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The Einstein-Smoluchowski promeasure versus the Boltzmann(-Peierls) equation
Physica
A 180 (1992)
336-358
North-Holland
The Einstein-Smoluchowski promeasure the Boltzmann(-Peierls) equation
Received
25 April
Beginning function
from
f(k.
inversion
of
1991
the specific model of the Boltzm~~nn~Peicrl~ the
I).
time-dependent
Boltrmann’t,
Boltzmann-Pcierls
relation.
equation
values of flk.
i\ convcnicnt
distribution study
(phonons.
carefully
in time
a kinetic
studied.
The
Ructuationh
distribution
equation
for the distribution
IS developed
function
f(k.
I)
magnons.
rotons.
etc.).
that the randomness
from
that
Einstein‘s
satisfica
The
hy the Einstein-Smoluchowski
approximation. of Ructuations
version
The
of the Green-K&o
exact formula
for
in this paper. cmcrgcs as a portion
equilibrium
in the moments
the thermal
assumption distribution promcasul-c
analog of Einstein‘s
promeasure of fl/,.
t).
CL* is used to
Some
approach to transport conductivity
the
the macro-
crucial
m the statistical
01 p, as being the infinite-dimensional
law in the Gaussian with
of
tw charactcrizcd
I) C:III
to think
the correlations
associated derived
The
into the present approach is simply
of the initial I_L,. It
theory
is to he vicwcd as ;I random variable characterizing
state of a gas of quasiparticles cntcring
versus
cocfticicnt
problems
proccsscs K,.
which
a~-e i\
of the general formalism
1. Introduction A convenient way of describing the gas composed of quasiparticles (phonons, magnons, rotons, etc.) [l-3] is to specify a one-point distribution function f(k, t). Since this function determines the number of quasiparticles with quasimomentum hk which is found in a system of volume V at time t (cf. the text before eq. (2.3a) in ref. [4]), one may suppose that f’(k, t) satisfies the space-independent, integro-differential equation, the so-called BoltzmannPeierls equation. Regardless of the program of obtaining such a kinetic equation for a one-point distribution function, in the previous paper (41 we set up a general 037%4371/92/$05.00
0
1992 - Elsevier
Science Publishers
B.V. All
rights
rcservcd
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
337
framework for a systematic theory of fluctuations occurring in quasiparticle gases “kept” in an overall equilibrium state. Thinking of the kinetic theory of gases [l, 2,5,6] and the Einstein thermodynamic theory of equilibrium fluctuations [7,8] as a unified structure, the physical picture which we have here in mind is not of course very similar to what one normally considers in the traditional description of both classical and quasiparticle gases: the distribution function f(k, t) is to be viewed as a random variable characterizing the macrostate of a system [4,9, lo]. In order to make this picture precise, we need prescribe the initial value f(k, 0) of f(k, t) and observe that, because every datum f(k, 0) evolves in time deterministically according to the Boltzmann (-Peierls) equation involved, any randomness will arise, then, only from the randomness in the statistical distribution of the initial values of f(k, t) in the ensemble of similarly prepared systems. In addition to these remarks, there remains the interesting problem of specifying in a sensible way what one means by a “statistical distribution” of f(k, 0). Although the initial experimental preparation of an ensemble may be quite arbitrary in principle [ll], the crucial assumption that enters into the present discussion is simply that the fluctuations of the initial distribution probability function f(k, 0) are characterized by the Einstein-Smoluchowski promeasure pue of scale parameter F. (As noted already in ref. [4], the reciprocal of the scale parameter E is proportional to the volume V of a system.) The first paper in this series [4] constructed p, and developed a manifestly infinite-dimensional analog of Einstein’s thermodynamic theory of equilibrium fluctuations [7,8]. The purpose of the second paper is to demonstrate how that approach may be used to formulate and answer the concrete physical problems in the context of Maxwell’s equations of transfer [12]. Before considering them in detail, we must now explain the notion of Maxwell’s equations of transfer and describe the general features of the moment method of solving the Boltzmann-Peierls equation [13]. If one multiplies both sides of the Boltzmann-Peierls equation by the IkenberryTchebychef functions forming a complete set [3,4] and then integrates the resulting expressions with respect to k, one obtains a system of infinitely many relations to be satisfied by the moments of f(k, t). Due to the completeness of the Ikenberry-Tchebychef functions [3,4], this system of relations, Maxwell’s equations of transfer [12], is formally equivalent to the Boltzmann-Peierls equation. At first sight, the equations of transfer seem to be a likely tool, but any attempt to study the time-behavior of the moments off(k, t) runs straight onto the nasty problem of evaluating the complicated collision integrals appearing on the rhs of those equations. And, even if one invents an exact method for the evaluation of the collision integrals, one may conclude very little a priori.
338
Z. Bunach,
Indeed,
S. Piekarski
as is well known,
say, the first II moments moments
alone
this should transfer
the equations
be true would
for the moment the “forward
promeasure
that govern
of f(k, t) do not become
until the remaining
m’ > m. Such a general called
I Einstein-Smoluchowski
higher
seem apparent of order
feature
coupling
order
the evolution
in general moments
m contains
the moments equations
of the equations
in time of.
equations
for these
are calculated.
once one realizes
of Maxwell’s
L’SBolrzmann eyuation
of order
of transfer
of moments”
That
that the equation
of
m’ where
is sometimes
[ 121.
Thus, what various authors have typically done (cf.. e.g., refs. [13, 141) has been to introduce some truncation scheme. and then to write down as a starting point of the discussion a closed system of equations for finitely many moments. Of course, this leaves the distribution function f‘(k, t) largely undetermined, since only the infinite set of moment equations can determine f(k, t). The assumptions based upon the truncation scheme, albeit drastic, are not unreasonable. And indeed, if nothing else, one may argue for their validity making use of the numerical experiments [14]. On the other hand. the exact solutions for Maxwell’s equations of transfer are possible, and it is very instructive to explore in some detail their properties in each nontrivial case. This paper will adopt the latter alternative. Precisely speaking, calling attention to the support and interpretation given by Gurevich [2] to a special case of the collision operator, we obtain exact equations describing detailed properties of the Ikenberry-Tchebychef expansion coefficients LI’“.’ of the distribution function [3,4]. Our equations are the quasiparticle analog [3] of those derived by Grad’s method of moments [13]. Then, since we do not consider the space-dependent effects”, the next stage in the analysis involves the further specification of the collision operator so as to arrive at the exact solution of Maxwell’s equations of transfer. This is of course nothing more than another way to look at the problem of solving the Boltzmann-Peierls equation. The above observations emphasize the essential similarities between the kinetic theories of quasiparticle and classical gases [12, 151. Now, the net upshot of the preceding paragraphs randomness in the statistical distribution of the initial
is as follows. values of f(k,
That the t) can be
described by the Einstein-Smoluchowski promeasure p+ is really a fact of crucial importance for, given only that f(k. t) satisfies the deterministic Boltzmann-Peierls equation, one may discuss the time evolution of I*, . In particular, observing that the collision operator is the infinitesimal generator of a strongly continuous semigroup {U(t); t 2 0} of operators [16] acting on the proper subspace N,, of Grad’s space H, one concludes that the reciprocal image of pF by U(t) is a promeasure on H,, [17-191. By the way, in the previous paper ” The distribution function f(k, I) represents directly before q. (2.31) in ref. [4]).
