Séminaire Jean Leray.
Sur les équations aux
dérivées partielles
N. N. B OGOLIUBOV
B. I. K HATSET
D. YA . P ETRINA
Mathematical description of equilibrium state of classical
systems based on the canonical formalism
Séminaire Jean Leray, no 3 (1969), p. 47-73
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47
MATHEMATICAL DESCRIPTION OF EQUILIBRIUM STATE
OF CLASSICAL SYSTEMS BASED ON THE CANONICAL FORMALISM
by N.N. BOGOLIUBOV, B.I. KHATSET, D.Ya. PETRINA.
Introduction
To derive
thermodynamic relations on the basis of statistical mechanics, one has
to study systems with infinite number of degrees of freedom. These systems arise
from finite ones when the number of particles N tends to infinity, which is followed
vN, v const.). This creates
by a proportional increase of the volume
rather difficult mathematical problems connected with the justification of the
To solve these problems the forualiems of either
limiting procedure as
canonical or grand canonical ensemble as well as that of partition functions are
=
used.
During the last two decades a considerable advance has been observed in this
field. In 1946 one of the authors [1] suggested some ways of the rigorous mathematical justification of the limiting procedure in statistical mechanics, using the
formalism of the Gibbs canonical ensemble, and developed a general method of determining the limiting distribution functions as formal series in powers of the density
In 1949 two of the authors [2] worked out the foundations of the rigorous mathe1;.
v
matical description of infinite systems in statistical mechanics. A comprehensive
presentation of the results obtained has been given in [3] in 1956. The works [2,3]
comprise the complete solution of mathematical problems arising when performing the
limiting process as N - co in the systems described by canonical ensemble for the
case of particle interaction through a positive two-body potential and sufficiently
small densities. It should be noted that the system of equations for the distribution
functions was essentially treated as an operator equation in a Banach space. However,
the methods developed in the works mentioned above did not call the attention of the
scientists.
In 1963
[4] again suggested
approach to studying the sets of
equations for the distribution functions. He used the grand canonical formalism,
which led him, in our opinion, to simpler problems involving the justification of
the limiting procedure. At the same time Ruelle succeeded in extending the class of
potential functions in question by exploiting the rather ingenious idea of symmetriRuelle
zation of the
canonical ensemble,
small
similar
starting equations for the distribution functions.
present paper is to give, on the basis of the theory of
rigorous mathematical description of the equilibrium state
of infinite systems of particles whose interaction potential
The purpose of the
(for
a
a
densities)
.
48
obeys Ruelle’s condition [4’ i and is not required to be positive. The methods developed in [2,3] and Ruelle’s method of symmetrization are mainly employed.
In Section 1 the
tions in the
of
is stated and the relations for the distribution func-
problem
finite volume
derived, which after taking the limit
become the well-known Kirkwood-Salsburg equations. Contrary to the grand canonical
case
formalism, there
a
are no
equations whatever for the distribution functions in the
of Gibbs ensemble in
case
are
a
finite volume.
after taking the infinite-volume limit.
son
with the
grand
only
compari-
The appropriate equations appear
This
brings about
new
problems
in
canonical formalism.
In Section 2 the theorem of existence and uniqueness of the solution of the
Kirkwood-8alsburg equations for tne potentials obeying Ruelle’s condition is proved,
and the densities are explicitly evaluated, for which the solution can be represented by iterations. The theorem on the al1alytical character of the density dependence of the limiting distribution functions is proved as well.
Section 3 presents the proof of existence of the limiting distribution functions
when the number of particles tends to infinity.
Finally,
in Section 4 the
uniqueness of these limiting functions is proved.
1. S ta temen. t of the Problem
1. Consider
system consisting of N identical particles enclosed in a threedimensional macroscopic volume
VN and interacting through central forces which
are characterized by the mutual potential
e2-(q). It is assumed that the position
of each particle is determined completely by its three cartesian coordinates
a
q l a ~ ~ 1,2,3),
f
point of departure is the common theory of equilibrium states based on the
Gibbs canonical distribution ’, and in the presentation of the subject we follows
4u~
the
works g 2 p 3 .
probability distribution
particles with the density
Introduce the
where
is the potential energy of the
system,
functions for the
positions of all
49
is the
ensemble.
