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The asymptotics of the correlation functions in the generalized model of Boltzmann-Enskog
Physica IlOA (1982) 329-338 North-Holland Publishing Co.
T H E ASYMPTOTICS OF T H E CORRELATION FUNCTIONS IN T H E GENERALIZED MODEL OF BOLTZMANN-ENSKOG B.I. SADOVNIKOV, N.G. INOZEMTSEVA, S.N. BOCHKOV Department o[ Quantum Statistics and Field Theory, Physics Faculty, Moscow State University, Moscow B-234, USSR
Received 28 April 1981
The long-timebehaviour of the autocorrelation functions for an ensemble of particles with pair interactions including a hard core and a long-range tail is considered. On the basis of the nonlinear generalized Boltzmann-Enskogkinetic equation we obtain the coefficientsand the t-3/2asymptotics for the time correlation functions.
The study of autocorrelation functions plays an important role in nonequilibrium statistical mechanics. On the basis of the functional hypothesis of Bogolubov 1) and cluster expansions in the B B G K Y hierarchy 2'3) a number of important results concerning the description of the approach of a system to equilibrium was obtained recently. Some peculiarities in the higher orders of hydrodynamic equations were pointed out. It occurred that the nonexponential decay of the autocorrelation functions discovered in the computer experiments on hard-sphere systems could be explained taking into account the nonlinear effects in the usual Boltzmann equation4). This fact emphasizes the importance of studying the region of applicability of usual kinetic equations for the reduced one-body distribution function. It is interesting to consider from this point of view the Boltzmann-Enskog equation which possesses, as was first proved by BogolubovS), exact microscopic solutions corresponding to the exact dynamics of the hard-sphere system and therefore may serve as a source for obtaining higher approximations in the density for the macroscopic properties of the system. This fundamental result and the method of ref. 5 can be extended to the generalized Boltzmann-Enskog equation where in addition to the hard-sphere interaction one has also a long-range component of the potential which is appropriate for the description of processes in realistic systems. In ref. 6 the corrections to the sound velocity and the eigenfunctions of the linearized Boltzmann-Enskog operator which takes into account the long-range interactions were determined. 0378-4371/82/0000-0000/$02.75 ~) 1982 North-Holland
330
B.I. S A D O V N I K O V
et al.
In the present paper we propose to analyze the asymptotic representation for autocorrelation functions on the basis of the model introduced. The kinetic equation of the model can be written in the form
Of(t, r, v) +vOf(to;'V)-na2 Ot
~ (v'
[(v'-v)o.][f(t,r,v*)f(t,r+a~r,v'*
)
v)tr~O
- f(t, r, v)f(t, r - ao., v')] dv do" I)o(t, r') dr'
m
f(t, r, v),
(1)
where p(t, r ' ) = f f ( t , r',v) dv', o. is a unit vector, a denotes the hard sphere diameter, m is the mass of the sphere, v * = v + o . ( ( v ' - v ) o . ) , v'* = v ' - o . ( ( v ' - v ) o . ) , th(r) is the long-range c o m p o n e n t of the potential. The autocorrelation functions considered are of the form
C~(t) =
V-'~
lim
V~oc
\k
j~(vk(O)) ~ j~(v,(t))~. /
I
(2)
n - N/V-const
Here V is a macroscopic volume of the dynamical system of N identical hard spheres, Vk(t) is the velocity of the kth particle at the time t, and the brackets ( . . . ) denote an average over a grand canonical equilibrium ensemble; ~ =
%, x): j~(v) = mvxvy;
j ~ ( v ) =~.(rnv ' 2 -5/3-t)vx;
1 [3 - kBT'
(3)
where T is the temperature. For an arbitrary interaction in the system the correlation functions can be transformed as follows:
C~(t)
n 2 f dvoq~(vo)j~(vo) f dr dv¢(v)j~(v)~b(t, r, v),
(4)
where tO(t, r, v) is the relative deviation of the distribution function f(t, r, v) from its equilibrium value q~(v): q~(v) = \27r /
e-"mo'v2'
f(t, r, v) = ,~(v)(l + tO(t, r, v)). As a consequence of the transition from (2) to (4) we have to impose a restriction on the initial condition
tO(O, r, v) = [nq~(vo)l-tN (r - ro)~(v - Vo) + f(r)
(5)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
331
corresponding to a 8-form of deviation from the equilibrium velocity distribution of a single p a r t i c l e . / ( r ) is an arbitrary function of the coordinates, and v0 can have any value from the interval (-0% oo). The deviation from equilibrium in the coordinate space is described by the function N ( r - r0) which is normalized by
(6)
f d r N ( r - r0) = 1.