the global
property
of the system
(cf. the text
2. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
339
[4] we introduced the Hilbert spaces H and H, without making any explicit reference to the Boltzmanm-Peierls equation. Leaving aside these technical issues, in our approach the crucial physics will be associated with the Ikenberry-Tchebychef expansion coefficients ~“‘~(t) of the distribution function f(k, t). To be completely explicit, one particularly important question to be investigated concerns the calculation of the unequaltime correlation functions of the random variables ~“‘~(t). In fact, the macroscopic quantities describing transport processes can be written as linear combinations of finitely many integrals calculated from the suitably chosen timecorrelation functions of ~2~~’ (t). This will, for example, enable one to construct a kinetic version [lo] of the Green-Kubo theory [20,21]. Indeed, adopting the results established in ref. [4], the desired calculations can really be justified on a formal level, and the exact formula for the thermal conductivity coefficient K= which we derive in this paper emerges as a portion of the general formalism. And finally, because the proposed form of the collision operator is not very complicated, one can actually address more difficult problems than simply albeit somewhat obtaining the exact formula for K=. It is straightforward, tedious, to construct the stochastic Boltzmann-Peierls equation and to discuss its fundamental properties. This is a problem that deserves some more attention in future investigations. In summary, it should be stressed that, for our model of the BoltzmannPeierls equation, all the physics associated with Maxwell’s equations of transfer and the Einstein-Smoluchowski promeasure p, can be treated exactly; it is only the intermediate calculations leading to the notion of an equilibrium promeasure that are to be considered in an approximate way [4]. By focusing upon conceptual issues, this paper will emphasize the essential connections between the kinetic theory of quasiparticle gases [l, 21 and the Einstein thermodynamic theory of equilibrium fluctuations [7,8]. Here we proceed as follows, Sections 2 and 3 describe a simple collision model and construct exact solutions of the kinetic equation by means of solutions of the system of equations for the moments of the distribution function f(k, t). Given the Einstein-Smoluchowski promeasure CL, on the proper subspace H,, of Grad’s space H (cf. ref. [4]), section 4 derives a variant of the time-correlation function formalism originally thought limited to a gas composed of classical molecules [5,6]. In this derivation, the role of Maxwell’s equations of transfer introduced in section 3 will be clearly evident. Then, in section 5 we adopt the Green-Kubo approach to transport processes [20,21] and calculate the thermal conductivity coefficient K=. While most of the explicit calculations are carried out without first specifying the detailed form of the Tchebychef functions Ap(z; T) [4], appendix A takes note of this restriction and explains, if not directly then at least through some useful inequalities and
340
Z. Banach,
theorems, model tion
S. Piekarski
a way of removing
(O(y)
iy = const).
3.1 is included
section
I Einstein-Smoluchowski
it for the isotropic
The auxiliary
as appendix
vs Boltzmann
and dispersionless material
the work
associated
equution
(phonon) with sec-
with final
remarks
of
[4]),
distribution
6.
Terminology
As we know
equation
and basic definitions (cf. the
text
before
function f(k, t) determines the number system of volume V at time t. The evolution
technical
B. We end
2. A model of the Boltzmann-Peierls 2.1.
promeasure
of f(k. t) is given
d,S= J(f)
eq.
(2.3a)
in ref.
of quasiparticles kinetic equation
the
with wave vector k in a that governs the time
by (2.la)
’
off due to collisions. where J(f) represents the rate of “increase” the behavior of f(k, t), it is necessary to determine J(S) explicitly. [3], we assume that the collision operator and t into another one, is characterized
[J(f)1 (k. t>= -
&
J(o), by
which transforms
To discuss As before
a function
of k
lf(k, t>-.M~>l
+
i dk, {g(k> k) f(k. t) [f(k 0 + 11
-
g(k k,) f(k, f>[f(k,, f) + 11)
(2. lb)
3
where w(y; T):=hL?(y)(k,T)-’
,
y := k,; ‘Ikl ,
y, := k,;‘lkJ
g := (kl-‘k
g, := (k,(-‘k,
,
s(k,
k*):=
L(Y) x(s”s*)
4, := (6&V/V)“3
, .
,
S(Y -Y.>
3 (2. Ic) (2. Id) (2.le)
Here T(T), L(y) and x(t), (51~1, are certain positive functions of their arguments, and S(y - y*) stands for Dirac’s delta. Without further comment, we shall use those symbols that either are reasonably standard or appear for the first time in ref. [4]. Here let us only recall that T is the kinetic-theory
Z. Banach,
S. Piekarski
/ Einstein-Smoluchowski
promeasure
vs Boltzmann equation
341
temperature corresponding to f and that f,(w) is the Bose-Einstein distribution function which has the same principal moment as f. The quantity 7 denotes in turn the effective (k-independent) relaxation time. In general, this relaxation time is a function of the temperature T: 7 = T(T). Of course, 2& and k, are the constants of Planck and Boltzmann, respectively, and fifl( y) represents the energy of a single quasiparticle with quasimomentum hk. The specification of the meaning of the symbols N and V is not necessary here, but can be found in section 2.1 of ref. [4]. Insofar as the aggregate of phonons in an insulating crystal at low temperature is concerned, the rate of change of f(k, t) that arises from the collisions associated with anharmonic contributions to the potential energy has already been calculated [2]. Unfortunately, the resulting nonlinear expression for this rate of change seems to be extremely complicated, and the idea to replace it by -Y’( f-h,), or by the (more realistic) proposition of Callaway#*, rests implicitly on the hypothesis that a large amount of detail contained in the definition of the collision operator J( 0) is not likely to influence significantly the values of many experimentally measured quantities. Obviously, after having introduced 7, the mean (k-independent) relaxation time, as a tool to solve analytically some problems standing behind (2.la), we might as well try to give the first term on the rhs of (2.lb) quite another interpretation. (In passing, it should be stressed that, in the present case, T is a constant.) With respect to the integral appearing on the rhs of (2.lb), the collision operator has such a structure if one considers, for example, the elastic scattering of phonons by different defects (imperfections) of a dielectric crystal. But we have explained already these questions, and of general interest in this regard is our discussion in section 3 of ref. [3]; cf. also Gurevich’s monograph [2]. Before going further, however, we would like to point out that S(k,, k) is nothing else than the transition rate for an elementary process which destroys a single quasiparticle in the mode k, and creates one characterized by the wave vector k. Of course, we assume that %(k,, k) = %(k, k,). Given (2.lc), the S-function allows only transitions satisfying the energy conservation law. In order to describe %(k, k,) completely, we must specify the scalar functions L(Y) and X(5), /5]~1, and to this aspect of the definition of %(k, k,) section 3.1 is primarily devoted. It is a simple matter to prove that the operator J(o) as given by (2.lb)-(2.le) reproduces correctly the essential features of the collision operators for quasiparticle gases [3].
