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On the retarded solution of the Liouville equation and the definition of entropy in kinetic theory
Physica 132A (1985) 74-93 North-Holland, Amsterdam
ON THE RETARDED SOLUTION OF THE LIOUVILLE EQUATION AND THE DEFINITION OF ENTROPY IN KINETIC THEORY R. DER Zentralinstitur fiir Zsotopen- und Strahlenforschung, DDR-7050 Leipzig, Permoserstr. 15, GDR
Received 25 January 1985
Zubarevs approach to non-equilibrium statistical mechanics, which consists in adding a source term to the Liouville equation so as to select its retarded solution, is reinvestigated. As discussed earlier, the source has to be modified by introducing a different set of thermodynamic parameters in order to achieve agreement with current theories of non-equilibrium statistical mechanics. By considering kinetic theory of a homogeneous classical gas it is demonstrated that the modification of the source is necessary in order to avoid unphysical divergencies and to maintain conservation laws. Moreover, a new definition of the non-equilibrium entropy in terms of the relevant observables is obtained which reveals several attractive features.
1. Introduction Among the many approaches dealing with the problem of deriving the evolution equations of macrophysics from the basic Liouville equation (LE), the theory of Zubarev’-3) takes a more or less singular position. In fact, while most of the well known approaches, the projection operator methods, for example, start from a conveniently formulated initial value problem of the LE, Zubarev formulates a boundary value problem by introducing an infinitesimal small source term into the LE. This one is intended at breaking just the time symmetry of the LE while leaving the macroscopic behaviour invariant. Thus, a non-equilibrium statistical operator (NESO) is constructed which is the retarded solution of the LE and is expected to correctly describe the macroscopic relaxation phenomena taking place in the system. Noteworthy among the many attractive features of Zubarev’s approach are its easy applicability to concrete physical situations and its close intuitive connection to Bogoljobov’s method of quasi-averaging4) which has proved very helpful in many branches of physics. The value of Zubarev’s approach has been demonstrated by a number of interesting applications (see refs. 3, 5, 6 and 0378-4371/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
7.5
others). Moreover, its connection with current theories of non-equilibrium statistical mechanics has been broadly discussed3*7). The relation between macroscopic evolution equations (MEE) as obtained from the initial value problem of the LE (see eq. (2.2) below) and the MEE obtained from the NESO was rediscussed recentlysV9). It was found that with the definition of the source term as given by Zubarev, the latter do agree with the former under very specific conditions only. However, it could be shown that this difficulty is removed by a self-consistent redefinition of the source term. The present paper aims at a further clarification of this point and at elaborating some consequences of the redefinition of the source term. For this purpose, we state in section 2 in a simpler language the arguments given in refs. 8 and 9 concerning Zubarev’s definition of the source term. In section 3 we consider a simple example, the derivation of kinetic equations for a classical homogeneous gas, in order to explicitly demonstrate these more general considerations. In the event, we also hope to demonstrate the usefulness of Zubarev’s approach, if the source term is appropriately redefined. Then, in section 4 we consider the more general problem of the definition of the non-equilibrium entropy. The new source term is argued to provide us with a modified information entropy which reveals several attractive features. This is demonstrated for the case of kinetic theory considered by showing that our entropy reduces to the full low density equilibrium entropy for t + m. A short summary and discussion is found in section 5.
2. Outline of Zubarev’s approach. Redefinition of the source term As in many theories of non-equilibrium statistical mechanics”), in Zubarev’s approach a central role is played by a generalized Gibbs state p4, called the quasi-equilibrium ensemble by Zubarev. This one is formulated in terms of the set P, {P: P,, . . . , P,,}, of relevant observables as p,(t)
= Q’,’
e-‘~=lpmFm(‘) = : p,{f7(t)},
(2.la)
where (2.lb) and (A) = 1 dl-A
or
(A)=
TrA
76
R. DER
in classical
or quantum
dimensional parameters,
phase space. The F,(t) denote conjugate thermodynamic which we will leave unspecified for the time being.
mechanics,
In terms of pq, the initial
problem
r
denoting
a point
(2.2a)
formulated
as’“)
P(O)= P,(O)
(2.2b)
7
so that the quantities are given by
of physical
interest,
i.e. the expectation
pm 0) = UJ, e-iL’p,(0)) = (Pm(t)p,(O)). The Liouvillean the commutator operator
in 6N-
of the LE
= 0
(a, + iL)p(t) is readily
value
respectively,
values
of the P,,,,
(2.2c)
is to be understood as usual in terms of the Poisson bracket of the Hamiltonian H with any phase space function
or or
A, respectively,
LA=i{H,A}
On the other source term
or
hand,
LA=i[H,A].
the NESO
is obtained
(2.3)
from
64 + iL)p,(t) = - 4~~0) -p,(f)> t
the solution
of the LE with
(2.4)
where we take E -+ 0 after taking the thermodynamic limit. Thus, if the NESO p, exists, it may be argued to exactly obey the LE in the limit E + 0. The solution
of (2.4) is given
as’).
p,(t) =
1 dt’ e~(E+iL)“p,(t -
t’)
F
(2Sa)
0
or by using partial
integration’):
p,(t) = p,(t) - i dt’ e-(E+iL)f’(iL - a,,)~Jt - t’) .
(2Sb)
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
II
Thus, p,(t) is given as a functional over F(t”), t”S f. We prefer to write p, here as
(2.6) so that by using the well known operator identity (A +
I?)-’ = A-l - A-%(,4
+ I?)-’
(2.7)
eq. (2.6) is easily seen to be the formal solution of eq. (2.4). For the following, it should be noted that for sufficiently large E and an arbitrary time t, p,(t) should exist for any choice of the parameters F,(t), m = 1,. . . , n, provided only p,(t) is an analytic function of t at the time considered. In Zubarev’s approach, the F,(t) are determined by the requirement that the expectation values of the Pk, k = 1,. . . , n, as obtained from both p, and pq agree exactly in the limit E + 0, i.e.
&(t) = ~~(P,P,(~)= (P,p,(O), k = 1, . . . , II.
(2.8)
Thus, assuming the set of equations $ =
(pk&{y)), k = 1,. . . , n,
could be solved for y so that one obtains y as a function of x: Y,=F,[x]=F,[x
,,...,
x,],
m=l,...,
n;
then we may obtain’) from (2.8) F,(t)=F,[P(t)],
m=l,...,n
(2.9)
and
In other words, eq. (2.8) may be used for the definition of the thermodynamic parameters F(t) iff there exists a set of functions p,(t), 1 = 1, . . . , n, which are the solutions of
78
R. DER
F,(t)
=
The point
lim P, E-t0 (
i;
+
of view taken
exist, in general. say, where
F +
Instead,
a,
P,ml)
.
(2.11)
in refs. 8 and 9 now is that such a solution we have to introduce
a second
does not
set of functions
p’(t),
we have
ml = l&y@&p'(r)l)
(2.12)
7
so that
P,(t)= P,[P’(t)] Then,
)
F;(t) = Ppyt)] .
(2.13)
introducing
P;w = P,PWl = &p’(~)l the modified
LE with source
(2.14)
reads
(a, + iL)p, 0) = - ~P,W - P:@)).