It should be noted that the mentioned arbitrariness is eliminated in the integration of Cs(t). So, in order to calculate the autocorrelation functions (4) it is necessary to find the solution of eq. (1) with the initial condition (5), quite similar to the case of the Boltzmann-Enskog equation7). In our case the function t0(t, r, v) obeys the equation ~tt + V~r - nzi(v)q,(t, r, v) ntl 2
[ ( v ' - v)o'l]'[to(t, r, v)to(t, r, v')l,v(v') dr' dtr
[ (v'-v)tr>~O
.~ n 1 a~ 0 [ dr'dv',p(v')qJ(t, r', v ' ) 6 ( l r - rt{) m ~ ( v ) Ov Or J ÷ n 1 O(~oto) a f dr'dv'~o(v')to(t, r ' , v ' ) 6 ( I r m,~(v) av Or
r'l),
(7)
where the o p e r a t o r s / i ( v ) and ~ act on the functions q,(t, r, v), ¢,(t, r, v') in the following way:
A(v)to(t,
r, v) = a 2
[
[ ( v ' - v)o'l[to(t, r, v*) + to(t, r + ao', v'*)
(o'-v)~r>~O
- to(t, r, v) - to(t, r - a~r, v')l,~(v') dr' dtr, ]'[to(t, r, v)to(t, r, v')] = to(t, r, v*)to(t, r + ao, v'*) - to(t, r, v)~O(t, r - ao', v').
(8) (9)
We should note that the effects of the finiteness of dimensions of the hard spheres have been considered earlier in refs. 6, 7, where we have demonstrated that the +_a~r shifts lead to corrections of higher order in the density of the system. Here when considering the influence of the long-range component in the pair interactions of the particles we should put
a~
I~1
/'
i.e. the dimension of the hard core of the colliding particles is much less than
332
B.I. SADOVNKIKO et al.
the distance o v e r w h i c h the potential ~b c h a n g e s a p p r e c i a b l y . T h a t r e a s o n w h y we shall use in eq. (7) the o p e r a t o r s A0(v) and ~'0, i.e.
Ao(v )to( t,
f
r, v) = a 2
[
is the
[(v' - v)~r][to(t, r, v*) + t0(t, r, v'*) - t0(t, r, v)
(v'-v)~r~>0
- to(t, r, v')]q~(v') d v ' d~r
(10)
w h i c h is the B o l t z m a n n o p e r a t o r ; and ~'0[to(t, r, v)to(t, r, v')] = to(t, r, v*)to(t, r, v ' * ) - tO(t, r, v)to(t, r, v').
(11)
It is natural to s t u d y the nonlinear eq. (7) b y m e a n s of p e r t u r b a t i o n t h e o r y , b e c a u s e the p e r t u r b a t i o n to(t, r, v) is b e l i e v e d to be small f o r large t, and in the vicinity of the point t = 0 the nonlinear part of the eq. (7) is also small as c o m p a r e d with the linear one as
0[to(0, r, v)to(0,
f
r, v')][(v'
v)~r]q~(v') dv' d~r = O.
( v ' - v )ct ~O
T h e last t e r m in the eq. (7) d e m a n d s a special consideration. W h e n t is small this t e r m c o n t a i n s the d e r i v a t i v e of 8(v - v0) but a c c o r d i n g to (4) the a u t o c o r relation f u n c t i o n c o n t a i n s the integration o v e r v0. Being integrated o v e r v0 the nonlinear t e r m O(q~to)/Ov v a n i s h e s , so the p e r t u r b a t i o n t h e o r y is valid near t=0. W e write the solution of eq. (7) as a series tO = to~0~+ 0,~ + . . ',
(12)
and we obtain b y virtue of (10), (11) and the a s s u m p t i o n linearized e q u a t i o n f o r to(0~: O0 ~°~
to~0)>>to,t the
OtoC°~ _ ~ .,,~o)_ n l O q ~
at + v--g-Zr - ,-~oq,
0 f dr' dv'~p(v')to~°)(t, r', v ' ) 6 ( J r - r' i) m q~ Ov Or
= o.
(13)
F o r the s e c o n d t e r m in the series (12) we o b v i o u s l y h a v e the n o n u n i f o r m equation: Oto"~
at + v -Oto - - ~"~r - ntis°to(l) = naZ
(
n 1 Oq~ 0
m ~p Ov Or
d r ' dv'q~(v')to~l)(t, r',
v')~(lr
- r' l)
[ ( v ' - v)o']'I'[to(°}(t, r, v)to(°)(t, r, v')]~o(v') dr' dcr
(v'-v)o,~O
+nl O(~to(°)) 0 f dr' dv'q~(v')toc°)(t, r', v')qb(lr - r'l). m ~ ov Or J
(14)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
333
Let us note that the initial conditions for the functions ~0~ and ~1) can be taken in the form 0t°)(0, r, r) = 4(0, r, v),
~btl)(0, r, v) = 0.