X2 Cf., for example,
section 6.1 in ref. [22].
2. Banach,
342
2.2.
General properties
If the distribution (Z.la)
i Einstein-Smoluchowski
S. Pirkarski
is written
promeasure
d,q
=
function
f(k,
t) obeying
the Boltzmann-Peierls
I 7
equation
as
(2.2) -
equation
of the collision operator
f(k,f) =h(w> [I + A(w) cp(z,g. t; 731substituting
L’S Boltzmann
k
into
(2.la)
CFE
4, .
(2.2)
yields (2.3a)
VT [ 161 dense subspace
where L is a linear mapping from a suitably chosen Grad’s space H,, onto H,, (H,, C H). If we set
D( 8) of
o_:=I+;j, I being
(2.3b)
the identity
(;b)(z,
operator,
then
we have
g, t; T) := -7k:)[A(z; T)]‘L[h(z: T)]
I
x dg. Wgog,) [dz, g., f; T)
- dz, g, t; T)].
k.
(2.3c)
To arrive at (2.2), we have replaced k by (z, g). This is legitimate because, as noted already in ref. [4], z may be viewed as a function of y and T; in section 3.2 of ref. [4] we denoted this function by i(y; T). And thus, one can interpret the quantity A(z; T) which appears on the rhs of (2.3~) by saying that h(z; T) is a solution for y of the equation f( y; T) = z. Granted these preliminary remarks, belongs addition, space
we should also remember [4] that A(o) : = w[ 1 + h,(w)] and that 9 to H,, if and only if ( 1, cp) ,{ = 0 (in this context, cf. eq. (2.2)). In I6 is a set of unit vectors in a three-dimensional Euclidean vector
IE. The verification
of eq. (2.3~) for (2.1)
and (2.2)
is immediate;
and,
indeed, one can prove that, regardless of the apparently nonlinear form of J(f), the 3 IS a linear operator on H,,: $9 E H,, for cp E D(L). To see why, it is sufficient to adopt for (2.lb) the symmetry properties of S(k, k_). We may use now the definition (2.3~) of ;‘, to deduce some of the basic properties of O_.By a standard manipulation. we arrive at (2.4a)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzrnann equation
343
provided that both cp and $ lie in the domain D(lL) of the operator II; thus ILis self-adjoint and strictly positive. Since L(y) > 0 and X( 6) > 0, the equality sign in (2.4b) holds if and only if $(z, g*) - $(z, g) = 0 almost everywhere on#3 9?2,‘xKxK. If there exist positive constants C1 and CZ such that y2L( y) c C, for y E .!?i!+ and#3 X([)sC,for[E[-l,l], thenfromCF=@‘,:={kE[E: O<(lc]<&} and the choice made above for ,3 it follows that L is a bounded operator on H,. When#” @!= @, := {k E [E: ]k) > 0}, the problem becomes much more complicated, since in this instance we have to extend the analysis to a broader class of operators, namely, one whose elements are either bounded or unbounded. Applying the Yosida-Hille theorem (cf. ref. [16], p. 219, we easily show that -T-‘IL is the infinitesimal generator of a strongly continuous semigroup {U(t); t B 0} of operators acting on a Hilbert space H,. In addition, if we choose to write ]]U(t)l],, for the norm of U(t), we obtain the following inequality:
In order to understand more completely some fundamental problems in the theory of stochastic kinetic equations, it is very useful to restrict attention to isotropic one-branch Debye models such that @ = EF [4]. However, these particular models constitute only a subclass of all possible ones (@ = @,, CF= @,) for which (2.5) holds, and so it is reasonable to obtain some definite results from (2.5), as well as from the additional assumptions concerning both L(Y) and x(5) (c f . section 3.1), without first having to specify @. In ref. [4] the Hilbert space H and its proper subspace H,, were introduced without explicitly invoking the Boltzmann-Peierls equation. We recall that the basic concepts of our previous paper [4] should be traced to those of Einstein [7]. Thus the fact that the collision operator U. is self-adjoint and strictly positive on the proper subspace H, of Grad’s space H appears as a very natural prerequisite for substantiating the eventual conclusions concerning the link between the kinetic theory of (quasiparticle) gases and the Einstein thermodynamic theory of equilibrium fluctuations. Clearly, since a,( 1, cp(o, t; T)), = 0 for cp(o, 0; T) E D(L) C Ho, we must seek a physically meaningful formalism for “isolated” systems (cf. sections 2.1 and 3.1 in ref. [4]). Indeed, the property of cp just mentioned is equivalent to the statement that the energy % of the gas is a constant. xi For the precise definitions of the segments 92 + := (0, y,) and 9?: := (z,, z,) in (0, m), see section 3.2 in ref. [4]. The definitions of the elementary cells EF and Q, are originally given in section 3.1 of ref. [4].
344
2. Bunach.
3. Maxwell’s
S. Piekarski
equations
I Einstein-Smoluchowski
of transfer
promeasure
vs Rolfzmann
equutiort
(a( y)Iy # const)
3.1. The explicit form for the transition rate The objective the moments
here will be to obtain
of f - A,. In order
the irreducible
equations
of transfer
to do this, one must first specify
for
the functions
L(y) and X( 5) (cf. (2.1~)). Although the inequality (2.5) guarantees that as t+x: the distribution function f( 0 , t) = fo[l + AP( 0, t; T)], cpE H,. approaches that appropriate to kinetic equilibrium, (CPP 1 f; Vi,, the
details
30
of this
(3.1)
3 decay
are
not
known
explicitly.