(2.15)
With this redefined source term, the MEE obtained from the new NESO can be shownBV9) to exactly agree for sufficiently late times with the corresponding MEE governing the time evolution of the expectation values obtained from the the information entropy defined in initial value problem (2.2). Moreover, terms of p: seems to be a better approximation to the non-equilibrium entropy than the one obtained from pq (see below). It should be mentioned that the difference between p(t) from the transient processes connected with the transition
and p’(t) results from the non-
dissipative state p,(t) or p:(t) to the macrostate p,(t) where dissipativity is fully established. From this follows already intuitively the intimate relation stressed in refs. 8, 9 and 11 between the initial slip occurring in the solutions (2.2) and the definition of p’. Moreover, the difference P - p’ can also be related to retardation effects occurring in the projection operator methods, for example. Quite generally, it may be stated that
F(t) - F(t) = O(lJ) where
t, is a time
)
5 = ;,
characterizing
the transient
processes
and
tR
is
a
typical
79
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
relaxation time. Thus, if f = 0, we reobtain Zubarev’s definition of the source term. Specific examples are given by the ideal gas but also by kinetic equations of a sufficiently dilute classical hard sphere gas where the Boltzmann or Enskog equation may be expected to be correct for all times t > 0 in (2.2). In the following section we shall demonstrate those general propositions by considering the special case of kinetic equations for a dilute classical gas with realistic interactions.
3. Kinetic theory Let us consider a homogeneous
system of N particles with Hamiltonian
H=Ho+H1=-p$V(i,j)
(3.la)
#
and Liouvillean L = 5 L,(i) + 5
i=l
(3.lb)
L,(i, j),
i
where V(i, j) = V(Xi, Xi) = V(lri - 51)s xi = {I;:,Pi) and
L(k)=
-ibpkc,L,(k,
I)= iF(-&-&).
k
The N-particle
distribution
function p,(t) be normalized as
Introducing
we find for the single particle momentum
cp(IA0 = fi(4 (PlP,(0) * Thus, the set P of relevant observables
distribution function the expression (3.2) is to be replaced by the phase space
80
R. DER
functions G(p), i.e. we have now to deal with a continuous set P.Consequently, the ansatz for pg reads p,{F(t)}
= p,(t)
= 0,’
e-Jd3PG(p)F(P.‘),
(3.3a)
so that from
dP7 t) = (6 (P)P,W> we obtain
WP, t) = -N In CP(P, 2). Therefore,
(3.3b)
we find explicitly (3.3c)
where n is the density, n = N/V. Thus, instead of eq. (2.10) we obtain the explicit expression (3.3~). Hence, we may write
(3.4) where cp’(p, t) is a function unknown to us so far. Further below, cp’ will be determined from the requirement that the average (e(p)p,) behaves well for F + 0 and all p. From eqs. (2.15) and (3.4) we obtain (3Sa) and cP(P, r, = (G(P)ir,
ir (t (P’(Pi, I))) i=I
.
(3.5b)
Writing
ri,=l-
F+i;+a,(‘L+a,)
(3.6)
LIOUVILLE
EQUATION
AND DEFINITION
OF ENTROPY
81
and 1
= i
(%U --
n=O n.1
e+iL+a,
1
(3.7)
E + iL
we obtain*
(P’(P)-P(P)=
mWV c-&w n=O
. -&P; >I. >+a,($(P)&P;
(3.8)
The strategy for evaluating the above expression is the following. We want to restrict ourselves to the dilute gas case so that we may assume the momentum relaxation is caused by uncorrelated sequences of binary collisions only. We use the exact relations iL iL -=-__-_
1 (iL)*
Efil?.
E
&E+iL
1 -= EfiL
1
(3.9a)
and
E
iL I 1 (iL)* (3.9b)
fz2e+iL
E
and find by standard methods (diagram or cluster expansions, for example)
x VI&) + 12,?(~))~‘(l)cp’(2~(p’(3) + ... where I,JE)
I&).
represents the integrodifferential
I
. .= ni2 dx,L,(k 0
E+
)
(3.10)
operator
i:(k L,(k, 0. , . II
(3.11)
7
and 1 may be considered as a dummy variable. The right-hand side of eq. (3.10) represents just the contribution of a single binary collision and that of an uncorrelated sequence of two such collisions. Higher order terms corresponding to sequences of k, k 3 3, binary collisions which diverge as ?-‘) for E +- 0 are not written explicitly. These terms will be discussed further below. * We supress the time argument r in rp(p, t) and
9’(p,
t)
for a while.
82
R. DER
For any reasonable, not too long ranged potential we may expand Z(e) for small E into a Taylor series: Z&)
(3.12)
= Z,J + SZ;! + * * * 7
where
z = !‘z(E),
I’ = lii
tJ,Z(&).
Thus”), we find Z is just the Boltzmann collision operator Z,&(Z)
so that (3.13a)
= 0(1/Q 9
where tf is the time of mean free flight. Moreover, particle scattering theory12S13)that
we conclude
from two-
(3.13b)
where t, characterizes the duration of a binary collision. Let us now return to eq. (3.8). Using (3.9), (3.10) and (3.12) we find the averages occurring in eq. (3.8) are given as a Laurent series in powers of E. Thus, if the limes E + 0 is to exist, the contributions corresponding to negative powers of E have to cancel each other. Considering for the time being terms of order E’ and c-l only and dropping all terms of order 5’ and higher, we find by means of (3.10) through (3.13)
+
u&.3+
Zl&,,
+ E-*Zl,2Z12,?)~‘(1)~‘(2)~‘(3) )
(3.14)
where I,, i = Z,,, + Z,, and p(i) = P(z+). Now, ie use (3.9) and (3.10) and
(@(P)b;) = 0
(3.15)
to conclude that the terms corresponding to II = 1,2, . . . of eq. (3.8) are at least of order E-* or t*. Thus, we obtain by introducing (3.14) (3.15) and (3.10) into (3.8):
LIOUVILLE
EQUATION
AND DEFINITION
OF ENTROPY
83
cd(l)CPU)=-G,f'P'mJ'c3+ ~-'@,b'(l>+ ~;,$9m'(a - 1,,2dwdw
v;.~L,~+ 1,,~1~~,~)(~‘(1)(~‘(2)40’(3)}.
(3.16)
Now, the &‘-term immediately yields the relation between rp’ and cp, i.e.
$4) = d(l) + ~;,*dw(P’(2)+ fx2)
(3.17a)
and hence
d(l) = dl)-
h4M2)+
fw2)
7
(3.17b)
where we remember p(i) = p(pi, t), for example. From the Cl-term we obtain
d’(l) = -v;,,cp’u)4(2) + ~~,,rp’W’(2) + (1;.242,3+ 11,21)12.t)~‘1(l)~‘(2)(0’(3) p where d’(l) = Q’(p,, d’(l)
=
t). Using (3.13b) we obtain from (3.18) immediately
4,24m4(2)
and by reintroducing
(3.18)
+
%9
(3.19)
this into (3.18) we find
4’ = 4,,rp’W’(2)+ ItzI;2,1(p’(l)lp’(2)~‘(3) + Q(5’).
(3.20)
This is the kinetic equation governing the time evolution of 9’. The corresponding equation for (p follows from (3.20) by first noting that by means of (3.17b) this can be rewritten as
d’(l) = b,*dl)&)+
W’).
(3.21)
Then, by taking the time derivative of (3.17a) we obtain immediately
d(l) = 1,,&d2)+
I;.2112,~~(l)~(2)rp(3)+ W’).
(3.22)
The kinetic equation obtained in this way is seen to be just the Boltzmann equation plus a correction term of the order of 5 = tc/ff which has been obtained earlier by several authors. In particular, Klimontovichi3) has stressed the point that the correction term is responsible for the conservation of the
R. DER
84
total
energy.
This
is clearly
born
out in the present
approach,
too.