(15)
The uniform equation (13) does not depend explicitly on t so its solution can easily be found
@{°)(t, r, v) = e-%~)'@(0, r, v),
(16)
where va
n 10~ a f dr' dv'~(v')~(r', v'),/,(Ir- r'l). m q~ Ov Or
(17)
The dependence of the right-hand side of the nonuniform equation (14) on t can be determined by means of (16) and (17). The solution of eq. (14) has the form Ik°)(t, r, v) = i e%~)~r'-'Q(t" r, v) dr',
(18)
0
where
Q(t, r, v) = na 2
f
(v'-v)o~O
x [ ( v ' - v)o.] T0[e-%(~)'~(0, r, v) e-%~')'~(0, r, v')] dr' do" +n__l O( m ~ T v ~ e-%t°'0(0' r, v)) × 0___0rf dr' dv'q~(v') e-%~°')%(0, r', v')4~(Ir - r'[). For the autocorrelation functions we have
Cs(t) = C~l)(t) + C~2)(t),
(19)
f
cgl,.) = .2 f dvoq~(vo)js(vo) dr do js(v)~(v)e-%'~)~(O, r, v),
(19a)
C"(t) = n2 f dvoq~(Vo)js(vo) f dr dv jdv)~(v) i e %""'-"O(t', r, v) dt'. 0
(19b)
334
B.I. SADOVNIKOV et al.
As a result of the equality
f dvoq~(vo)j~(vo) = 0 eq. (19a) may be written as
C~,l)(t) = n
f
dr dv js(v)~(v)
e
w+I~t'N(r- r0)j~(v).
(20)
-
The function N ( r - r 0 ) is normalized by condition (6), and expression (20) contains the integration over r, so all the terms which contain the operator O/Or can be omitted. Then, we finally obtain
C~'(t) =
n(j~(v), °'~°(~"j~(v)),
where the scalar product is defined with the weight q~(v). This is the standard expression exponentially decreasing in time for the autocorrelation functions in the linearized Boltzmann modelT). When proceeding to the expression (19b) which describes the nonlinear effects of the model considered, we note that the structure of Q(t, r, v) (18) allows us to decompose C~:)(t) into the two parts
C~2)(t) = C~2)(t) + C~2~(t), C~2,)(t) =
(21)
na~ f dk f dvw(v)j~(v) f e(' ")~'o(~'dt' 0
× f dv'q~(v')To{exp[-t'(ik(v'-v)-n/~o(V)-n~,o(V')-ik(cr,(v)-rr,(v')).] \-~-(-~ N~ I 6(v - v') + nf, N~j~(v') + where fk, Nk are the Fourier components of the functions f ( r ) and N ( r respectively. The operator 7rAy) is defined by the expression
rrk(v)rl(v)
n 10¢f m q~ Ov
dv'q(v')~(k),
r0), (22)
where 4~(k) is the Fourier transform of the long-range component 4'. C~:(t) =
n f dvoq~(Vo)j~(Vo)f drdvq~(v)j~(v)
t
X f e"-""~o'°' dt'--~
0~(q~(v) e
%(vw(~vo)N (r - ro)6(V - Vo) + nf(r) ) )
0
o ×~r f dr'dv'¢(v')e
%(v'W(~N(r'-ro)6(v'-vo)+n[(r'))
•
(21b)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
335
If we use when integrating over r in (21b) the equality f f(r)g(r)dr = f f(k)~(-k)dk, where f and ~ are the Fourier transforms of f and g, we can write instead of (21b) t
=
f d, f
dv'q~(v') 0
1 a x --~-~v{C(v ) e x p [ - t (ik(v - v') - nAo(V) - nAo(v') - ik('trk(v) - 7r-k(v')))]}
[j~(v) .- 2B(v - v') + n.f,N~js(v') + nf'~Njjs(v))(-ik~(k)). × ~-~-~IN,
(23)
The asymptotic behaviour of C~(t), C~(t) when t - ~ is determined by the form of the singularities of the function C~2~)(p)=f~ePtC~2~)(t)dt in a finite region of the complex p-plane. A consistent calculation of C~2~)(p) gives
C~2)(p)=n ( j ~ ( v ) f dv'q~(v')~ dk . , (21r)3 p-nAo(v) [
Pk(v,v')
× fo p-+ ik(v - Ir~(v) + ~r-~(v')- v') - nAo(v) - nAo(v')
]),
(24a)
where
(A(v), B(v)) = f q~(v)A*(v)B(v) dr,
.. 2o, P~(v, v') = ~js(v') ~k otv
(
(25)
- v') + n[~NTj~(v') + n[Y, N d s ( v ) .