While
the
problem
is
unsolved today for the genera1 functions L(y) and %( 0, some steps have been made toward solving it (cf., e.g., section 5 in ref. [3]), and our aim here and in section 3.2 is to present in outline the explicit solution in the special case when O(y) iy # const and when L(y) is given by
L(y) =
y-’ i
II 0
b,,[n’(y>]” ,
(3.2)
where O’(y):=dfl(y)/dy and 6,,, 0~ v G r, appear in the role of constant coefficients. Also, we assume that %( [), (51 G 1, can be expanded in a series of Legendre polynomials [23] P,( 5). v = 0, 1.2, . . . , x.
X(S) =
c d,Jy(S)
“=(I
since the analysis conclusion. The propositions
becomes (3.2)
(3.3)
3
simpler and (3.3)
thereby
yet without
are useful
loss of generality
for the reason
of the
they were useful
before [3], namely, that they allow us to calculate exactly the effect of the collision operator O_ on Y”( 0) S, [ A( 0 ; T); T], and thus yield definite and rigorous special results instead of the approximations with which we must remain content if we insist upon using any other propositions. (Concerning the precise definitions of Y”(g), S,(y; T) and A(z; T), see sections 4.2 and 3.2 in ref. [4]. However, to make this development selfsufficient, we recall that, for a( y)/y # const, &( y; T) := [w’(y; T)]’ and s Ifi+ ,(Y: T) := [Y/w(Y; T)l[o’(y; w’(y; T) being the derivative of VI”. w( y; T) := W(y) (k,T)-’ with respect to y. In addition, Y”(g) := [(2a + 1)!!/4ncu!]“‘( 0”s) are Ikenberry’s tensorial harmonics. Finally, h(z; T) is a solution for y of the equation w’(y; T) = z; y = A(z; T).)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
345
The symmetric traceless moments N*,P of f - f,, are defined as follows: N
I
a,’:= dk YYg) 4x T) S,(Y; 7’) (f-f,) CF
=
WP, yaqJm
(Y,p = 0, 1,2,. . . ) a,
T); T]), ,
(3.4a)
cPEf&.
(3.4b)
We introduce (3.4a) just so as to arrive at the equations of transfer for NaXPin which (3.2) and (3.3) cause no difficulty in the rigorous determination of ([Lq, Y”S,), from finitely many moments N*,O of f-f”. Directly from (2.2), (2.3), and (3.2)-(3.4) we find that
d
f
N”.P = _ _1 N”.P -4&c,
c, :=do-
7
i
b, (yj”
“=O
Nn,p+2v ,
No” = 0 ,
(3.5a)
1
-d 2a+*
0..
These particular equations of transfer are relations among moments that the kinetic theory is designed to deliver, and, if r 9 1, they suggest the way in which the essential phenomena of relaxation may occur approximately when a more realistic form for L(y) is used. As observed already in the introduction, the equation of transfer for N”,P necessarily connects the moment N”,’ of order p to the moments Na,p+2” of order /? + 2~ where v = 1, . . . , r. In one very drastic limiting case, namely, when the elastic scattering effects associated with different defects in a crystal were neglected entirely (3 = 0), one could of course obtain the quasiparticle analog of the classical model originally proposed by Bhatnagar, Gross, and Krook [24]. This simplified approach is not discussed here, however, because, irrespective of the forward coupling of our equations of moments, we can evaluate exactly the time evolution of N”,P by solving (3.5a). In (3.5a) we set Noa0 = 0; then the Bose-Einstein distribution function &(w) has the same principal moment as f(k, t): 1 dk fin(y) MO) = jdk C LF
O(Y) f(k, t) .
(3.6)
To go into the detail regarding the R(y) = cy case (c being a strictly positive constant) would needlessly duplicate the ideas already presented, but the discussion given in appendix B for Q(y) = cy satisfies the purpose of this
346
Z. Banach. S. Piekarski
paper:
to show that the method
suffices
to obtain
incorporating
a systematic
all results
The exact solution
3.2.
It is clear
that
/ Einsteirt-Smoluchocc?ski promeusure vs Boitzmcmrt equation
eqs.
of sections
3.1 and 3.2, applied
description
of the trend
in the theory
based
(3.5a)
D(5) C H,,), then elementary general solution of (3.5a) is =
=
toward
from the outset
# cy,
equilibrium,
upon
0(y)
= cy.
of Maxwell’s equations o,f transfer have
a unique
because they are ordinary linear differential If we denote by cp,,(z, g; T) the function
~“.~(r)
to R(y)
inspection
dk [:‘( y; T) Y”(g)
~:,(cp,,, j:‘[A(o;
for all r. r = 0, 1.2,.
solution
.
equations of first order in the time. cp(z. g, t: T) in which t = 0 (cp,,E
shows that for a general
w( I’; T) S,(y;
T) [f(k,
value
of r the
0) -.6,(w)]
T); T] SJA(o; T); T])f,.
(3.7a)
where i;(y;
T):=exp{
-
[ti~~~(y; T)l).
(3.7b)
~
(3.7)
[R’(y)]“S,(y;
from
T)=
(3.5a)
we have made
(Yj”
(3.7c)
‘. use of [4]
S,+,,,(y: T).
(3.8)
So as to exclude once and for all the r/rcV G 0 case. we shall assume henceforth. without restatement, that ccI 2 0; e.g., d,, > 0 and d, = 0 when I_Y> 0. The results (2.5) and (3.7) express the relaxation theorem. They show. if not directly then by inference, that as t 3x, the values of N”.“(t) all decrease “exponentially”
to the values
appropriate
to kinetic
equilibrium.
namely.
0.