In fact,
writing r;,,l,,,cptI)cp(2)(p(3)
= a&(~(l)~(2)
+ ~(52)
and using
I d3p,H,(P,)~
1,
Z(P(lb
(2)= 0
(3.23)
3
= p2/2m we find from
where
H,(p)
kinetic
energy
tn t)
K(t) = N j- d3p Ho( of the N-particle
system
(3.22) for the time derivative
K(t)
of the
(3.24)
the expression
(3.25)
As shown
in the appendix,
we may write this as
0 = K(t) + V(t) = a, lim(Hp,(t))
(3.26)
,
E’O
where
is just the potential
energy
of the system
in the kinetic
state given by p,(t).
The question of the conservation of total energy may also be considered directly in the framework of eqs. (2.4) and (2.15). Averaging eq. (2.15) with the Hamiltonian H yields obviously
I$ := a,(H&) = _&(H(& -pi)). Using
(3.28)
eq. (3.6) we obtain
(
E~,=E H
E + ii +
W + %)P;) =---&W-J~P.$~
a
I
I
8.5
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
so that
(E+ Gm
(3.29)
= ~%woP;) 7
where we used
a,Uf,p;) = 0 since 9’ is normalized. From the appendix together order 5’ and higher,
with eq. (3.17b) we find, neglecting terms of
where (3.30)
K+ V=E=lim(Hp,). t-+0
Introducing this into eq. (3.29) yields ii, = 0 +
O(E2)
so that SE = const. Using the fact, that eq. (2.15) is homogeneous find finally
in time we
JGc =0 what is valid for small values of E in the approximation considered, sufficiently small values of 5. However, if applying the same procedure to eq. (2.4) we find
E&) = -&V(t))
(3.31) i.e. for
(3.32)
so that for E # 0 in Zubarev’s original approach the total energy is not conserved while it obviously is with the redefined source term as proposed in the present paper. In concluding this section, we briefly summarize the main points of the derivations given above. We started from the general expression (3.5) or (3.8) where up’denotes an unknown function of p and r. We assume the momentum relaxation to result mainly from uncorrelated sequences of binary collisions of finite duration t,, so that the averages occurring in eq. (3.8) are obtained as a
R. DER
86
Laurent series in positive and negative powers of E. Consequently, the righthand side of eq. (3.8) may be rearranged into a Laurent series, too, see eq. (3.16). Then, the term in co already yields the relation (3.17) between cp and cp’. Moreover, if p of eqs. (3.5) and (3.8) is to exist, the coefficients of all powers c-l, 1 = 1,2, . . . ) have to vanish. This yields in order E-I the kinetic equations (3.20) for (D’and (3.22) for cp. By a straightforward but tedious algebra it may be shown that the relations between (p and p’ obtained also guarantee the cancellation of all terms of order &-I, 1 = 2,3, . . . ) so that we may conclude (4~~) exists with the above choice of p’ for c-,0. The inverse is also true: if we use a function p’ in (3.5) which is not a solution of the kinetic equation (3.20) ’ then (4~~) does not exist for E + 0. In particular, we obtain unphysical divergencies if we use Zubarev’s prescription (2.8) since this amounts just to putting cp’= cp what obviously is not consistent in different powers of E-‘. Therefore we are led to the conclusion that the redefinition of the source term is not only necessary in order to get the correct retardation terms but is also of principal importance to avoid unphysical divergencies. A further consequence of the replacement of p4 by p: is investigated in the following section.
4. On the definition of entropy in a non-equilibrium
state
As proposed by Zubarev, we might write for the entropy S(r) of the system at a given instant of time t
s&d = -$$P,(~) In ~~0))= - (~~0)ln P&N . In the case of the kinetic theory considered,
(4.1)
we find from (3.3) and (4.1)
S,(t) = A%(t) - N(ln it + 1) ,
(4.2)
where
(4.3) i.e. we obtain just Boltzmann’s
kinetic theory definition of the entropy.
‘Note that eq. (3.20) differs from eq. (3.22) by terms of order 5.
This
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
87
one, however, is well known to suffer from the drawback that its stationary value is just given by the entropy of the ideal gas, i.e. s, = N ln (2?rmkT)3’2 es/* . n This is true not only if the time evolution of (p(p, t) is governed by the Boltzmann equation but also when the non-ideality correction is included as in eq. (3.22). The ansatz (4.1) may be justified by using information theoretical arguments’). On the other hand, we can relate it to the full entropy of the system
se= -(P, InPA
(4.4)
by noting that
S, = lim SE E-m since
In the light of the discussion given in ref. 9 we might argue that pg follows from p, under the condition of a maximum damping of the non-macroscopic information. Applying this principle to the solution
of eq. (2.15) yields a different expression for the non-equilibrium
entropy, i.e. (4.5)
In particular, if using (3.4) we reobtain eq. (4.2) where h is to be replaced with h’:
h’(t) = - 1 d3p p’(p, 2) In P’(P, t) .
(4.6)
R. DER
88
Using eq. (3.17b), h’ can be written as a functional difference cp - cp’ to be small we may write it as
of cp. Assuming
the
h’(c) = h'[cp(r)l = h[cp(t)l+ j- d3p,U’,,,CP(P~~ t)d~2, t>>ln v(p17t>+ fTt2), (4.7) where
and
cp-1(l)~~,2cp(l)cp(2)+ Ok+)
In p’(l) = In (Iwas used. Moreover, functional
=
we find by means
of eq. (3.21) for the entropy
production
S:(t) the
S,(t)+ j- dp, (P-‘(P,,WI,&‘I~ t)cp(p2,~))~‘,,,cp(Pl~ t)cp(p,,t) + m3. (4.8)
From the properties clear that
~;be,l = 0 where
of the Boltzmann
collision
operator
I it is immediately
(4.9)
9
qe4 is the Maxwellian
cp,,(p) = (25~rkT))~‘~ For the stationary
S:, = S&L,I
value
e-p2’2mkT. of SA we find
= Se, - (2mkV’
so that from the appendix
(4.10)
j- d3p, P:~;,~(P,,(P,~,(P~)
we conclude
>
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
89
m
SA = S, + Nn27r
I
dr r2(e-V(r)‘kT- l)V(r)/kT,
(4.11)
0
where V(r) is the two particle interaction potential. Therefore, Sk is given by the entropy of the ideal gas plus the contribution of the correlation entropy in leading order of the density which is well known from equilibrium statistical mechanics. From this result, we may conclude that the ansatz (4.5) for the entropy of the kinetic state seems really to be favoured over the original ansatz (4.1) proposed by Zubarev if one aims at a theory which is consistent up to order 5 inclusively. Unfortunately, a proof that the entropy production Si is positive is still lacking but this drawback is shared by S, if we use eq. (3.22) with the non-ideality correction of order 5 included. The ansatz (4.5) for the entropy of a non-equilibrium state may intuitively be understood in the following sense. From information theory, pi(t) = p,{F’(t)} is interpreted as the ensemble of maximum indeterminacy with given values of the thermodynamic parameters F;(t), or the corresponding average values F;(t), k = 1, . . . , n. The state p,(t) with corresponding average values p(t) is obtained from pi(t) by
- oE being independent of pi and hence of P, P’. Thus, anything we have to know are the values p’(t) so that p:(t) may be regarded as representing the macroscopic knowledge necessary in order to obtain just the state p,(t) belonging to the average values Pk(f), k = 1, . . . , n. Therefore, it seems quite natural to choose the expression (4.5) as the corresponding information entropy which hopefully represents a good approximation to the entropy of the non-equilibrium state considered.
