
where the mapping F :
[0,1]
x E
--7
E is differentiable with respect to the
first argument and such that
F(O,
x)
= x and
F(1,
x)
=
F(x)
(the Radon-
Nikodym derivative
d(~:)
does
not
depend on the choice
of
such a mapping).
This
proposition follows from
Proposition
2.2
due
to
results
of
[12].
326
Remark
2.2.
Proposition
2.3 is a version
of
the
classical Liouville
theorem
on
the
conservation
of
the
phase
volume;
Proposition
2.2
can
be
regarded
as
an
infinitesimal version of
the
same
Liouville
theorem.
Remark
2.3.
Derivatives
and
logarithmic
derivatives
can
also
be
defined
for
measures
not
being
finite so
that
analogues
of
Propositions
2.2
and
2.3
remain
valid.
Definition
2.2.
The
Liouville
equation
with
respect
to
measures
on
the
phase
space E of a
Hamiltonian
system
is
the
equation
: (t) = DH(v(t))
with
respect
to
functions
of
a
real
argument
taking
values
in
some space of
cylindrical
measures
on
E;
here,
L'H
is
an
operator
on
the
space
of
measures
adjoint
to
the
Liouville
operator
on
the
space
of
functions.
If
we
want
to
be
more
precise
we
will
speak
of
measures
on
(E,
Ag(E)).
Theorem
2.1.
The Liouville equation with respect to measures
on
the
phase space E has the following
form
: (t) =
-(v(t))'(IH')
==
_(3v(t)
(I(H'(.),
·)v(t)
The
right
hand
side
of
this
equation
is
the
derivative of
the
measure
v(
t)
along
the
vector field
IH'
(.),
thus,
the
theorem
follows from
Proposition
2.2
and
the
definition
of
the
Liouville
operator
on
the
space
of
functions.
Remark
2.4.
The
Hamiltonian
function H is usually
the
sum
of
a series
which
may
converge
not
everywhere;
but
in
what
follows we do
not
discuss
convergence conditions for
the
corresponding
series.
3.
Systems
of
equations
with
respect
to
finite-dimensional
distributions
of
probabilities.
In
this
section we consider a general scheme
of
the
passage from
the
Liouville
equation
with
respect
to
measures
on
an
infinite-dimensional space
to
an
equivalent infinite
system
of
equations
with
respect
to
functions
on
finite-
dimensional spaces.
In
this
and
in
the
next
sections
we
assume
that
the
symplectic space E
has
a symplectic basis B = {ek : k =
1,2,
... }
and
is
endowed
with
the
Euclidean
structure
with
respect
to
which
the
elements
of
E form
an
orthonormal
basis. We also assume
that
every subspace of E
generated
by
a finite family
of
elements
of
B is
equipped
with
the
Lebesgue
327
measure generated by
its
Euclidean
structure
(this measure depends only
on
the
initial symplectic
structure).
For each finite set {ek
1
, ... ,
ek
n
}
of elements from
8,
constituted
a sym-
plectic basis of a finite-dimensional (symplectic) subspace of
E,
let
Fk1,
...
,k
n
be
this
subspace,
and
let
Gk1,
...
,k
n
be
the
symplectic quotient space of E
by
its
closed subspace generated by those elements of 8 which are
not
con-
tained
in
the
linear
span
of {ekll ... ,ekn}.
Let
9(=
9(8))
be
the
set of
all quotient spaces
Gk1,
...
,k
n
.
The
Liouville
equation
with
respect
to
mea-
sures on
(E, Ag (E))
is
equivalent
to
the
system of equations
with
respect
to
the
restrictions of these measures
to
(some) subalgebras
Ac(E),
where
G E
9(8).
Under
the
assumption
that
the
measures on these
sub
algebras
may
be
determined by densities on
the
spaces
Gk1,
...
,k
n
, or, equivalently
(because of
the
natural
isomorphism between
Fk1,
...