The rates at which they do so are governed by the functions T<”( y; T) of y and T. We call these functions either the times of relaxation for N”‘“(t) or the P-independent relaxation times. (By way of digression, a very simple model based upon cu = 0 contains the most important features of the BoltzmannPeierls operator J(o), but it presents some shortcomings: this model does not allow the collision frequency to depend on the magnitude ,fQ of the wave vector k; JkJ = &y.) Of course. given (2.5) and (3.7), we have not proved anything so strong as the assertion of the ultra-narrow trend toward equilibrium, according to which in a grossly homogeneous condition if k E CF. then
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
347
lim,,,1f(k t) -&,(~)I= 0. CTruesdell and Muncaster (cf. ref. [12], p. 363) have purposely left unspecified the precise sense of the limit f 3 fO.) It is to be remarked that, although a knowledge of all the moments is theoretically equivalent to a knowledge of the distribution function f(k, t) = f,(w) ]I + A(o) cp(z, g, t; Ul, no direct expression of the latter is obtained from eq. (3.7a). This explains our interest in considering only such solutions (~(2, g, t; T) of (2.3a) as can be expanded in a series of the IkenberryTchebychef functions#4 Y”(g) A,(z; T). Thus, setting cp,(z, g; T) := (~(2, g, t; T), we obtain r
cPr =
x
c c a”,‘(t)0
Y”A, ,
(3.9a)
v=o y=o
(ao3”(t) - NO’O(t) )
A+‘.O(t)= 0)
;
(3.9b)
eq. (3.9a) is satisfied in the sense of the norm joIn in H [4]. The expansion coefficients a”,‘(t) have a simple representation in terms of q,. Indeed, if we “multiply” each side of (3.9a) by#” VV(z; T) Ya( g) A,@; T) and then integrate with respect to g E K and z E 3 (:, we find that P”(t)
=
(q~,,Y”A,), >
9,
E
W) .
(3.10)
Since Ap(z; T) is a linear combination of S,[A(z; T); T] with coefficients depending upon T [4], since 0 d y d p, and since [P( y; T) are independent of p, it follows from (3.4a), rp = q,, N”*P = N”.P(t), (3.7a) and (3.10) that
c”(t) = (~0, 5;Y”A,),
3
‘POE
W) .
(3.11)
Thus, while in general we are not able to calculate q, explicitly, for the very special circumstances of a model gas of quasiparticles and a function L(y) of the form (3.2) we need only specify the initial value ‘poE D(L) of q, and determine cp(y; T) from (3.7b) and (3.7~). It is this fact which makes the search for exact solutions of the type (3.9) and the exact, time-dependent theory of fluctuations feasible. We have presently no hint toward proving the corresponding relaxation L4If we orthogonalize the set {S,[h(z; T); T]; p =O, 1,. , m}, we obtain a sequence of Tchebychef functions A,(z: T) which, apart from the sign, is uniquely determined by the following conditions: {A, IA,} = S,,. Here the symbol SPYdenotes the Kronecker delta, and { 0 10) is a scalar product in the Hilbert space U.*(%~) of all real-valued functions which are square-integrable on 9,: C (0, m) with weight AZA’~‘fo(l +fO); A’(,?; T) := ah(z; T)/az. For more details, see sections 4.2 and 3.2 in ref. [4]. X5W(z;
T) := (A(z; T)]‘A’(z;
T)w*( A(z; T); T) fO[w(A(z; T); 7’)]{ 1 +h,[w(
A(r; T); T)]}.
snql fs~osua~ ruopue.~ asaql 30 sanlm palDadxa aql awlnw2 01 ww3!p lou s! l! ‘([p] ‘3al u! (E’P) .ba .33) [“rl :x]g =: [x]“g 30 uop!uyap aqi 01 y3eq 8u+a3ax .(%-I “‘f~) uo slosuai u~opuw p2~!.1puqIC:, 4u!aq SE papmh aq uw (~)~.,n@ (I&D pue (J)~.,,D wql as!ldms ou SC awo3 plnoqs I! ‘“)H uo ‘ri amseawo~d !ysMoq~nlo~s-u!alsu!~ aql 01 %up.~ome ch 30 uo!inq!.wp ~r?3!is!iels E [p] .3a1 u! pawnsse aheq am am!s ‘IahaMoH ‘x) aa.kiap 30 losual ssalaml z+lawurtCs pauyap-l[aM 3~~0s s! ‘0 4 3 ‘I1 ( d vn~ ;j ‘d, ) pz!l!u! ualz@ r! JO~ = (I)$/ 1,n ~l~luenb aq1 “‘fj 3 (a)0 3 h uo!i!puo~ (.maisi(s aql 30 i(%aua aql salouap g i(lgur?nb aql lr?ql 11waJ am .lsuon = 8 _pawos!., :uop!puo:, sno!Aqo ??u!Mo~lo3 aql u! paujeluo3 uo~1em~o3u~ aqi paldope amq aM ([l’~) put? (6’~) 8u!i\!lap u! ieqi 1~3 aqi sm~yuo3 ICldru!s 11 ‘paapul +u!s!ld_ms aq IOU plnoqs luawa3cldaI s!ql ‘am03 30 ‘(H # ‘)H ‘H 3 “H) “‘H 1(q H azzldal lsnur 3~ ‘0 = (~)~,.,,n a.mq ias am am!s ‘JaAaMoH ‘H a3eds sSpwt) uo amstmuo~d !ysMoq3nIours-u!alsu!~ aql pauyap amq aM [b] ‘3al u! ‘Zugeads iClas!aald) .wn!.tqg!nba lewaql u! suo!lenlmy luapuadap-aury aql %!qgsap suogmn3 uo!lt?laJlo3 3ql alelnDIe3 01 s! uogoas s!ql3o asodmd aql “d, 30 cm!l u! uognlom aql su.~ailo% ieql (c’z) uogmba ~egua~a33!p-o.Galu~ aqi 8u!sn .pau!umlap pur? pauyap aq hem salqc!.nzA uropuw alqw8alu!-almbs 30 suogmn3 UO!~~I~JIO~ lsaldw!s aql q3!qM LI! SiCBM 3UIOS alaq] UMOqS aAl?Lj 3M .[b] ‘331 30 t, Uql3,aS u! “JJ uo ‘ri amsealuold !ysMoq3nlours-u!alsu!~ aql Apnls 01 u@aq lsly ah
‘patomp
Aplew!-rd
am
YJOM s!ql 3o suog3as Zu!u!swai aql ‘smalqold asaql 0~ .ald!m!ld s,uumuz~~o~ 30 [L] uo!maldlaw! sGu!alsug uodn passq suogenlmg 30 hoaql luapuadapu! -am!1 aql %~!lap!suo~ alaM auo 31 se uo!ssnmp aql u! paaDold uaql pm
‘rl ampow! AIUO paau auo ‘(I)+,,D 30 suogenlmg uxn!lqq!nba aql alenpz,ta 01 laplo u! 'splo~ lay10 UI .([p] 'Jai u! 1‘p uogztas ‘33) “H uo uo!imn3 je~!.tpuqh u! pay!lsnI aq 111~ auo ‘uogenba LIB qsns (11’~) .ba’Jo suogmgdru! 1mgzmd aql ‘[p] I? %U!aq st? “( ua,@ ‘paapuI
“v~,A d_‘j “d)) 8u!leaJi .1ea13 al!nb aq plnoqs
“H uo ‘r/ amseauold um!lqg!nba asoqm put? (ii)7
suorlXIn3
!ysMoq3n~omS-u!aisu!~ aqi Icq paq!map air? suo!lenlmg uogmn3 P se ‘d, 30 ‘6, anlm ~E!J!u! aql 8ugaldlaluI ‘(E’E) pue (z*(;) icq ua@ aie (3)~ asoqm c lowado