5. Conclusions
The results obtained above for the special case of kinetic theory are hoped to demonstrate sufficiently clearly the following aspects concerning Zubarev’s theory of the NESO. On the one hand, it has to be emphasized that the source term of eq. (2.4) has to be modified in the sense that the thermodynamic parameters Fk(t), k = 1,. . . , n are to be replaced by a set F;(t) with corresponding expectation values P;(r), 1= 1, . . . , n. The necessity for doing so follows clearly
90
R. DER
from the example investigated. In fact, p,(t) if considered as a functional of the thermodynamic parameters is well behaved for E + 0 only if these are just the functions F;(t) = Fk[p’(t)], the set I”(t) being related in a unique way with p(t), see eqs. (3.17) in kinetic theory, for example. The modification of the source term has also been shown to guarantee the conservation of total energy at finite values of E, see eqs. (3.31) and (3.32). On the other hand, the modified NESO approach is demonstrated to yielding in a rather direct way non-trivial physical results. Moreover, it leads to an explicit expression for the entropy Si of the non-equilibrium state in terms of the expectation values of the relevant observables, S: attaining its stationary value at the correct equilibrium entropy. These facts clearly support the point of view that Zubarev’s theory which is deeply rooted in basic physical principles is an interesting and fruitful approach to the general problem of describing relaxation phenomena in systems with reversible microscopic dynamics. As compared to other theories of non-equilibrium statistical mechanics, it must be noted that macroscopic evolution equations (MEE) as obtained from the NESO are intrinsically local in timess9). This is also shown by the derivations given in section 3, i.e. by the way in which the time derivative is eliminated in the course of the cancellation of the secularly divergent terms. The same procedure also applies to the higher order (n 2 1) terms in eq. (3.8) so that ultimately one always obtains MEE which are of first order in the time derivative of p(t). Nevertheless, all of the retardation effects are fully contained in the theory as exemplified by the correction term in eq. (3.22). Quite generally, in using the methods of refs. 8, 9 and 11, it can be shown that the MEE obtained from eq. (2.15) agree exactly with those obtained by Robertsoni4) or Grabert”), for example, in their memory renormaiized form given recently”). This also marks the limitation of the applicability of both equations (2.4) and (2.15). In fact, it can be shown by the techniques developed in refs. 9 and 11 that p, is well behaved for small E only, if the transients occurring during the transition from pi to p, decay in time more rapidly than any inverse power of t. In other words, the NESO can exist only if corresponding retarded theories are memory renormalizable.
Appendix
i) The potential energy : The potential energy of the system in the state p,(t) (3.5)-(3.7) given as
is by use of eqs.
91
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
- cv,) V(t)= v[cp (01= (W$ - C n!
(f4-&
n=O
Let us introduce the integrodifferential
J&E)
= i G 1 dl d2 V(l, 2)
OL+
a,)~$ .
64.1)
operator J:
1 E + iL(l, 2)
64.2)
L,(lT 2)
where J dl . . . = J dx, . . . , for example. Neglecting contributions from sequences of more than two binary collisions we may expand 1 iLpA = J&)(P’(l)P’(2) H*--> E + iL
(
= J,,cP’(~)~P’(~)+ (J&,~
+ ~-lJ~,~(~)I~~,~(&)(~r(l)(~‘(2)(0’(3)
+ J,,,(LI,,+
~-1~,~.~))~‘(1)~‘(2)~‘(3).
(A-3)
Using 1 _=---E+iL
1
1
iL
E
Ec+iL
and I
dpcp’(p, t) = 1 9
so that a,(H,p;) = 0 , we also find
(
a, K-
1 E + iL
P:, = - (-G,7L12,3+E-~J!,~I~~,~)(P’(~)~‘(~)(P’(~). >
In the light of the arguments given in section 3 we therefore (A.3) and (A.4)
V(t) =
$1dl d2 VU, WU)d@)
+ J~,240’(l)d(2),
where eq. (3.17b) was used finally. Using L,(i)cp(i) = 0 we may rewrite V(t) as
(A.4)
obtain from (A.l),
(A-5)
92
R. DER
dl d2 V(1,2) e-‘““~‘)cp(l)cp(2).
= lim g 1-z 2
In
equilibrium,
we
obtain
from
this
by
(A.61
means
of two-particle
scattering
theory’2*‘3) and eq. (4.10)
V[peq] = 2rNn
drr2V(r)e~V”“kr.
(4.7)
I 0
ii) Derivation of eq. (3.26): The integral term occurring in eq. (3.25) has been broadly discussed by Klimontovichr3). Following his arguments, we rewrite this term in the following way:
= - g
ljn~;
j dl d2 V(l, 2)L(l,
2)
1 F + iL(l, 2)
L(l, 2)(~ (1)~ (2)
WV
We used the homogeneity of the system so that we may replace L, with L in eq. (3.11) everywhere. Moreover, we exploited the symmetry of the expression with respect used
I
to x,, x2 which allows to replace
dl d2 H,(l, 2)L(l, 2). . . = -
H,(l)
by iH,(l,
dl d2 V(l, 2)L(l,
2). Eventually,
we
2). .
Now, we write
a --_=
(iL)2
a& E + iL
and find by means
(iL*)
i&L
= l-F-
(E + iL)2 of two-particle
c + iL
(E + iL)2
scattering
theory
(4.9)
LIOUVILLE
EQUATION
AND DEFINITION
OF ENTROPY
i’L(1V2)L,(l, 2) = 0,
93
(A.lO)
if the potential is not too long ranged. Introducing (A.9), (A.lO) into (A.8) yields
Iii&j-dld2 V(‘,2)[E+i;~l 2)-+Wd2).
=-
(A.ll)
3
As above, we find from the normalization
a,
dl
d2
VU,
IMP,,
+,4~2,9
=
of 9 that
0.
I
Thus, from eqs. (3.25), (A.6) and (A.ll) we immediately
obtain eq. (3.26).
iii) Derivation of eq. (4.11): This equation is immediately obtained by using eqs. (A.6) and (A.7) in (A.ll). References 1) D.N. Zubarev, Non-Equilibrium Statistical Thermodynamics (Nauka, Moscow, 1971), in Russian; English translation (Consultants Bureau, New York, 1974); German translation (Akademie-Verlag, Berlin, 1976). 2) D.N. Zubarev, Physica 56 (1971) 345. D.N. Zubarev and M.Yu. Novikov, Theor. Mat. Fiz. 13 (1972) 113,3 (1970) 126. 3) D.N. Zubarev, in: Itogi Nauka i Techniki, Vol. XV, Moscow 1980, in Russian. 4) N.N. Bogoljubov, Physica Suppl. 26 (1960) 1. 5) D.N. Zubarev, Teor. Mat. Fiz. 53 (1982) 1004. D.N. Zubarev and V.G. Morozov, Physica 120A (1983) 411; Teor. Mat. Fiz. 60 (1984) 270. 6) G. Riipke and V. Christoph, Phys. Stat. Sol. (b) 59 (1979) K15. G. Riipke, Teor. Mat. Fiz. 46 (1981) 279. G. Riipke and F.E. Hiihne, Phys. Stat. Sol. (b) 107 (1981) 603. F.E. Hiihne, H. Wegener and G. Rijpke, Phys. Lett. 99A (1983) 367. 7) D.N. Zubarev, Physica 56 (1971) 345. 8) R. Der and G. Rdpke, Phys. Lett. 95A (1983) 347. 9) R. Der, ZfI-Mitteilungen 79, 1983 and in preparation. 10) See, for example, H. Grabert, Projection Operator Methods in Non-equilibrium Statistical Mechanics, Springer Tracts in Modem Physics, vol. 95 (Springer, Berlin, 1982). 11) R. Der, Preprint ZfI-46 and Physica l32A (1985) 47. 12) R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley-Interscience, New York, 1975). 13) Yu. L. Klimontovich, Kinetic Theory of Nonideal Gases and Plasmas (Nauka, Moscow, 1975). 14) B. Robertson, Phys. Rev. 144 (1966) 151, 153 (1%7) 391.