Introduce
As
V~
usual,
where
T:1e physical system considered is
conf irrational integral.
v
now a
sequence of distribution functions
is assumed to be
a
is the volume per
object of investigation
In the limit of
an
a
Gibbs canonical
[1]
vsl,
sphere whose volume is also denoted by
is the densi ty of particles. The principal
v
in statistical
infinite volume tl1e
physics
is
density 1
presented by the limiting functions
is assumed to ioe constant.
v
studying the problem associated with the limiting procedure mentioned above we
employ the set of equations which tbe limiting functions
Below we shall derive some relations wl1ich are necessary for obtaining the appropriate
9 - - - 9CJ.S Vv)
equations.
and transform it
2. Consider the expression
vl11ere
Substituting
(2.1)
into
(1.1)
one
obtains
as
follows
50
where
Introduce also the distribution functions
and
quantities
with
in terms of which
Similarly,
~3.1 ~
is written
for the functions
as
one
obtains the relations
51
3. Suppose for
exist in
some sense
limit in
a
(lIe
a
while that the limits
and 1irite the relations
formal way.
do not indicate the
It is
easily
seen
(5.1)
and
(6.1)
that the relations
after
(5~1)
performing the
become
integration limits when the integration is carried out over
the whole three-dimensional space). Aiter taking the limit the relation (6.1) will
be of the form
52
Should
be valid for
~8.~ ~ ~ t9.~ ~
tiie relations
will become the well-known
Kirkwood-Salsburg equations [5]
To
complete
(11.1)
i’ox tl1e
of S
case
the definitions that all
= 1 ,
are
It follows from
put
0
symmetric functions of variables
we
position to discuses the problems arising in the mathematical
description of the system in equilibrium state. To present a complete justification
for the mathematical description of such systems 9 based on the Gibbs canonical
4. Me
are now
in
a
using sequences of d.istribution functions,
solve the following three problema :
distribution and
to
of
1)
to prove that the limits
2)
to prove that the
taking
3)
the limit
(’~.~ ~
exist in
a
think it necessary
certain sense,
limiting distribution functions
do not
depend
on
the imy
(that is, (10.1) holds
9
sufficiently small densities
the Kirkwood’-Salsburg set oxn equations.
to prove that for
solution of
we
1
v
there exists
a
unique
B’!e shall solve the above mentioned problems in the following order : first,
problem 3 - ancl then 1
and 2.
2. The existence theorem for the solution
oi the Kirkwood-Salsburg set of
equations
1. In the present section
(11.1)
shall consider the
we
for the distribution functions
ciently small
densities 1
v
t:1BY
vie have the set of equations
possess
..
Kirkwood-Salsburg equations
and prove that for suffi-
a
L
unique solution.
53
The solution of tlais set will be determined in
introduced
Banach space
[2,3,43
Consider the linear space whose elements
follows.
as
a
of measurable bounded functions
are
which is
tile columns
i
operations of addition and multiplication by
becomes a Banach space B for the following norm of
number.
with ordinary
a
space
the element
where
A
In
was
is
[2,31
positive constant which is
a
the equivalent
introduced,
which
li
was more
is
The set of equations
where
to be determined.
norm
for the distribution functions
The operator
This linear
convenient when
~~ ~ ~ 9
~.
can now
the
Mayer-Montroll equations
23).
formally defined
(1 .2)
studying
in the space
B
through the relation
be written down in the form
54
Below
tor
shall prove the existence and
We
equation
(4.2)
and for
it is
in the space
sufficiently
known,
and its
in order that
wl1icl’l converges in the
conditions
are
assumptions concerning the potential
(4.2) possess
operator a(v)K be
unity.
Tlae solution
of the space
norm
valid for
some
equation
the
be less that
norm
under
small 1.
it is sufficient that
B
B
uniqueness of the solution of the opera-
sufficiently
wl1ic11 is ass1..l11led to be
a
B .
small vv
F
It is
is
a
unique solution
B
defined in the whole space
represented by the series
easily
seen
in the case of
real-valued function in
in
E3
[2,3J
tl1at these
non-negative potential
such
’A
’I
shall
complex)
assume
first that
a(v)
and
are
independent (generally speaking,
parameters with
and evaluate the
norm
of the
operator
a(v)I( .