ayl 103 alay
mop
aA2eq aM se ‘(L t;i)“~
mu!1 uo!lr?xe~al luapuadapu!-d aqi icq I! i!mgap 01 sn MOM 01 se aIdm!s OS aq 01 q3eoldde aql laadxa 01 aIqeuosea1 mu s! 11 Inq ‘ICpsca pawlsuowap aq um um!lqq!nba ‘@I%]3
.suopenba
01 puail 3!laug
aql30 waioaql leiaua% e ‘(s’z) iaq$o pm ,,suo!l~ialu! 30 s~el,,
&mbau! aql qi!M iaqlo 103 swaioaql
Z. Banach,
S. Piekarski
E,[a”‘P(t)] = 0
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
349
(4. la)
)
(4. lb) where
E := 4nl3N and#6 E(CZ(CZ)denotes, as we know [3,4], the natural projection of the ath tensorial power IE” of lE onto the irreducible subspace Ker,Tr of symmetric traceless tensors of degree (Y. The symbol { 0 ) 0 } represents the scalar product for L2(!ZZi) as determined by eq. (A.2) in appendix A of ref. [4]:
(4. lc) The result (4.lb) can be viewed in a slightly different way if one introduces a somewhat wider class of random variables Xi, (t, r+G) E [0, m) x HO, on (H,,, pL,), (4.2a) 9, :=
U(t) cp
9
clef&.
(4.2b)
Here {U(t); t 2 0) is a one-parameter semigroup on H, induced by -T-IL.. We shall refer to the collection of random variables {X’,; (t, $) E [0, m) X H,,} as a generalized random field over [0, ~0)x H,,. In view of (4.2), X$,X$? defines a cylindrical function on Ho [17-191, and since this function is integrable on Ho with respect to CL,, we obtain
-w&q
= (dJ,,U(t + $1+2>,.
(4.3)
Our interest in studying X’,(q) lies in the fact that pL,is not a measure on H,,. A theory based upon the fluctuating kinetic equation suggests in turn that the total time-correlation function of {Xi; (t, I/J)E [0, ccl)x Ho}, denoted by X>,X&, should be invariant under the distance-preserving transformation 3, defined by S,(XLIXS,,) := X~~“X>~“, where a E (0, to). Indeed, considering a situation in which the time evolution of cpr is governed by the stochastic Boltzmann-Peierls equation, one can prove that the equilibrium second moment of the generalized random field converges in the well-defined limit, not to EF[Xk,X$J as given by (4.3), but to (4.4) X6We assume that a crystal of volume V has N atoms (cf. section 2.1 in ref. [4]).
3.50
Z. Bonuch,
Eq. does
(4.4) not
However, the
is consistent
contradict
/ Einstein-Smoluchowski
space
positive,
promeusure
with the laws of statistical
the
principles
of large
since the Einstein-Smoluchowski
Hilbert
strictly
S. Piekurski
H,, for which the proof
the
of (4.4),
assemblies
although
possible,
Due to this fact, here we will not present
it. Rather, (4.4),
between
(4.3)
and
[5,6]
of discrete
equation
and thus particles.
~~ is not a measure
operator
important
difference
mechanics
promeasure collision
vs Bolrzrnattn
k is self-adjoint is extremely
in order
difficult.
to “remove”
we shall restrict
on and the
our attention
henceforth to the s = 0 case. Such an assumption is altogether natural, because in the calculations WC propose in section 5 the extra condition (t - s( = t + s plays a special role. In any event, it is reasonable to set down the following. H~pothesi.s.
According
to (4.3)
and (4.4).
we may note
that
if ]t - sl # I + s.
then the total time-correlation function is not governed by the linearized (linear) Bohzmann(-Peierls) equation. To calculate X:,,,Xi,, without going into detailed microscopic investigations, we need only discuss the mathematical properties of the stochastic Boltzmann(-Peierls) equation. Given eq. (4.3) for s = 0. the next stage in the analysis is to use this equation to estimate the thermal conductivity coefficient K., . Thus, in section 5 we shall discuss the Green-Kubo approach to transport processes (20,21]. Before doing that, we look at the problem concerning the evolution in time of the Einstein-Smoluchowski promeasure p, of scale parameter E. We recall that p, is a family {p,,,?: K E %\j(H,,)} defined as follows: (i) ,@(N,,) is a collection of (closed) subspaces of H,, of finite dimension together with the partial ordering relation C ; (ii) p?,,<. R E .\:1(H,,), is a Bore1 probability measure on R as given by eq. (4.1) in ref. (41; (iii) if $(H,,) 3 R C S E.\:,(H,,). then P,,~~(G) = I*+,,~(0) for every Bore1 subset G of R, where L) is a (Borel) cylinder in S induced by G and the orthogonal projection of S onto R. (For the precise definition of p,, see section 4.1 in ref. [4].) Let D, be a cylinder in U(t) H,, = H,, associated with U(t) G and the orthogonal projection of U(t) H,, onto U(r) R (G C R E;\:(H,,)). Moreover, we assume here that pi,r(,) ,? (U(r) G) is the probability of finding the system in the “macrostate” cp~E D, at time t where t 2 0. Clearly, observing that ‘p, = U(t) +I,,, we wish to regard this probability as unchanged in the course of time; thus we set
PL:, ,0(t) G) = PL,.dG) n(r)
Alternatively,
(4.5a)
we obtain (4.5b)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
351
A family & := {/-&:R~WU
is called the reciprocal image of p, by U(t). Since T(t) is a continuous linear mapping from H,, onto itself, one can prove that pLf,is a promeasure on H, (cf. the text at the top of p. 498 in ref. [17]). In rough summary of these results, we may say that the procedure we have used in deriving (4.5b) is meant to replace the theory based upon pL,and (2.3a) by the theory formulated in terms of &. In this context, it should be stressed that & cannot be extended to a measure on H,.