Pk(v, v') "n'_h(v'))-
x ~v ~ ( V ) p + ik(v - 1r~(v) - v' +
)) nAo(v) - nAo(v')
"
(24b)
As there is a nonzero value among the eigenvalues of the operator A0(v) it is obvious that the point p = 0 is the singularity at the extreme right of the functions (24a, b), and thus, the behaviour of the quantities (24a, b) near the point p = 0 determines the leading term in the asymptotics of the autocorrelation functions when t ~ . All the eigenfunctions of the operator ~0(v) which correspond to the zero eigenvalue are orthogonal to the currents j~(v). As a result the structures j s ( v ) / ( p - nAo(v)) in (24a, b) do not give the singularity in the point p = 0 and so it is enough to study an expression of the
336
B.I. SADOVNIKOV et al.
form
( j~(v)
f dr dk ~(v')To
-nA0(v)' 3
(2703
P~(v,v')
p + Sk(v)- S-~(v')/'
(26)
where
S~(v) = ik(v - Try(v)) - nAo(v).
(27)
The eigenfunctions of the operator S~(v) corresponding to the eigenvalues that tend to zero when I k l ~ 0 were found in ref. 6. The first approximation in Ikl gives
d/~](v)--6~',
j=3,4,5;
a =
~-~,
~=/3n~0,
,
where t)~ ~ is the orthonormal set of the eigenfunctions of the operator /i0 corresponding to the hydrodynamic solutions of the linearized Boltzmann equation with the wave vector kT). We note that the functions $~2) are not mutually orthogonal. Let us decompose the quantity Pk(v, v') into the eigenfunctions of S,(v), Sk(v') retaining only the term with the functions (28):
PE(V, V') = ~ P ~'J)(b~(v)(k~k(v').
(29)
Lj
In order to determine the leading singularity in the point p = 0 it is enough to consider the expressions
c~?(p)~ (i~(v)
dk
× (p + z~l+ z~k)-~), C(2)~,~_ n [ j~(v)
(30a) dk
-
1
where z~ ~ are the eigenvalues of ~k(v) corresponding to the eigenfunctions ~0k(v). It is necessary to emphasize that P~'~) does not contain the quantities f, as the one-body currents jdv) are orthogonal to all the functions ~0k(v) we use. The quantities zJ,° for small values of Ikl were calculated in ref. 6:
zk,,2).~÷ -
i
X/a-r-S-~_ Ikl+lkl2(~T+2~,)
VI3m
z¢'-~ [klZ~T, Zk4'5'~- Ikl2~,
3
"
(31)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
337
where ~x, ~n are the thermodiffusion and the shear viscosity coefficients in the usual Boltzmann model. We note that the structure of the singularity in p = 0 is determined by integration in (30a, b) in the domain where k is small. The leading term in (30a, b) which determines the asymptotics of the correlation functions corresponds to those terms of the sums over (i, j) where Z~i) + Z0-'~~ ]k] 2. As the numerator of the integrand in (30b) contains k, C~2?(p) C~(p) in the vicinity of the point p = 0 and therefore C~21)(t)-> C~(t) when t ~ oo. Hence for the study of the asymptotic behaviour of C~2)(t)it is sufficient to consider the expression
~-nao(v) .I
xf
dk
~ ' 7"O(~'~(V)~k(v')P~"°exp[-- t(z~)+ z0-'~)]),
(32)
where X'0 signifies the sum over the values of the indices (i, j) determined by the condition (z~) + z ~ ) ~ const x k 2. If we put in (32) z~) + z°_~= k2(X~)+ )t°_[) we shall find c~2)(t) =
t_3~2(i,(v)' f oak f dv',p(,,') •
J4=J
X X'. 'ro~i~(v)t~-)k(~')P (i'(~)k J)()~ +47r)~3/2 ~))-3/2~]. IJ
(33)
Thus, in the nonlinear generalized Boltzmann-Enskog model with the longrange component of the potential the autocorrelation functions also have the nonexponential asymptotics t -312.One can calculate the constants h~ ~, X°-[ in the framework of the theory of Enskog. The effects corresponding to the long-range component ~b lead to a modification of coefficients in (29), (33).
Acknowledgement The authors are much indebted to Academician N.N. Bogolubov for valuable suggestions and advice.
References 1) N.N. Bogolubov, Studies in Statistical Mechanics, I, J. de Boer and G.E. Uhlenbeck, eds.
(North-Holland, Amsterdam, 1962). 2) J.R. Dorfman and E.G.D. Cohen, Phys. Rev. A6 (1972) 776; Phys. Rev. AI2 (1975) 292.
338 3) 4) 5) 6)
B.I. SADOVNIKOV et al.
M.H. Ernst and J.R. Dorfman, Physica 61 (1972) 157. J.T. Ubbink and E.H. Hauge, Physica 70 (1973) 297. N.N. Bogolubov, Theor. Math. Fiz. 24 (1975) 242; JINR preprint E4-8789 (1975). S.N. Bochkov, N.G. Inozemtseva and B.I. Sadovnikov, Communication of the Joint Institute for Nuclear Research, P17-81-10 (1981). 7) B.I. Sadovnikov and N.G. lnozemtseva. Physica 94A (1978) 615.