,k
n
and
Gk1,
...
,kJ,
on
Fk1,
...
,k
n
,
one
can
compare
the
system
of equations
with
respect
to
the
den-
sities
with
the
Bogolyubov
system
of equations
b
;
but
these two systems are
equivalent only if
the
space E is finite-dimensional; in general case
they
are
different (see below).
For any measure
f.1
E
Mg(E)
and
any finite set {ek
1
, ... ,
ek
n
}
of elements
of
8 let ft,
...
,k
n
denote
the
corresponding density of
the
marginal measure
on
the
space
Gk1,
...
,k
n
.
Theorem
3.1.
Let v E
Mg(E)
and {ekll ... ,ekn},
{erll
...
,e
rm
} C
8,
let
the Hamiltonian function
H
be
Gr1,
...
,r
m
-cylindrical and let
hr1,
...
,r
m
be
the
Hamiltonian vector field on
E generated
by
this function, that is,
hr1,
...
,r
m
I(H'(·)).
If{erll
...
,e
rm
} C {ekll ...
,ekn},
then
bIn
nonequilibrium
statistical
mechanics,
the
Bogolyubov
system
(or
chain)
of
equations
(we
do
not
consider
here
the
Bogolyubov
equations
describing
the
equilibrium
states
2 ,
which
have
the
similar
form
but
a
different
meaning)
is
an
infinite
sequence
of
equations
with
respect
to
time-dependent
functions
on
n-particle
phas
e
spaces.
In
the
classical
papers
some
similar
but
finite
systems
of
equations
are
deriv
ed
from
the
classical Li-
ouville
equations
describing
finite
sets
of
particles;
after
that
a
formal
operation
called
the
passage
to
the
thermodynamical
limit,
in
which
the
number
of
particles
and
the
volume
occupied
by
them
tend
to
infinity, while
the
particle
density
remains
constant,
is
perform
ed
.
Although
the
functions
on
n-particle
phase
spaces
that
form
solutions
to
finite
syst
e
ms
of
Bogolyubov
equations
are
proportional
to
the
densities
of
the
cor-
responding
probability
distributions,
under
the
passage
to
the
thermodynamical
limit
(in
which
the
number
of
equations
in
the
system
tends
to
infinity),
the
proportionality
coefficients
tend
to
infinity;
this
is
the
formal
reason
for
the
fact
that
solutions
to
Bo-
golyubov
infinite
systems
of
equations
consist
of
functions
not
proportional
to
densities
of
any
finite-dimensional
probability
distributions.
328
f
v'h
(X)
-
k1,.·.,k
n
-
j(f{
V
}U{k
k
})'(
... X
S1
'
...
,
Xs
... )hrl
...
r
(",XSl'
...
,
Xs
... )dxs1···dx
s
Tl,···,T
rn
1,···,
n
p,
,Tn
P P
(equalities
of
the first type can
be
and
are considered as special cases
of
equalities
of
the second type).
Remark
3.1.
If
{e
r
"
... , e
rm
} n
{ekl'
... , ek
n
}
= 0
then
fk:~
..
,kjx)
==
o.
Let 1t =
I::
Hrl
,
...
,r
n
,
where 1t
r1
,
...
,r
n
is a G
r1
,
...
,r
n
-cylindrical function
for each finite
set
{r1'
... , rn}
of
positive integers
and
the
summation
is over
all such
sets
of
positive integers.
Let
also
hr1
,
...
,r
m
=
I(H~"
...
,
rIJ·)),
Theorem
3.2.
A
function
1/(')
taking values
in
Mg(E)
is a solution
of
the Liouville equation with respect to measures
if
and
only
if
the
functions
t
f--+
gr"
...
,r
n
(t), defined by the equalities
gr"
...
,r
n
(t)
=
f::,(,t).,r
n
satisfy the
following infinite
system
of
differential equations:
g~"
...