ON THE RETARDED SOLUTION OF THE LIOUVILLE EQUATION AND THE DEFINITION OF ENTROPY IN KINETIC THEORY R. DER Zentralinstitur fiir Zsotopen- und Strahlenforschung, DDR-7050 Leipzig, Permoserstr. 15, GDR
Received 25 January 1985
Zubarevs approach to non-equilibrium statistical mechanics, which consists in adding a source term to the Liouville equation so as to select its retarded solution, is reinvestigated. As discussed earlier, the source has to be modified by introducing a different set of thermodynamic parameters in order to achieve agreement with current theories of non-equilibrium statistical mechanics. By considering kinetic theory of a homogeneous classical gas it is demonstrated that the modification of the source is necessary in order to avoid unphysical divergencies and to maintain conservation laws. Moreover, a new definition of the non-equilibrium entropy in terms of the relevant observables is obtained which reveals several attractive features.
1. Introduction Among the many approaches dealing with the problem of deriving the evolution equations of macrophysics from the basic Liouville equation (LE), the theory of Zubarev’-3) takes a more or less singular position. In fact, while most of the well known approaches, the projection operator methods, for example, start from a conveniently formulated initial value problem of the LE, Zubarev formulates a boundary value problem by introducing an infinitesimal small source term into the LE. This one is intended at breaking just the time symmetry of the LE while leaving the macroscopic behaviour invariant. Thus, a non-equilibrium statistical operator (NESO) is constructed which is the retarded solution of the LE and is expected to correctly describe the macroscopic relaxation phenomena taking place in the system. Noteworthy among the many attractive features of Zubarev’s approach are its easy applicability to concrete physical situations and its close intuitive connection to Bogoljobov’s method of quasi-averaging4) which has proved very helpful in many branches of physics. The value of Zubarev’s approach has been demonstrated by a number of interesting applications (see refs. 3, 5, 6 and 0378-4371/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
7.5
others). Moreover, its connection with current theories of non-equilibrium statistical mechanics has been broadly discussed3*7). The relation between macroscopic evolution equations (MEE) as obtained from the initial value problem of the LE (see eq. (2.2) below) and the MEE obtained from the NESO was rediscussed recentlysV9). It was found that with the definition of the source term as given by Zubarev, the latter do agree with the former under very specific conditions only. However, it could be shown that this difficulty is removed by a self-consistent redefinition of the source term. The present paper aims at a further clarification of this point and at elaborating some consequences of the redefinition of the source term. For this purpose, we state in section 2 in a simpler language the arguments given in refs. 8 and 9 concerning Zubarev’s definition of the source term. In section 3 we consider a simple example, the derivation of kinetic equations for a classical homogeneous gas, in order to explicitly demonstrate these more general considerations. In the event, we also hope to demonstrate the usefulness of Zubarev’s approach, if the source term is appropriately redefined. Then, in section 4 we consider the more general problem of the definition of the non-equilibrium entropy. The new source term is argued to provide us with a modified information entropy which reveals several attractive features. This is demonstrated for the case of kinetic theory considered by showing that our entropy reduces to the full low density equilibrium entropy for t + m. A short summary and discussion is found in section 5.
2. Outline of Zubarev’s approach. Redefinition of the source term As in many theories of non-equilibrium statistical mechanics”), in Zubarev’s approach a central role is played by a generalized Gibbs state p4, called the quasi-equilibrium ensemble by Zubarev. This one is formulated in terms of the set P, {P: P,, . . . , P,,}, of relevant observables as p,(t)
= Q’,’
e-‘~=lpmFm(‘) = : p,{f7(t)},
(2.la)
where (2.lb) and (A) = 1 dl-A
or
(A)=
TrA
76
R. DER
in classical
or quantum
dimensional parameters,
phase space. The F,(t) denote conjugate thermodynamic which we will leave unspecified for the time being.
mechanics,
In terms of pq, the initial
problem
r
denoting
a point
(2.2a)
formulated
as’“)
P(O)= P,(O)
(2.2b)
7
so that the quantities are given by
of physical
interest,
i.e. the expectation
pm 0) = UJ, e-iL’p,(0)) = (Pm(t)p,(O)). The Liouvillean the commutator operator
in 6N-
of the LE
= 0
(a, + iL)p(t) is readily
value
respectively,
values
of the P,,,,
(2.2c)
is to be understood as usual in terms of the Poisson bracket of the Hamiltonian H with any phase space function
or or
A, respectively,
LA=i{H,A}
On the other source term
or
hand,
LA=i[H,A].
the NESO
is obtained
(2.3)
from
64 + iL)p,(t) = - 4~~0) -p,(f)> t
the solution
of the LE with
(2.4)
where we take E -+ 0 after taking the thermodynamic limit. Thus, if the NESO p, exists, it may be argued to exactly obey the LE in the limit E + 0. The solution
of (2.4) is given
as’).
p,(t) =
1 dt’ e~(E+iL)“p,(t -
t’)
F
(2Sa)
0
or by using partial
integration’):
p,(t) = p,(t) - i dt’ e-(E+iL)f’(iL - a,,)~Jt - t’) .
(2Sb)
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
II
Thus, p,(t) is given as a functional over F(t”), t”S f. We prefer to write p, here as
(2.6) so that by using the well known operator identity (A +
I?)-’ = A-l - A-%(,4
+ I?)-’
(2.7)
eq. (2.6) is easily seen to be the formal solution of eq. (2.4). For the following, it should be noted that for sufficiently large E and an arbitrary time t, p,(t) should exist for any choice of the parameters F,(t), m = 1,. . . , n, provided only p,(t) is an analytic function of t at the time considered. In Zubarev’s approach, the F,(t) are determined by the requirement that the expectation values of the Pk, k = 1,. . . , n, as obtained from both p, and pq agree exactly in the limit E + 0, i.e.
&(t) = ~~(P,P,(~)= (P,p,(O), k = 1, . . . , II.
(2.8)
Thus, assuming the set of equations $ =
(pk&{y)), k = 1,. . . , n,
could be solved for y so that one obtains y as a function of x: Y,=F,[x]=F,[x
,,...,
x,],
m=l,...,
n;
then we may obtain’) from (2.8) F,(t)=F,[P(t)],
m=l,...,n
(2.9)
and
In other words, eq. (2.8) may be used for the definition of the thermodynamic parameters F(t) iff there exists a set of functions p,(t), 1 = 1, . . . , n, which are the solutions of
78
R. DER
F,(t)
=
The point
lim P, E-t0 (
i;
+
of view taken
exist, in general. say, where
F +
Instead,
a,
P,ml)
.
(2.11)
in refs. 8 and 9 now is that such a solution we have to introduce
a second
does not
set of functions
p’(t),
we have
ml = l&y@&p'(r)l)
(2.12)
7
so that
P,(t)= P,[P’(t)] Then,
)
F;(t) = Ppyt)] .
(2.13)
introducing
P;w = P,PWl = &p’(~)l the modified
LE with source
(2.14)
reads
(a, + iL)p, 0) = - ~P,W - P:@)).