It should be noted that the condition
Then it follows from
Hence,
v
(3.2)
that
~(q) ~
0
implies that
55
If
now we
put ii
2e
=
and take into account the
get
(8.2)
inequalities
and
(9.2),
we
0-T
At the
same
time
It remains to ci1eck
the definitions
only the
(5.2)
the
immediately follows
from
equations
assumptions (7.2) and (8.2), the
unique solution in a neighbourhood of
and with the
0
of
B .
since
It is tlms shown that
Kirkwood-Salsburg set
The latter
condition
(2.2)
and
is defined in the whole space
K
that the operator
vre see
(1.2)
has
a
0 ,
=
developed by two of the authors in 1949 [2,3].
We shall show now that it may also be applied to the more general case when the
requirement that the potential C2(q) be nonnegative is replaced by the following
The method described above
[4] s
Ruellets condition
There exists_
the following
a
positive constant
Ea
such that for all
it results that for any
there exists at least
In view of the
(1.2)
one
symmetry
in
index
i
point
and an
(1J., ... ,1S)
belonging
may,9 after
Ruelle [4],
to
11 3s
such that
of the functions
symmetric form.
a
S
is valid :
inequality
From this
equations
was
Let
one
TT4»
denote the operator acting
wri te
on
the
through the formula
It follows from
(14.2)
that there exist measurable functions
iant under t:1e rotation group, leaving the values in the
vi (q1, ...,qS),
interval [0,1],
invar-
and such
that
with the
B?.
can
inequality
(14.2)
be considered
as
0eing valid if ~~ . ~~1 9...
the partition of the unity.
~q-;~ ~
0 .
The set of functions
56
Now
we
define the operator
n
by the relation
In view of the
may
symmetry of the functions
be represented in the following form
~(q)
With the assumption that
equations
obeys the conditions
(8.2)
~13.2~ p we shall
(15.2) for suffi-
and
establish the existence and uniqueness of the solution to the set
ciently small 1 e As above,
parameters and the inequality
~~ 5,2 ~
The set
is
now
determined
a(v)
(8.2)
B
the
norm
Bearing
as
independent complex
(15.2)
(4.2) provided
(3.2) :
ossesses
a
that the
operator
a
holamorphic function of
unique solution in the Banach
of the
assertion, it suffices
operator K defined i~y
in mind that the estimate
which follows from
a(v)
(14.2),
we
(10.2)
obtain from
in the
domain (1 7.2 ) .
to show that with the conditions
(16,2 ~
is
now
(15.2) :
is less than
replaced by
IC
Ida
.
To prove the first
(17.2)
considered
is assumed to be valid.
by the symmetrized relations
for
This solution is
are
v
may be written in the form
I. The se t of equations
space
and
(1.2 ~
1.
57
2e2i&#x3E;+1
Putting
A
estimate
required :
=
and
taking
into account
(17.2)
we
obtain from
(18.2)
the
We have thus proved the existence and uniqueness of the solution of the set
(15.2),
or, which is the same of the equation
solution is
a
holomorphic function of
represented by the series
v
(6.2)
now
to
we
as a
function of
(17.2)
our case
Let
To this
may be omitted.
I. The numbers
[3].
lie shall
a (v)
only
quantities
such that
a.:
for
in this domain since it may be
v
uniformly
with
regard
to
a(v)
v
as a
end,
we
real
(positive)
case
variable and
our
and
the condition
shall find the estimate for
)a(v) I
a(v)
2
,
elementary inequality
special
1+a &#x3E; 0 .
of
a (v)
quantity
the following
which holds not
This
obey the
Consider the
use
domain (17.2).
of interest of the limiting distribution functions
show that in this
us
in the
~x~ ~ ~ 7.2 ~ .
have to consider
v .
and
which converges
in each closed domain contained
4. Timing
a(v)
(4.2)
but also for
arbitrary
58
B--.f
use
of
(21.2)
one
obtains from
the
following estimate
N
whence for
a.(v) ==
Since, by definition;,
the inequality
lim
the inequality
for any z ,
~
then
ar(v)
2
aD (v)
also
satisfy
N
(19.2). ~ N-~oo
for
From Lemma lit results
the first
(20.2)
that
holds,
the second condition of
is ,
(17.2)
implies
one.