,kn
(t)(·) =
where the
summation
is over all finite sets
{r1'
... ,
rm}
of
natural numbers
for
which
{r1'
... , r
m}
n {k1' ... , k
n
}
#-
0,
and,
in
accordance with what has
been said above,
if
{r1'
... ,
rm}
n {k1' ... , k
n
}
=
{r1'
... ,
rm},
then
the integral
sign is assumed to
be
missing.
This
follows from
Theorem
3.1
and
Remark
3.1.
4.
Bogolyubov's
systems
of
equations.
In
this
section we describe a relationship between
an
infinite
system
of
equations similar
to
the
Bogolyubov system,
and
the
Liouville
equation
with
respect
to
functions
taking
values in
the
space of infinite measures.
Below
the
symbol
M=(E)
denotes a vector space of measures,
on
the
phase
space E ,
taking
values in
the
extended
real line [-00,00]. We assume
that
the
domains
of
such measures is a ring
of
Borel
subsets
but
we
will
not
discuss here
in
details such domains; instead,
we
specify
the
measures by in-
tegrals,
with
respect
to
them,
of
some Borel functions whose restrictions
to
329
each
of
the
subspaces
{F
1'1
,
...
,1'n}
are
continuous
and
compactly
supported.
Those
integrals, in
turn,
are
defined as
the
limits of sequences of integrals
over
the
subspaces
{F1'"
...
,1'
n
}'
If
the
measures
on
the
subs
paces
{F1'"
...
,1'
n
}
are
given by
their
densities
'l/J1'"
...
,1'
n
'
then
we say
that
the
family of func-
tions
{'l/J1'
"
..
.,Tn
:
{rl,
...
,r
n
}
C
N}
determines
a measure v
on
E.
A family
of such functions is said
to
be
compatible
if, for
any
finite
set
{rl,
... ,
rn}
and
its
proper
subset
{rl,
... ,
rd,
the
following holds:
'l/J1'"
...
,1'k
(Xl ,
..
. ,
Xk)
=
limvol(V)
-H
)()
vOI(V)
Iv
'l/J1'l
,
..
.,Tn
(Xl, ... , Xn)dXk+I ... dxn; here V is a ball cen-
tered
at
zero in F
rk
+
1
,
...
,1'
n
and
vol (V) is
its
volume.
Suppose
that
the
operator
L'H,
which is adjoint
to
the
Liouville
operator,
is defined
on
Moo(E).
Theorem
4.1.
Let, for every finite set
{rl,
... ,
rn}
of
natural numbers,
gr"
...
,1'
n
be
a function
of
a real argument taking values
in
the space
of
(bounded continuous nonnegative) functions on F
1'"
...
,1'
n
'
and, for each
t,
the family
of
functions
g1',
,
...
,1'n
(t) is compatible and determines a unique
measure
v(t)
E Moo(E). Then the function v(-) is a solution
of
the Liou-
ville equation with respect to measures,
if
the family
of
functions
g1'"
..
,1'n
is
a solution
of
the
system
of
equations
+
00
L
. 1
hm
- - -
N---+oo
Nm-p
{j"
..
,jp}={1'"
...
,r",}n{k
"
...
,k
n
}
{1'"
...
,1'",}\{k
"
..
,kn}n{I,
...
,N
}
Remark
4.1.
The
function v(·) being a solution
of
the
Liouville
equation
does
not
imply
that
the
family
of
functions
g1'"
...
,r
n
(.)
is a solution
of
the
system
of
equations from
the
theorem. For example
it
is so if all measures
v(t)
are
probabilities
(then
we need
to
apply
Theorem
3.1).
Remark
4.2.
The
difference between
the
system
of equations in
Theorem
3.2
and
that
in
Theorem
4.1 is
that
the
latter
contains a generalized Ce-
saro
mean
[15]
(instead of one of
the
sums), which
may
exist even when
the
corresponding series diverge. Moreover
the
functions
g1'"
...
,1'",
are
not