(2.15)
With this redefined source term, the MEE obtained from the new NESO can be shownBV9) to exactly agree for sufficiently late times with the corresponding MEE governing the time evolution of the expectation values obtained from the the information entropy defined in initial value problem (2.2). Moreover, terms of p: seems to be a better approximation to the non-equilibrium entropy than the one obtained from pq (see below). It should be mentioned that the difference between p(t) from the transient processes connected with the transition
and p’(t) results from the non-
dissipative state p,(t) or p:(t) to the macrostate p,(t) where dissipativity is fully established. From this follows already intuitively the intimate relation stressed in refs. 8, 9 and 11 between the initial slip occurring in the solutions (2.2) and the definition of p’. Moreover, the difference P - p’ can also be related to retardation effects occurring in the projection operator methods, for example. Quite generally, it may be stated that
F(t) - F(t) = O(lJ) where
t, is a time
)
5 = ;,
characterizing
the transient
processes
and
tR
is
a
typical
79
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
relaxation time. Thus, if f = 0, we reobtain Zubarev’s definition of the source term. Specific examples are given by the ideal gas but also by kinetic equations of a sufficiently dilute classical hard sphere gas where the Boltzmann or Enskog equation may be expected to be correct for all times t > 0 in (2.2). In the following section we shall demonstrate those general propositions by considering the special case of kinetic equations for a dilute classical gas with realistic interactions.
3. Kinetic theory Let us consider a homogeneous
system of N particles with Hamiltonian
H=Ho+H1=-p$V(i,j)
(3.la)
#
and Liouvillean L = 5 L,(i) + 5
i=l
(3.lb)
L,(i, j),
i
where V(i, j) = V(Xi, Xi) = V(lri - 51)s xi = {I;:,Pi) and
L(k)=
-ibpkc,L,(k,
I)= iF(-&-&).
k
The N-particle
distribution
function p,(t) be normalized as
Introducing
we find for the single particle momentum
cp(IA0 = fi(4 (PlP,(0) * Thus, the set P of relevant observables
distribution function the expression (3.2) is to be replaced by the phase space
80
R. DER
functions G(p), i.e. we have now to deal with a continuous set P.Consequently, the ansatz for pg reads p,{F(t)}
= p,(t)
= 0,’
e-Jd3PG(p)F(P.‘),
(3.3a)
so that from
dP7 t) = (6 (P)P,W> we obtain
WP, t) = -N In CP(P, 2). Therefore,
(3.3b)
we find explicitly (3.3c)
where n is the density, n = N/V. Thus, instead of eq. (2.10) we obtain the explicit expression (3.3~). Hence, we may write
(3.4) where cp’(p, t) is a function unknown to us so far. Further below, cp’ will be determined from the requirement that the average (e(p)p,) behaves well for F + 0 and all p. From eqs. (2.15) and (3.4) we obtain (3Sa) and cP(P, r, = (G(P)ir,
ir (t (P’(Pi, I))) i=I
.
(3.5b)
Writing
ri,=l-
F+i;+a,(‘L+a,)
(3.6)
LIOUVILLE
EQUATION
AND DEFINITION
OF ENTROPY
81
and 1
= i
(%U --
n=O n.1
e+iL+a,
1
(3.7)
E + iL
we obtain*
(P’(P)-P(P)=
mWV c-&w n=O
. -&P; >I. >+a,($(P)&P;
(3.8)
The strategy for evaluating the above expression is the following. We want to restrict ourselves to the dilute gas case so that we may assume the momentum relaxation is caused by uncorrelated sequences of binary collisions only. We use the exact relations iL iL -=-__-_
1 (iL)*
Efil?.
E
&E+iL
1 -= EfiL
1
(3.9a)
and
E
iL I 1 (iL)* (3.9b)
fz2e+iL
E
and find by standard methods (diagram or cluster expansions, for example)
x VI&) + 12,?(~))~‘(l)cp’(2~(p’(3) + ... where I,JE)
I&).
represents the integrodifferential
I
. .= ni2 dx,L,(k 0
E+
)
(3.10)
operator
i:(k L,(k, 0. , . II
(3.11)
7
and 1 may be considered as a dummy variable. The right-hand side of eq. (3.10) represents just the contribution of a single binary collision and that of an uncorrelated sequence of two such collisions. Higher order terms corresponding to sequences of k, k 3 3, binary collisions which diverge as ?-‘) for E +- 0 are not written explicitly. These terms will be discussed further below. * We supress the time argument r in rp(p, t) and
9’(p,
t)
for a while.
82
R. DER
For any reasonable, not too long ranged potential we may expand Z(e) for small E into a Taylor series: Z&)
(3.12)
= Z,J + SZ;! + * * * 7
where
z = !‘z(E),
I’ = lii
tJ,Z(&).
Thus”), we find Z is just the Boltzmann collision operator Z,&(Z)
so that (3.13a)
= 0(1/Q 9
where tf is the time of mean free flight. Moreover, particle scattering theory12S13)that
we conclude
from two-
(3.13b)
where t, characterizes the duration of a binary collision. Let us now return to eq. (3.8). Using (3.9), (3.10) and (3.12) we find the averages occurring in eq. (3.8) are given as a Laurent series in powers of E. Thus, if the limes E + 0 is to exist, the contributions corresponding to negative powers of E have to cancel each other. Considering for the time being terms of order E’ and c-l only and dropping all terms of order 5’ and higher, we find by means of (3.10) through (3.13)
+
u&.3+
Zl&,,
+ E-*Zl,2Z12,?)~‘(1)~‘(2)~‘(3) )
(3.14)
where I,, i = Z,,, + Z,, and p(i) = P(z+). Now, ie use (3.9) and (3.10) and
(@(P)b;) = 0
(3.15)
to conclude that the terms corresponding to II = 1,2, . . . of eq. (3.8) are at least of order E-* or t*. Thus, we obtain by introducing (3.14) (3.15) and (3.10) into (3.8):
LIOUVILLE
EQUATION
AND DEFINITION
OF ENTROPY
83
cd(l)CPU)=-G,f'P'mJ'c3+ ~-'@,b'(l>+ ~;,$9m'(a - 1,,2dwdw
v;.~L,~+ 1,,~1~~,~)(~‘(1)(~‘(2)40’(3)}.
(3.16)
Now, the &‘-term immediately yields the relation between rp’ and cp, i.e.
$4) = d(l) + ~;,*dw(P’(2)+ fx2)
(3.17a)
and hence
d(l) = dl)-
h4M2)+
fw2)
7
(3.17b)
where we remember p(i) = p(pi, t), for example. From the Cl-term we obtain
d’(l) = -v;,,cp’u)4(2) + ~~,,rp’W’(2) + (1;.242,3+ 11,21)12.t)~‘1(l)~‘(2)(0’(3) p where d’(l) = Q’(p,, d’(l)
=
t). Using (3.13b) we obtain from (3.18) immediately
4,24m4(2)
and by reintroducing
(3.18)
+
%9
(3.19)
this into (3.18) we find
4’ = 4,,rp’W’(2)+ ItzI;2,1(p’(l)lp’(2)~‘(3) + Q(5’).
(3.20)
This is the kinetic equation governing the time evolution of 9’. The corresponding equation for (p follows from (3.20) by first noting that by means of (3.17b) this can be rewritten as
d’(l) = b,*dl)&)+
W’).
(3.21)
Then, by taking the time derivative of (3.17a) we obtain immediately
d(l) = 1,,&d2)+
I;.2112,~~(l)~(2)rp(3)+ W’).
(3.22)
The kinetic equation obtained in this way is seen to be just the Boltzmann equation plus a correction term of the order of 5 = tc/ff which has been obtained earlier by several authors. In particular, Klimontovichi3) has stressed the point that the correction term is responsible for the conservation of the
R. DER
84
total
energy.
This
is clearly
born
out in the present
approach,
too.
In fact,
writing r;,,l,,,cptI)cp(2)(p(3)
= a&(~(l)~(2)
+ ~(52)
and using
I d3p,H,(P,)~
1,
Z(P(lb
(2)= 0
(3.23)
3
= p2/2m we find from
where
H,(p)
kinetic
energy
tn t)
K(t) = N j- d3p Ho( of the N-particle
system
(3.22) for the time derivative
K(t)
of the
(3.24)
the expression
(3.25)
As shown
in the appendix,
we may write this as
0 = K(t) + V(t) = a, lim(Hp,(t))
(3.26)
,
E’O
where
is just the potential
energy
of the system
in the kinetic
state given by p,(t).