T H E ASYMPTOTICS OF T H E CORRELATION FUNCTIONS IN T H E GENERALIZED MODEL OF BOLTZMANN-ENSKOG B.I. SADOVNIKOV, N.G. INOZEMTSEVA, S.N. BOCHKOV Department o[ Quantum Statistics and Field Theory, Physics Faculty, Moscow State University, Moscow B-234, USSR
Received 28 April 1981
The long-timebehaviour of the autocorrelation functions for an ensemble of particles with pair interactions including a hard core and a long-range tail is considered. On the basis of the nonlinear generalized Boltzmann-Enskogkinetic equation we obtain the coefficientsand the t-3/2asymptotics for the time correlation functions.
The study of autocorrelation functions plays an important role in nonequilibrium statistical mechanics. On the basis of the functional hypothesis of Bogolubov 1) and cluster expansions in the B B G K Y hierarchy 2'3) a number of important results concerning the description of the approach of a system to equilibrium was obtained recently. Some peculiarities in the higher orders of hydrodynamic equations were pointed out. It occurred that the nonexponential decay of the autocorrelation functions discovered in the computer experiments on hard-sphere systems could be explained taking into account the nonlinear effects in the usual Boltzmann equation4). This fact emphasizes the importance of studying the region of applicability of usual kinetic equations for the reduced one-body distribution function. It is interesting to consider from this point of view the Boltzmann-Enskog equation which possesses, as was first proved by BogolubovS), exact microscopic solutions corresponding to the exact dynamics of the hard-sphere system and therefore may serve as a source for obtaining higher approximations in the density for the macroscopic properties of the system. This fundamental result and the method of ref. 5 can be extended to the generalized Boltzmann-Enskog equation where in addition to the hard-sphere interaction one has also a long-range component of the potential which is appropriate for the description of processes in realistic systems. In ref. 6 the corrections to the sound velocity and the eigenfunctions of the linearized Boltzmann-Enskog operator which takes into account the long-range interactions were determined. 0378-4371/82/0000-0000/$02.75 ~) 1982 North-Holland
330
B.I. S A D O V N I K O V
et al.
In the present paper we propose to analyze the asymptotic representation for autocorrelation functions on the basis of the model introduced. The kinetic equation of the model can be written in the form
Of(t, r, v) +vOf(to;'V)-na2 Ot
~ (v'
[(v'-v)o.][f(t,r,v*)f(t,r+a~r,v'*
)
v)tr~O
- f(t, r, v)f(t, r - ao., v')] dv do" I)o(t, r') dr'
m
f(t, r, v),
(1)
where p(t, r ' ) = f f ( t , r',v) dv', o. is a unit vector, a denotes the hard sphere diameter, m is the mass of the sphere, v * = v + o . ( ( v ' - v ) o . ) , v'* = v ' - o . ( ( v ' - v ) o . ) , th(r) is the long-range c o m p o n e n t of the potential. The autocorrelation functions considered are of the form
C~(t) =
V-'~
lim
V~oc
\k
j~(vk(O)) ~ j~(v,(t))~. /
I
(2)
n - N/V-const
Here V is a macroscopic volume of the dynamical system of N identical hard spheres, Vk(t) is the velocity of the kth particle at the time t, and the brackets ( . . . ) denote an average over a grand canonical equilibrium ensemble; ~ =
%, x): j~(v) = mvxvy;
j ~ ( v ) =~.(rnv ' 2 -5/3-t)vx;
1 [3 - kBT'
(3)
where T is the temperature. For an arbitrary interaction in the system the correlation functions can be transformed as follows:
C~(t)
n 2 f dvoq~(vo)j~(vo) f dr dv¢(v)j~(v)~b(t, r, v),
(4)
where tO(t, r, v) is the relative deviation of the distribution function f(t, r, v) from its equilibrium value q~(v): q~(v) = \27r /
e-"mo'v2'
f(t, r, v) = ,~(v)(l + tO(t, r, v)). As a consequence of the transition from (2) to (4) we have to impose a restriction on the initial condition
tO(O, r, v) = [nq~(vo)l-tN (r - ro)~(v - Vo) + f(r)
(5)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
331
corresponding to a 8-form of deviation from the equilibrium velocity distribution of a single p a r t i c l e . / ( r ) is an arbitrary function of the coordinates, and v0 can have any value from the interval (-0% oo). The deviation from equilibrium in the coordinate space is described by the function N ( r - r0) which is normalized by
(6)
f d r N ( r - r0) = 1.