Fiiially,
consider the character of the
density dependence of the limiting
distribution functions.
THEOREM II. The distribution functions
to 1
in
Proof
and 1v
neighbourhood of the
a
[3]. By
Theorem
in the domain
zero
(17.2),
are
tInt
of the
point 1 =
are
--I*s, by virtue
a(v)
is
a
of
holomorphic functions of
Lemma I, for
holomorphic function
(6.2)
demonstrated
whence
respect
of 1
in
a(v)
some
0 .
Equation (4.2) is translation-invariant
possesses tiis property.
Therefore, F1 (q, v)
in Section 4,
The solution of
series
holomor- h-ic with
point.
I,
It is thus sufficient to show that
neighbourhood
F S (q, fqs iv)
since each term of the
=
it will be
59
Now, by making
where the function
show that in
use
x
in
a
in
holomorphic
a
and §
neighbourhood
o
a
and the function
of the
point 1
dX
It follows from
(28.2)
§j /
neighbourhood
in
(17.2).
fora’/
the
2
,
=
some
that
is
holomorphic
us
partial
(23.2)
with
Let
can be
respect
to
0 .
v
..
n.....,
0
obtain
in the domain
I
a(v)
The following expression is obtained for
a
0
we
The latter condition implies that the relation
solved with respect to
v
is
some neighbourhood 0:( the
v
, ~a
2013
of the Kirkwood-Salsburg equations,
= 1
,
da :
point 1 =
at the p
a
of the
zero
point.
0
and, as a consequence,
This completes the proof.
v
3. Existence of the limiting distribution functions
1~ In the present section
we are
studying the problem
When solving this problem, it is convenient to
are
use
the relations
the basis for deriving the Kirkwood-Salsburg equations.
Consider thus the relations
I stated in Section 1.
(5*1)y (6.1)
which
60
(6.1)) due to the functions
2, (15.2)). These relations will 0e
which follow from
(see
Sec.
sional Euclidean space
the whole
The
Let
column f
assumed valid in the whole
following notations
are
are
3S-dimen-
defined in
(l.1)y (4.1).
introduced
the opera tor acting
in
the Banach space
B
on an
arbitrary
by the formulas
Here the expressions
Denote by
Uith the operators
in the
being symnetric
The functions
by the formulas
KS(N-l)
FS(N-l(q1,...,qs, VN)
the
are
following operator
1 ~’~~’~~ ~
following compact
and columns
form
also defined in the whole
E 3S -b
’
relations
(1.3)
are
represented
61
shall investigate
2.
One
can
v
for
show that the
2e2-o+l I
°
the properties of the operator
now
of the
norms
operators
K
- l)
and
I , 1 -1, - S (N 5-
||K(N-2)||
k
1,
unity,
argument is similar to that employed i11 the proof of Theorem I if
account that the quantities
obey the inequalities
aN-,
are
less
rr*"
1
1
21 ’a
restriction
same
A
The number
as
appearing in the
The
one
can
separate any finite number
whose limits
as
N ft
are
one
1
N-,,
S
The
takes into
norm
definition is subjected to the
positive sequences
1 *)
2 fv
-- 1- 2I.
*)
i above ’oy
bounded from
by 2
k
(1
(VTT)
in Section 2.
are
-l)
s
l+1
Therefore, using the diagonal
dia,-process,
converging subsequences
of
.eoee by
a¡
(v) , 0 .. ,a , ( v) j
3
shall
N-l a N-l (VN)
N
see
below
there exists
themselves converge
, * .
=
a1(v).