The question of the conservation of total energy may also be considered directly in the framework of eqs. (2.4) and (2.15). Averaging eq. (2.15) with the Hamiltonian H yields obviously
I$ := a,(H&) = _&(H(& -pi)). Using
(3.28)
eq. (3.6) we obtain
(
E~,=E H
E + ii +
W + %)P;) =---&W-J~P.$~
a
I
I
8.5
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
so that
(E+ Gm
(3.29)
= ~%woP;) 7
where we used
a,Uf,p;) = 0 since 9’ is normalized. From the appendix together order 5’ and higher,
with eq. (3.17b) we find, neglecting terms of
where (3.30)
K+ V=E=lim(Hp,). t-+0
Introducing this into eq. (3.29) yields ii, = 0 +
O(E2)
so that SE = const. Using the fact, that eq. (2.15) is homogeneous find finally
in time we
JGc =0 what is valid for small values of E in the approximation considered, sufficiently small values of 5. However, if applying the same procedure to eq. (2.4) we find
E&) = -&V(t))
(3.31) i.e. for
(3.32)
so that for E # 0 in Zubarev’s original approach the total energy is not conserved while it obviously is with the redefined source term as proposed in the present paper. In concluding this section, we briefly summarize the main points of the derivations given above. We started from the general expression (3.5) or (3.8) where up’denotes an unknown function of p and r. We assume the momentum relaxation to result mainly from uncorrelated sequences of binary collisions of finite duration t,, so that the averages occurring in eq. (3.8) are obtained as a
R. DER
86
Laurent series in positive and negative powers of E. Consequently, the righthand side of eq. (3.8) may be rearranged into a Laurent series, too, see eq. (3.16). Then, the term in co already yields the relation (3.17) between cp and cp’. Moreover, if p of eqs. (3.5) and (3.8) is to exist, the coefficients of all powers c-l, 1 = 1,2, . . . ) have to vanish. This yields in order E-I the kinetic equations (3.20) for (D’and (3.22) for cp. By a straightforward but tedious algebra it may be shown that the relations between (p and p’ obtained also guarantee the cancellation of all terms of order &-I, 1 = 2,3, . . . ) so that we may conclude (4~~) exists with the above choice of p’ for c-,0. The inverse is also true: if we use a function p’ in (3.5) which is not a solution of the kinetic equation (3.20) ’ then (4~~) does not exist for E + 0. In particular, we obtain unphysical divergencies if we use Zubarev’s prescription (2.8) since this amounts just to putting cp’= cp what obviously is not consistent in different powers of E-‘. Therefore we are led to the conclusion that the redefinition of the source term is not only necessary in order to get the correct retardation terms but is also of principal importance to avoid unphysical divergencies. A further consequence of the replacement of p4 by p: is investigated in the following section.
4. On the definition of entropy in a non-equilibrium
state
As proposed by Zubarev, we might write for the entropy S(r) of the system at a given instant of time t
s&d = -$$P,(~) In ~~0))= - (~~0)ln P&N . In the case of the kinetic theory considered,
(4.1)
we find from (3.3) and (4.1)
S,(t) = A%(t) - N(ln it + 1) ,
(4.2)
where
(4.3) i.e. we obtain just Boltzmann’s
kinetic theory definition of the entropy.
‘Note that eq. (3.20) differs from eq. (3.22) by terms of order 5.
This
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
87
one, however, is well known to suffer from the drawback that its stationary value is just given by the entropy of the ideal gas, i.e. s, = N ln (2?rmkT)3’2 es/* . n This is true not only if the time evolution of (p(p, t) is governed by the Boltzmann equation but also when the non-ideality correction is included as in eq. (3.22). The ansatz (4.1) may be justified by using information theoretical arguments’). On the other hand, we can relate it to the full entropy of the system
se= -(P, InPA
(4.4)
by noting that
S, = lim SE E-m since
In the light of the discussion given in ref. 9 we might argue that pg follows from p, under the condition of a maximum damping of the non-macroscopic information. Applying this principle to the solution
of eq. (2.15) yields a different expression for the non-equilibrium
entropy, i.e. (4.5)
In particular, if using (3.4) we reobtain eq. (4.2) where h is to be replaced with h’:
h’(t) = - 1 d3p p’(p, 2) In P’(P, t) .
(4.6)
R. DER
88
Using eq. (3.17b), h’ can be written as a functional difference cp - cp’ to be small we may write it as
of cp. Assuming
the
h’(c) = h'[cp(r)l = h[cp(t)l+ j- d3p,U’,,,CP(P~~ t)d~2, t>>ln v(p17t>+ fTt2), (4.7) where
and
cp-1(l)~~,2cp(l)cp(2)+ Ok+)
In p’(l) = In (Iwas used. Moreover, functional
=
we find by means
of eq. (3.21) for the entropy
production
S:(t) the
S,(t)+ j- dp, (P-‘(P,,WI,&‘I~ t)cp(p2,~))~‘,,,cp(Pl~ t)cp(p,,t) + m3. (4.8)
From the properties clear that
~;be,l = 0 where
of the Boltzmann
collision
operator
I it is immediately
(4.9)
9
qe4 is the Maxwellian
cp,,(p) = (25~rkT))~‘~ For the stationary
S:, = S&L,I
value
e-p2’2mkT. of SA we find
= Se, - (2mkV’
so that from the appendix
(4.10)
j- d3p, P:~;,~(P,,(P,~,(P~)
we conclude
>
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
89
m
SA = S, + Nn27r
I
dr r2(e-V(r)‘kT- l)V(r)/kT,
(4.11)
0
where V(r) is the two particle interaction potential. Therefore, Sk is given by the entropy of the ideal gas plus the contribution of the correlation entropy in leading order of the density which is well known from equilibrium statistical mechanics. From this result, we may conclude that the ansatz (4.5) for the entropy of the kinetic state seems really to be favoured over the original ansatz (4.1) proposed by Zubarev if one aims at a theory which is consistent up to order 5 inclusively. Unfortunately, a proof that the entropy production Si is positive is still lacking but this drawback is shared by S, if we use eq. (3.22) with the non-ideality correction of order 5 included. The ansatz (4.5) for the entropy of a non-equilibrium state may intuitively be understood in the following sense. From information theory, pi(t) = p,{F’(t)} is interpreted as the ensemble of maximum indeterminacy with given values of the thermodynamic parameters F;(t), or the corresponding average values F;(t), k = 1, . . . , n. The state p,(t) with corresponding average values p(t) is obtained from pi(t) by
- oE being independent of pi and hence of P, P’. Thus, anything we have to know are the values p’(t) so that p:(t) may be regarded as representing the macroscopic knowledge necessary in order to obtain just the state p,(t) belonging to the average values Pk(f), k = 1, . . . , n. Therefore, it seems quite natural to choose the expression (4.5) as the corresponding information entropy which hopefully represents a good approximation to the entropy of the non-equilibrium state considered.