It should be noted that the mentioned arbitrariness is eliminated in the integration of Cs(t). So, in order to calculate the autocorrelation functions (4) it is necessary to find the solution of eq. (1) with the initial condition (5), quite similar to the case of the Boltzmann-Enskog equation7). In our case the function t0(t, r, v) obeys the equation ~tt + V~r - nzi(v)q,(t, r, v) ntl 2
[ ( v ' - v)o'l]'[to(t, r, v)to(t, r, v')l,v(v') dr' dtr
[ (v'-v)tr>~O
.~ n 1 a~ 0 [ dr'dv',p(v')qJ(t, r', v ' ) 6 ( l r - rt{) m ~ ( v ) Ov Or J ÷ n 1 O(~oto) a f dr'dv'~o(v')to(t, r ' , v ' ) 6 ( I r m,~(v) av Or
r'l),
(7)
where the o p e r a t o r s / i ( v ) and ~ act on the functions q,(t, r, v), ¢,(t, r, v') in the following way:
A(v)to(t,
r, v) = a 2
[
[ ( v ' - v)o'l[to(t, r, v*) + to(t, r + ao', v'*)
(o'-v)~r>~O
- to(t, r, v) - to(t, r - a~r, v')l,~(v') dr' dtr, ]'[to(t, r, v)to(t, r, v')] = to(t, r, v*)to(t, r + ao, v'*) - to(t, r, v)~O(t, r - ao', v').
(8) (9)
We should note that the effects of the finiteness of dimensions of the hard spheres have been considered earlier in refs. 6, 7, where we have demonstrated that the +_a~r shifts lead to corrections of higher order in the density of the system. Here when considering the influence of the long-range component in the pair interactions of the particles we should put
a~
I~1
/'
i.e. the dimension of the hard core of the colliding particles is much less than
332
B.I. SADOVNKIKO et al.
the distance o v e r w h i c h the potential ~b c h a n g e s a p p r e c i a b l y . T h a t r e a s o n w h y we shall use in eq. (7) the o p e r a t o r s A0(v) and ~'0, i.e.
Ao(v )to( t,
f
r, v) = a 2
[
is the
[(v' - v)~r][to(t, r, v*) + t0(t, r, v'*) - t0(t, r, v)
(v'-v)~r~>0
- to(t, r, v')]q~(v') d v ' d~r
(10)
w h i c h is the B o l t z m a n n o p e r a t o r ; and ~'0[to(t, r, v)to(t, r, v')] = to(t, r, v*)to(t, r, v ' * ) - tO(t, r, v)to(t, r, v').
(11)
It is natural to s t u d y the nonlinear eq. (7) b y m e a n s of p e r t u r b a t i o n t h e o r y , b e c a u s e the p e r t u r b a t i o n to(t, r, v) is b e l i e v e d to be small f o r large t, and in the vicinity of the point t = 0 the nonlinear part of the eq. (7) is also small as c o m p a r e d with the linear one as
0[to(0, r, v)to(0,
f
r, v')][(v'
v)~r]q~(v') dv' d~r = O.
( v ' - v )ct ~O
T h e last t e r m in the eq. (7) d e m a n d s a special consideration. W h e n t is small this t e r m c o n t a i n s the d e r i v a t i v e of 8(v - v0) but a c c o r d i n g to (4) the a u t o c o r relation f u n c t i o n c o n t a i n s the integration o v e r v0. Being integrated o v e r v0 the nonlinear t e r m O(q~to)/Ov v a n i s h e s , so the p e r t u r b a t i o n t h e o r y is valid near t=0. W e write the solution of eq. (7) as a series tO = to~0~+ 0,~ + . . ',
(12)
and we obtain b y virtue of (10), (11) and the a s s u m p t i o n linearized e q u a t i o n f o r to(0~: O0 ~°~
to~0)>>to,t the
OtoC°~ _ ~ .,,~o)_ n l O q ~
at + v--g-Zr - ,-~oq,
0 f dr' dv'~p(v')to~°)(t, r', v ' ) 6 ( J r - r' i) m q~ Ov Or
= o.
(13)
F o r the s e c o n d t e r m in the series (12) we o b v i o u s l y h a v e the n o n u n i f o r m equation: Oto"~
at + v -Oto - - ~"~r - ntis°to(l) = naZ
(
n 1 Oq~ 0
m ~p Ov Or
d r ' dv'q~(v')to~l)(t, r',
v')~(lr
- r' l)
[ ( v ' - v)o']'I'[to(°}(t, r, v)to(°)(t, r, v')]~o(v') dr' dcr
(v'-v)o,~O
+nl O(~to(°)) 0 f dr' dv'q~(v')toc°)(t, r', v')qb(lr - r'l). m ~ ov Or J
(14)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
333
Let us note that the initial conditions for the functions ~0~ and ~1) can be taken in the form 0t°)(0, r, r) = 4(0, r, v),
~btl)(0, r, v) = 0.