»
some
over
as
N
For the time
only
-&#x3E;
oo.
one
tzat for each sequence
limiting point,
Moreover, it
being,
we
shall
that
is, the
will be shown that
mean
sequences
a (v) = a1 (v)
=
...
under the sequences
of their
converging subsequences and the limiting process will be performed
these sU0sequences as N - aJ ~ although we shall write FI -* oo ,
Let
space
(Section 4)
B
denote the operator
through the formula
acting
on an
arbitrary element
f
of the Banach
62
(R)
be the characteristic function of the
and the origin
as a
VI~
centre and
The function
tiess
Let
One
T (R-r)
more
is
VN
-L
with
radius
and characteristic
required to possess the following proper-
as
be the
operator acting
property of the operator
being fixed,
sphere with radius
rN
as
If
l
the
sphere
on an
arbitrary element
KS(N-l)
s
is
feB
given by the following lemma.
then tlw sequence of operators
converges
9 relative
Proof. Consider .for any
f
to the norm to
expression
by the formula
zero as
N -&#x3E; oo.
63
where
N0
Let
is
as
D, y,
yet
an
arbitrary finite number.
denote the last three terms
respectively.
The following estimates
hold
-
where
Being the remainder of the absolutely converging series, the quantity E1
can be made arbitrarily small for sufficiently large but finite
NO independently
of N. The quantity E2 (rN)
br
can
made arbitrarily small for sufficiently large
N due to the integral
(No)
.
being absolutely convergent. The difference
by
obeys the following estimate :
of the first two
expressions, denoted
64
with
E
3 (N)
(here No
is
a
fixed finite
Thus we obtain
which is
This
for
being arbitrarily
equivalent
large enough N
since
number),
finally
to the
following estimate
immediately implies
which
that the sequence of
means
zero as
Let
N -
-
n
III
norm
norms
f
with
n+ = fn+2 = ... = C
fn+1
given by
Consider the operator
The
Lemma is proved.
norm.
denote the space consisting of columns
B
and with the
relative to the
m
converges to
of these operators
into
are
B :
less than unity.
The operators
are
the
sums
of
a
which converge to
finite number of operators
zero
relative to the
norm.
Therefore the operators
65
also converge to
zero
relative to the
It is
norm.
easily
seen
that tl’1e
following
estimate is valid :
Below
where
n, j
manner as
we
shall also Yieed the
are
~5.3 ~
integers, j
11..
following inequalities
These inequalities
relative to the
and T
Therefore,
verified in the
same
in Lemma 2.
a.e(v)K)
Remark. The operator
zero
are
R ~
for any
0
tiae operator
norm.
=
0
’1’(R-r)K(N-t)
for tl1.e elements
N ,9
a~ (v)Il£ / 0 ,
arbitrarily
large
does not converge to
there is
and
with
always
norm
acts from
fS =
an
0
B
into
S 9 N-Q .
for
element f
such that
of
i s finite.
3. Consider
now
where the operator K
to the
%~~
the relation
(N-l-j)
(43)* Applying
acts a£-ler
the
it
ope:rator
repeatedly
we
get
X (N-,e -j -1and,
according
definition
Tale columns
differ from each otl1.er for
by definition.
I~ ~ Nz ,
66
We have further
where
use
has been made of the
It follows from
what
follows)
(S .3) 9 (9.3)
the
following formulas
inequality
Bearing in mind
from
(7.3)
The latter
B
N
that for large enough
will be assumed in
1 holds,
1 ,
1 IF li1
the following estimate for
inequalibj implies that
(which
1
norms are
bounded
the columns
uniformly
obtain
I
~~~’~~Q ~
belong to the Banach space
for
and their
we
with
regard
to
N and Z .
67
Consider the columns
which appear when
iterating the relations
and belong to the space
B
THEOREM
III.
For any
tends to
zero
in the space
Proof.
Let
us
as
yt follows from the estimate
and
f ixed - o
B
(see (9.1))
as
sequence
1B1 -7 00 .
represent the difference
in the form
.
n
The
norm
small for
X
of the column
large enough
and the columns
The inequalities
1’1.
does not exceed
By virtue
9
one
~5.3 ~ , ~~.3 ~ imply
2- k -
and
can
of the definitions of the
has
be made
arbitrarily
operators
X(N-.e) 11
68
Hence
It is
easily verified that
Using the above inequalities
we
obtain
The filial estimate is given by
Choosing
inequality
n and
(20.3)
N
sufficiently large,
arbitrarily small.
This
one
means
can
that
make the
right ’Land side
of the
69
Theorem is
proved.