5. The thermal
conductivity
The actual calculation of a time-correlation function is a very difficult task in general. One of the appealing aspects of our model of the Boltzmann-Peierls equation is that it gives not only a formally exact representation of various time-correlation functions, but that it also does this in terms of integrals for which the thermal conductivity coefficient K~ can be evaluated explicitly. Notice that our model is not simply a mean free path model. In the Green-Kubo theory [20,21], the thermal conductivity coefficient K~ correlation function can be expressed in terms of the unequal-time E, t q(t) o q(O)1by
(5.1)
dt ~,MtPq(0)1,
V being the volume of a gaseous system and q(t), t 2 0, being the random heat flux vector. To illustrate how (5.1) may be applied, directly from (4.lb) and (3.7b) we see that
dt EJPB(t)
0
tPp&VI = @a+ l){A, h&J .
(5.2)
So as to obtain the explicit formula for K-,., we consider now only the case in which ?A?’ = (0, W) and R(y) = cy (the isotropic and dispersionless (phonon) model). Then the dependence of q(t) upon ~“~(t) is characterized by the following particularly simple expression (cf. also eqs. (4.13) in ref. [4]):
q(t) =
2
(g)“* (k3g2
Given (5.1)-(5.3),
the calculations,
al’o(t).
though elementary
(5.3) and straightforward,
352
2. Bunach,
are lengthy, technical
S. Piekarski
I Einstein-Smoluchowski
promeasure
so we shall only set down the final result for
steps involved
in arriving
KS Boltzmunn equation
K., ,
not describing
the
at it:
(5.4a)
where
7.J‘?):= 7(T) [ 1 + V (J:=
-,
N
F n%(T) c, .s,, I,,,(+$z)“]’.
c, = cl,,-
$d,
(5.4b)
(5.4C)
The real difficulty lies in evaluating the integral on the rhs of (5.4a). However, the successful analysis of this integral now seems to be possible, for eqs. (3.2) and (3.3) have removed all the problems that dogged the GreenKubo approach to transport coefficients, and machine calculation should be able to deliver for the integral (5.4a) approximations as accurate as desired. Certainly the more serious problem is that of obtaining the accurate expression for T(T). Indeed, it should be evident that the presence of the defects and imperfections in a crystal does not contribute very much directly to the total thermal conductivity coefficient K~, because the ;{-contribution is generally much smaller than the contribution arising from the mean free path approximation. In other words, so long as the total number of defects and imperfections in a crystal is small enough for the relaxation time ~~(2) to remain independent of z, we can replace TV by T(T) in eq. (5.4a). Information about the way the coefficient T(T) depends upon T can then be found by analyzing the results described in detail in the experimental papers (cf., e.g., section 2.6 in the review paper by Beck et al. [15]). Nevertheless. it is important to remark once more that. using the GreenKubo method, we derived, for our blurred (but nontrivial) image of the “true” collision operator J(a). a formally exact representation of K~ which is identical with that found in the first Chapman-Enskog solution [2] of the BoltzmannPeierls equation. This then gives a justification for basing the computation of transport coefficients peculiar to the kinetic theory of quasiparticle gases on the time-correlation functions.
6. Final remarks We went beyond the Einstein theory of fluctuations (71. which deals explicitly only with a finite number of degrees of freedom, inasmuch as we could
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
353
consider gaseous systems described by continuous distribution functions f(k, t). Since a complete knowledge of the distribution function f(k, t) is formally equivalent to a knowledge of all its moments, there would appear to be no reason a priori to expect that the system in question should be represented in terms of a finite set of state variables. One would instead be led to anticipate the possibility of a much more complicated (and interesting) description. That these sorts of complications actually do arise has in fact been demonstrated in ref. [4] for the special case of the quasiparticle gas. The connection between the thermodynamic theory of equilibrium fluctuations and the kinetic theory of gases was stipulated by means of a postulate, the essential ingredient of which was the precise definition of the EinsteinSmoluchowski promeasure EL,on the proper subspace H, of Grad’s space H [4]. This promeasure paved the natural way for the calculation of the mean-square fluctuation in the value of the random variable a”,‘(t) determined from f(k, 2). At the same time, our construction of pu, was largely self-contained and did not rely on specialized statistical knowledge. This means, of course, that we decided to detach the infinite-dimensional analog of Einstein’s theory from its connections with statistical mechanics. Given this state of affairs, we are aware of the fact that the quantitative discrepancies between the approach taken by the present paper and statistical mechanics vanish asymptotically for large (macroscopic) systems, and that the two formalisms can be used interchangeably. Grad’s method of moments [13] has a much wider applicability than the most widespread approach in the literature would suggest. The reason for this is that in the kinetic theory of gases it is imperative to use Grad’s ideas as a point of departure, since these ideas are of interest in obtaining the generalized continuum-like equations#‘. However, in addition to the problem of allowing for the continuum models of less traditional form, there exist a number of other theoretical issues that one might raise. Thus, for example, of great interest for our purpose here is the relevance of Grad’s procedure [ 13,3] to the Einstein-Smoluchowski promeasure pL, on H,, [4]. Indeed, Grad’s method of moments implies that one can introduce in a sensible way a particularly important class of cylindrical functions on (H,,, pE), thereby simplifying the calculations enormously. To summarize, in the conventional presentation there is a clean-cut distinction between the Einstein theory of equilibrium fluctuations and the kinetic theory of gases. Actually, however, the application of pu, and Grad’s method of moments bridges the gap between the two points of view. Thus, it would also seem natural to apply the techniques developed in our two papers to the #’ These
equations
are not based
on constitutive
relations
of the Euler
or Navier-Stokes
type.
3.51
Z. Bunuch.
investigation
S. Piekarski
of fluctuation
/ Einstein-Smoluchowski problems
promeusure
related
to both
vs Bolrzmann
classical
eyuution
and relativistic
[ 26,271.
gases
Appendix
A. Comments
on the Tchebychef
polynomials
When R(y) = cy, c being a strictly positive constant. the Tchebychef function A,(z; T) becomes a polynomial in .z E .% i of precise degree p. We shall consider here only the still more special case in which !8 (r = (0, x). As we have shown in ref. (41, the system {A,; p = 0, 1.2,. . . x} is orthonormal, that is. % {A,(Ay)
the weight
vv(2;
If
dz N’(z; I-) A&;
=
function
T) A&;
VV(z; T) being
given
T) = 6,, 3
(A.1)
by
gj‘ (ey):
T):=
(A.21
denotes the highest coefficient of ACc(z; T), we may assume that has been so chosen that K,(T) >O. Although the isotropic and dispersionless (phonon) model (R(y) = cy, 8 ,; = (0, x)) is so simple as to be almost K~(
T)
K,(T)
degenerate, our treatment of it illustrates problems associated with the mathematical and solved.
the usual way in which specific properties of A,(z; T) are set up
If we write (A.3a) w,,(z) := -74”(z; T) [‘NJ’(z; T)]
’ ,
(A.3b)
w,(z) := VV(z; T) [%“(z; T)]
’
(A.3c)
9
we obtain
w,,(z)c 1
3
w,(z)S5+3z’.