(15)
The uniform equation (13) does not depend explicitly on t so its solution can easily be found
@{°)(t, r, v) = e-%~)'@(0, r, v),
(16)
where va
n 10~ a f dr' dv'~(v')~(r', v'),/,(Ir- r'l). m q~ Ov Or
(17)
The dependence of the right-hand side of the nonuniform equation (14) on t can be determined by means of (16) and (17). The solution of eq. (14) has the form Ik°)(t, r, v) = i e%~)~r'-'Q(t" r, v) dr',
(18)
0
where
Q(t, r, v) = na 2
f
(v'-v)o~O
x [ ( v ' - v)o.] T0[e-%(~)'~(0, r, v) e-%~')'~(0, r, v')] dr' do" +n__l O( m ~ T v ~ e-%t°'0(0' r, v)) × 0___0rf dr' dv'q~(v') e-%~°')%(0, r', v')4~(Ir - r'[). For the autocorrelation functions we have
Cs(t) = C~l)(t) + C~2)(t),
(19)
f
cgl,.) = .2 f dvoq~(vo)js(vo) dr do js(v)~(v)e-%'~)~(O, r, v),
(19a)
C"(t) = n2 f dvoq~(Vo)js(vo) f dr dv jdv)~(v) i e %""'-"O(t', r, v) dt'. 0
(19b)
334
B.I. SADOVNIKOV et al.
As a result of the equality
f dvoq~(vo)j~(vo) = 0 eq. (19a) may be written as
C~,l)(t) = n
f
dr dv js(v)~(v)
e
w+I~t'N(r- r0)j~(v).
(20)
-
The function N ( r - r 0 ) is normalized by condition (6), and expression (20) contains the integration over r, so all the terms which contain the operator O/Or can be omitted. Then, we finally obtain
C~'(t) =
n(j~(v), °'~°(~"j~(v)),
where the scalar product is defined with the weight q~(v). This is the standard expression exponentially decreasing in time for the autocorrelation functions in the linearized Boltzmann modelT). When proceeding to the expression (19b) which describes the nonlinear effects of the model considered, we note that the structure of Q(t, r, v) (18) allows us to decompose C~:)(t) into the two parts
C~2)(t) = C~2)(t) + C~2~(t), C~2,)(t) =
(21)
na~ f dk f dvw(v)j~(v) f e(' ")~'o(~'dt' 0
× f dv'q~(v')To{exp[-t'(ik(v'-v)-n/~o(V)-n~,o(V')-ik(cr,(v)-rr,(v')).] \-~-(-~ N~ I 6(v - v') + nf, N~j~(v') + where fk, Nk are the Fourier components of the functions f ( r ) and N ( r respectively. The operator 7rAy) is defined by the expression
rrk(v)rl(v)
n 10¢f m q~ Ov
dv'q(v')~(k),
r0), (22)
where 4~(k) is the Fourier transform of the long-range component 4'. C~:(t) =
n f dvoq~(Vo)j~(Vo)f drdvq~(v)j~(v)
t
X f e"-""~o'°' dt'--~
0~(q~(v) e
%(vw(~vo)N (r - ro)6(V - Vo) + nf(r) ) )
0
o ×~r f dr'dv'¢(v')e
%(v'W(~N(r'-ro)6(v'-vo)+n[(r'))
•
(21b)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
335
If we use when integrating over r in (21b) the equality f f(r)g(r)dr = f f(k)~(-k)dk, where f and ~ are the Fourier transforms of f and g, we can write instead of (21b) t
=
f d, f
dv'q~(v') 0
1 a x --~-~v{C(v ) e x p [ - t (ik(v - v') - nAo(V) - nAo(v') - ik('trk(v) - 7r-k(v')))]}
[j~(v) .- 2B(v - v') + n.f,N~js(v') + nf'~Njjs(v))(-ik~(k)). × ~-~-~IN,
(23)
The asymptotic behaviour of C~(t), C~(t) when t - ~ is determined by the form of the singularities of the function C~2~)(p)=f~ePtC~2~)(t)dt in a finite region of the complex p-plane. A consistent calculation of C~2~)(p) gives
C~2)(p)=n ( j ~ ( v ) f dv'q~(v')~ dk . , (21r)3 p-nAo(v) [
Pk(v,v')
× fo p-+ ik(v - Ir~(v) + ~r-~(v')- v') - nAo(v) - nAo(v')
]),
(24a)
where
(A(v), B(v)) = f q~(v)A*(v)B(v) dr,
.. 2o, P~(v, v') = ~js(v') ~k otv
(
(25)
- v') + n[~NTj~(v') + n[Y, N d s ( v ) .