As a consequence of Theorem III
we
obtain that
4. Uniqueness of the limiting distribution functions
1. In this Section
rune ti ons, i . e.,
THEOREM IV.
we
we
shall prove the uniqueness of the
shall prove that
For small
v
and the relations
(1.4)
Proof. According to
F~
the
=
F
and
limiting distribution
valid.
are
(12.3)
we
have
Consider the difference
It follows from
Below
we
(3.4)
shall establish the
Bearing
in mind
(5.4)
following estimate:
one
gets from
(4-4)
limiting distribution
functions
are
identical
70
According
to
Lemma 1,
It follows from
Choosing
the quantity
yields
Since
=
...
that for the
norm
sufficiently large ,
6 (v)
in
(6.4)
of
obey the inequality
Fl+2
1
the following estimate holds
easily
one can
be made less than
can
inequality (8.4)
arbitrarily small, that is,
6(v)
is
=
(13.3)
A
a0(v)
the quantities
and the
show that for small
unity.
The
inequality
is valid for any
=
Fl+1
and,
in
enough 1
I , the
general, F
v
(6.4)
norm
=
I? 1 I? a
Fl+1 = ... .
Now the inequalities
and
are
(5-4)
that for small
imply
identical, and,
in
general,
v
a(v) = a1(v)
=
the quantities
al(V)
&#x3E; ( v) ....
Theorem is proved.
It follows from Theorem IV that the column
F
satisfies the Kirkwood-Salsburg
equation
satisfy
2. The functions
Let
us
take tiae limit m
(9.4)
and. show that the
the relations
following relation holds
71
To this
end, consider the identity
and evaluate the first and the second terms of the
second term
The
obey
inequality
the
above
(10.4)
are
constant.
Using also
now
to
that the second term of
implies
~~ 1.4~
is thus
~~ 1,4~
tends to
zero as
The
proved.
(10.4)
As it is
implies that
proving the inequality
the formula
N
obeys the following estimate
easily seen, the functions
translation-invariant and therefore the functions
The relation
Proceed
(11.4).
of
following estimate
The first term of
The relation
right-hand side
4
(5.4).
=
From
1
,
(12.3)
l &#x3E; 0 .
it follows that
are
72
we
obtain the required estimate
3 0 B’le shall show
Indeed,
(5.4) :
that the sequence
now
if there Bvere two
a(v)
subsequences converging to
than corresponding to them there would oe two columns
the
IV,
Bearing
in mind that
a(v) - a~ (v)
the
in the
same
6(v) 1
from (15.4)
where
and
Bearing
F’r
F
and
we
obtain for the difference
obeying, by
equations
Theorem
This,
unique limiting point.
and al (v) respectively,
the
F, (qjv)
=
and-
1
=
1
,
following estimate
as
way
for
in Theorem
sufficiently
a(v)
this in
mind,
’A
IV,y implies
small 1.
v
tlie
of the
validity
It follows from
(13-4)
inequality
that
F z
a1(v).
=
we
finally
obtain from
(10.4)
The uniqueness of the limiting distribution functions being taken into account,
Theorem III
(Section 3)
For any fixed
j&#x26;
N
can
oe formulated
and 1
v
-uniformly
tends to
zero as
limiting
distribution functions
-&#x3E; oo
2013==0 .
v
12,"
as
follows.
sequence
Observe that the
with
are
regard to q1,...,qS.
holomorphic functions
in
v
.
some
neighbourhood
73
REFERENCES
~.* ~~~~~~~~
~~~~~~~~~~~~, ~.-~., 1946.
1. H.H. ~ ~~
~ ~ ~ ~~
~~~~~~~~~~~~ ~~~~~~ ~ ~~~~~~~~~~~~ ~~
~~~~~~.
2. H. ~ ,S.~ .~HCP,194 6,321.
3. B.I. X A ~~T.*
H~y~~~~
~~~~~~~
~~~~~~~~~~~~~
~~~~~~~~~~~~~ ~~~~~~~~~. ~~~~~~.
~~~~~,, 1956, 3, 113, 139.
4. D. RUELLE .
5. ~. ~~~~,
*) 1. N.N.
Ann. of
Phys., 1963, 25, 209.
~~~~~~~~~~~~~~ ~~~~~~~~.
Bogoliubov , 3. B.I.
~~, ~., 1960.
5. T. Khill.