5. Conclusions
The results obtained above for the special case of kinetic theory are hoped to demonstrate sufficiently clearly the following aspects concerning Zubarev’s theory of the NESO. On the one hand, it has to be emphasized that the source term of eq. (2.4) has to be modified in the sense that the thermodynamic parameters Fk(t), k = 1,. . . , n are to be replaced by a set F;(t) with corresponding expectation values P;(r), 1= 1, . . . , n. The necessity for doing so follows clearly
90
R. DER
from the example investigated. In fact, p,(t) if considered as a functional of the thermodynamic parameters is well behaved for E + 0 only if these are just the functions F;(t) = Fk[p’(t)], the set I”(t) being related in a unique way with p(t), see eqs. (3.17) in kinetic theory, for example. The modification of the source term has also been shown to guarantee the conservation of total energy at finite values of E, see eqs. (3.31) and (3.32). On the other hand, the modified NESO approach is demonstrated to yielding in a rather direct way non-trivial physical results. Moreover, it leads to an explicit expression for the entropy Si of the non-equilibrium state in terms of the expectation values of the relevant observables, S: attaining its stationary value at the correct equilibrium entropy. These facts clearly support the point of view that Zubarev’s theory which is deeply rooted in basic physical principles is an interesting and fruitful approach to the general problem of describing relaxation phenomena in systems with reversible microscopic dynamics. As compared to other theories of non-equilibrium statistical mechanics, it must be noted that macroscopic evolution equations (MEE) as obtained from the NESO are intrinsically local in timess9). This is also shown by the derivations given in section 3, i.e. by the way in which the time derivative is eliminated in the course of the cancellation of the secularly divergent terms. The same procedure also applies to the higher order (n 2 1) terms in eq. (3.8) so that ultimately one always obtains MEE which are of first order in the time derivative of p(t). Nevertheless, all of the retardation effects are fully contained in the theory as exemplified by the correction term in eq. (3.22). Quite generally, in using the methods of refs. 8, 9 and 11, it can be shown that the MEE obtained from eq. (2.15) agree exactly with those obtained by Robertsoni4) or Grabert”), for example, in their memory renormaiized form given recently”). This also marks the limitation of the applicability of both equations (2.4) and (2.15). In fact, it can be shown by the techniques developed in refs. 9 and 11 that p, is well behaved for small E only, if the transients occurring during the transition from pi to p, decay in time more rapidly than any inverse power of t. In other words, the NESO can exist only if corresponding retarded theories are memory renormalizable.
Appendix
i) The potential energy : The potential energy of the system in the state p,(t) (3.5)-(3.7) given as
is by use of eqs.
91
LIOUVILLE EQUATION AND DEFINITION OF ENTROPY
- cv,) V(t)= v[cp (01= (W$ - C n!
(f4-&
n=O
Let us introduce the integrodifferential
J&E)
= i G 1 dl d2 V(l, 2)
OL+
a,)~$ .
64.1)
operator J:
1 E + iL(l, 2)
64.2)
L,(lT 2)
where J dl . . . = J dx, . . . , for example. Neglecting contributions from sequences of more than two binary collisions we may expand 1 iLpA = J&)(P’(l)P’(2) H*--> E + iL
(
= J,,cP’(~)~P’(~)+ (J&,~
+ ~-lJ~,~(~)I~~,~(&)(~r(l)(~‘(2)(0’(3)
+ J,,,(LI,,+
~-1~,~.~))~‘(1)~‘(2)~‘(3).
(A-3)
Using 1 _=---E+iL
1
1
iL
E
Ec+iL
and I
dpcp’(p, t) = 1 9
so that a,(H,p;) = 0 , we also find
(
a, K-
1 E + iL
P:, = - (-G,7L12,3+E-~J!,~I~~,~)(P’(~)~‘(~)(P’(~). >
In the light of the arguments given in section 3 we therefore (A.3) and (A.4)
V(t) =
$1dl d2 VU, WU)d@)
+ J~,240’(l)d(2),
where eq. (3.17b) was used finally. Using L,(i)cp(i) = 0 we may rewrite V(t) as
(A.4)
obtain from (A.l),
(A-5)
92
R. DER
dl d2 V(1,2) e-‘““~‘)cp(l)cp(2).
= lim g 1-z 2
In
equilibrium,
we
obtain
from
this
by
(A.61
means
of two-particle
scattering
theory’2*‘3) and eq. (4.10)
V[peq] = 2rNn
drr2V(r)e~V”“kr.
(4.7)
I 0
ii) Derivation of eq. (3.26): The integral term occurring in eq. (3.25) has been broadly discussed by Klimontovichr3). Following his arguments, we rewrite this term in the following way:
= - g
ljn~;
j dl d2 V(l, 2)L(l,
2)
1 F + iL(l, 2)
L(l, 2)(~ (1)~ (2)
WV
We used the homogeneity of the system so that we may replace L, with L in eq. (3.11) everywhere. Moreover, we exploited the symmetry of the expression with respect used
I
to x,, x2 which allows to replace
dl d2 H,(l, 2)L(l, 2). . . = -
H,(l)
by iH,(l,
dl d2 V(l, 2)L(l,
2). Eventually,
we
2). .
Now, we write
a --_=
(iL)2
a& E + iL
and find by means
(iL*)
i&L
= l-F-
(E + iL)2 of two-particle
c + iL
(E + iL)2
scattering
theory
(4.9)
LIOUVILLE
EQUATION
AND DEFINITION
OF ENTROPY
i’L(1V2)L,(l, 2) = 0,
93
(A.lO)
if the potential is not too long ranged. Introducing (A.9), (A.lO) into (A.8) yields
Iii&j-dld2 V(‘,2)[E+i;~l 2)-+Wd2).
=-
(A.ll)
3
As above, we find from the normalization
a,
dl
d2
VU,
IMP,,
+,4~2,9
=
of 9 that
0.
I
Thus, from eqs. (3.25), (A.6) and (A.ll) we immediately
obtain eq. (3.26).
iii) Derivation of eq. (4.11): This equation is immediately obtained by using eqs. (A.6) and (A.7) in (A.ll). References 1) D.N. Zubarev, Non-Equilibrium Statistical Thermodynamics (Nauka, Moscow, 1971), in Russian; English translation (Consultants Bureau, New York, 1974); German translation (Akademie-Verlag, Berlin, 1976). 2) D.N. Zubarev, Physica 56 (1971) 345. D.N. Zubarev and M.Yu. Novikov, Theor. Mat. Fiz. 13 (1972) 113,3 (1970) 126. 3) D.N. Zubarev, in: Itogi Nauka i Techniki, Vol. XV, Moscow 1980, in Russian. 4) N.N. Bogoljubov, Physica Suppl. 26 (1960) 1. 5) D.N. Zubarev, Teor. Mat. Fiz. 53 (1982) 1004. D.N. Zubarev and V.G. Morozov, Physica 120A (1983) 411; Teor. Mat. Fiz. 60 (1984) 270. 6) G. Riipke and V. Christoph, Phys. Stat. Sol. (b) 59 (1979) K15. G. Riipke, Teor. Mat. Fiz. 46 (1981) 279. G. Riipke and F.E. Hiihne, Phys. Stat. Sol. (b) 107 (1981) 603. F.E. Hiihne, H. Wegener and G. Rijpke, Phys. Lett. 99A (1983) 367. 7) D.N. Zubarev, Physica 56 (1971) 345. 8) R. Der and G. Rdpke, Phys. Lett. 95A (1983) 347. 9) R. Der, ZfI-Mitteilungen 79, 1983 and in preparation. 10) See, for example, H. Grabert, Projection Operator Methods in Non-equilibrium Statistical Mechanics, Springer Tracts in Modem Physics, vol. 95 (Springer, Berlin, 1982). 11) R. Der, Preprint ZfI-46 and Physica l32A (1985) 47. 12) R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley-Interscience, New York, 1975). 13) Yu. L. Klimontovich, Kinetic Theory of Nonideal Gases and Plasmas (Nauka, Moscow, 1975). 14) B. Robertson, Phys. Rev. 144 (1966) 151, 153 (1%7) 391.