Pk(v, v') "n'_h(v'))-
x ~v ~ ( V ) p + ik(v - 1r~(v) - v' +
)) nAo(v) - nAo(v')
"
(24b)
As there is a nonzero value among the eigenvalues of the operator A0(v) it is obvious that the point p = 0 is the singularity at the extreme right of the functions (24a, b), and thus, the behaviour of the quantities (24a, b) near the point p = 0 determines the leading term in the asymptotics of the autocorrelation functions when t ~ . All the eigenfunctions of the operator ~0(v) which correspond to the zero eigenvalue are orthogonal to the currents j~(v). As a result the structures j s ( v ) / ( p - nAo(v)) in (24a, b) do not give the singularity in the point p = 0 and so it is enough to study an expression of the
336
B.I. SADOVNIKOV et al.
form
( j~(v)
f dr dk ~(v')To
-nA0(v)' 3
(2703
P~(v,v')
p + Sk(v)- S-~(v')/'
(26)
where
S~(v) = ik(v - Try(v)) - nAo(v).
(27)
The eigenfunctions of the operator S~(v) corresponding to the eigenvalues that tend to zero when I k l ~ 0 were found in ref. 6. The first approximation in Ikl gives
d/~](v)--6~',
j=3,4,5;
a =
~-~,
~=/3n~0,
,
where t)~ ~ is the orthonormal set of the eigenfunctions of the operator /i0 corresponding to the hydrodynamic solutions of the linearized Boltzmann equation with the wave vector kT). We note that the functions $~2) are not mutually orthogonal. Let us decompose the quantity Pk(v, v') into the eigenfunctions of S,(v), Sk(v') retaining only the term with the functions (28):
PE(V, V') = ~ P ~'J)(b~(v)(k~k(v').
(29)
Lj
In order to determine the leading singularity in the point p = 0 it is enough to consider the expressions
c~?(p)~ (i~(v)
dk
× (p + z~l+ z~k)-~), C(2)~,~_ n [ j~(v)
(30a) dk
-
1
where z~ ~ are the eigenvalues of ~k(v) corresponding to the eigenfunctions ~0k(v). It is necessary to emphasize that P~'~) does not contain the quantities f, as the one-body currents jdv) are orthogonal to all the functions ~0k(v) we use. The quantities zJ,° for small values of Ikl were calculated in ref. 6:
zk,,2).~÷ -
i
X/a-r-S-~_ Ikl+lkl2(~T+2~,)
VI3m
z¢'-~ [klZ~T, Zk4'5'~- Ikl2~,
3
"
(31)
THE ASYMPTOTICS OF THE CORRELATION FUNCTIONS
337
where ~x, ~n are the thermodiffusion and the shear viscosity coefficients in the usual Boltzmann model. We note that the structure of the singularity in p = 0 is determined by integration in (30a, b) in the domain where k is small. The leading term in (30a, b) which determines the asymptotics of the correlation functions corresponds to those terms of the sums over (i, j) where Z~i) + Z0-'~~ ]k] 2. As the numerator of the integrand in (30b) contains k, C~2?(p) C~(p) in the vicinity of the point p = 0 and therefore C~21)(t)-> C~(t) when t ~ oo. Hence for the study of the asymptotic behaviour of C~2)(t)it is sufficient to consider the expression
~-nao(v) .I
xf
dk
~ ' 7"O(~'~(V)~k(v')P~"°exp[-- t(z~)+ z0-'~)]),
(32)
where X'0 signifies the sum over the values of the indices (i, j) determined by the condition (z~) + z ~ ) ~ const x k 2. If we put in (32) z~) + z°_~= k2(X~)+ )t°_[) we shall find c~2)(t) =
t_3~2(i,(v)' f oak f dv',p(,,') •
J4=J
X X'. 'ro~i~(v)t~-)k(~')P (i'(~)k J)()~ +47r)~3/2 ~))-3/2~]. IJ
(33)
Thus, in the nonlinear generalized Boltzmann-Enskog model with the longrange component of the potential the autocorrelation functions also have the nonexponential asymptotics t -312.One can calculate the constants h~ ~, X°-[ in the framework of the theory of Enskog. The effects corresponding to the long-range component ~b lead to a modification of coefficients in (29), (33).
Acknowledgement The authors are much indebted to Academician N.N. Bogolubov for valuable suggestions and advice.
References 1) N.N. Bogolubov, Studies in Statistical Mechanics, I, J. de Boer and G.E. Uhlenbeck, eds.
(North-Holland, Amsterdam, 1962). 2) J.R. Dorfman and E.G.D. Cohen, Phys. Rev. A6 (1972) 776; Phys. Rev. AI2 (1975) 292.
338 3) 4) 5) 6)
B.I. SADOVNIKOV et al.
M.H. Ernst and J.R. Dorfman, Physica 61 (1972) 157. J.T. Ubbink and E.H. Hauge, Physica 70 (1973) 297. N.N. Bogolubov, Theor. Math. Fiz. 24 (1975) 242; JINR preprint E4-8789 (1975). S.N. Bochkov, N.G. Inozemtseva and B.I. Sadovnikov, Communication of the Joint Institute for Nuclear Research, P17-81-10 (1981). 7) B.I. Sadovnikov and N.G. lnozemtseva. Physica 94A (1978) 615.