Given the Laguerre polynomials functions Z@(z; T) defined by
(A.3d) L:)(z)
[23], one
can easily
show
that
the
(A.4)
Z. Banach,
S. Piekarski
are orthonormal
W&l
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
355
with respect to ‘W(z; 7’):
= 1 dz W(z; T) ~,(z;
T) .Z,,(z; T) = SPY.
(A.9
0
For each integer p, p 3 0. we define -yPas follows: -rP := ([AplApl)“2
(-4.6)
;
thus rP > 0 and rp G ({ApjAp})l’Z = 1. If we replace now Ap(z; T) by a linear combination of {L&(z; T); 0 =Sv s p} with constant coefficients cpV, eq. (A.6) becomes ri = c& + csr + . * . + c&, and from Cauchy’s inequality, -yPs 1, and (A.4) we conclude that#’ [23]
where, for convenience, we set L?/(z) := 0. Using the inequality (3) as formulated on p. 334 in ref. [23], property (A.7) shows us immediately that for all z in the finite interval (0, ,I, l> 0, there is a positive constant C, not depending upon z E (0, I), such that
(‘4.8) Regardless of (A.7) and (A.8), we also know [25] that the following relation holds for any three consecutive orthonormal polynomials A,, y = p - 1, p, p + 1: zA,
= ‘Q(‘$+,)
-1 Ap+, + npAp +
K~-,(K~)-‘A~_,
;
(A.9a)
here np is a constant not depending upon T. Hence, applying the definition of { 0 1o} (cf. eq. (A.2) in appendix A of ref. [4]), we obtain in view of (A.l) and (A.9a)
Kp(‘Q+~)-’ = b$h$+J
9
no = {zAplAs>.
xXFor the definition and properties of L:‘(z), on p. 334 in ref. [23].
(A.9b)
see pp. 295-302 of ref. [23] and the inequality (3)
Z. Banuch,
356
S. Piekarski
We now record
I Einstein-Smoluchowski
two easy but important
promeasure
consequences
vs Boltzmann equation
of (A.3d),
(A.7)
and
(A.9b). Theorem.
There
exists a positive
“/AK@ +I ) ‘a(p+q”,
C such that
na sC(j3
Proof. To prove (A.lO), ([Lw,(AZ])‘~‘([ZW,(A~])‘~~. that the rate of growth
constant
+ 1)“.
(A. 10)
it is sufficient to observe that ({ZAP (A y } j s The theorem then follows from (A.7) and the fact
of w,(z)
is no greater
than
n
that of z’; cf. (A.3d).
In section 3.2 we defined a solution of the Boltzmann-Peierls equation (2.3) as one that is determined completely by (3.9) and (3.11). The inequality (2.5) supports the existence of a broad class of such solutions, and so it is permissible to conclude that, for each choice of the initial value ‘p(,E D(L) C H,, of ‘p,, t E [O, r), the series (3.9a) converges in the sense of the norm ) 0 It, in H to an element ‘p, E D(L) C H,,; in addition, if ‘poE D(L), then lim,__, j(p,j,, = 0. Aside from these observations, it is natural to ask whether the series (3.9a) converges pointwise for (z, g) E !Z ,T x IK, and, if so, whether it converges to a classical solution of (2.3). We shall not present the details of these problems here. because such problems belong to mathematical physics rather than to statistical mechanics. All we shall intend to state is that the elementary properties of Ap(z; T) listed above permit us to establish quite easily sufficient conditions for the pointwise convergence of the series (3.9a). By way of digression, our method of dealing with A,@: T) can be immediately generalized to the case of relativistic gases [26,27].
Appendix
B. Comments
on the isotropic
and dispersionless
(phonon)
model:
($3: = (0, m), L&(y)ly = c = const) It
was
= h(z;
shown
in
T) = (k,T/kc)z
VW@; T) = (%I3 Now we define
appendix
A
of
ref.
[4]
that
y = (kL3T/Z1c)w(y; T)
and that (ezy;,2
the function
L(y)
(B.1) as follows:
L(y)=y-2 b ,, , 0 CVSY,
U3.2) being
some fixed nonnegative
coefficients.
We next introduce
the
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
symmetric traceless moments N”*@E Ker,Tr N a-0 .-.-
i
dk
vs Boltzmann equation
357
or f - f,,:
Y”(iL4.y; T)lB+l(f-fo)=
$(cp,Yap%,.
(B.3)
LF Starting with (2.3), by use of (B.2) and (3.3) we may derive the equations of transfer for N*.‘:
The above result also shows that eqs. (3.7) remain valid for L?(y) = cy if in their statement S,(y; T) = Sp[h(z; T); T] are replaced by [o(y; T)]” = 2’: N”%)
=
I dk iP(y; T) ya(g) l! = Gh,? 47WH 9
[w(Y;
T)lp+‘[fW) -.M~)l
cpE W) .
(B.5)
Since in the special case of an isotropic and dispersionless (phonon) model the Tchebychef function AB(z; T) is a polynomial with respect to z of precise degree /3 (in which the highest coefficient K~(T) is positive), we easily arrive at
a”.‘@)= boo,, lFY”A,), >
‘POEW).
03.6)
References [t] [2] [3] [4] [5] [6] [7] [8] [9] [lo] Ill] [12] [13] [14]
R.E. Peierls, Quantum Theory of Solids (Clarendon, Oxford, 1955). V.L. Gurevich, Kinetika Fononnykh Sistem (Nauka, Moscow, 1980). Z. Banach and S. Piekarski, J. Math. Phys. 30 (1989) 1816. Z. Banach and S. Piekarski, Physica A 180 (1992) 309, this volume. P. Resibois and M. De Leener, Classical Kinetic Theory of Fluids (Wiley-Interscience, New York, 1977). R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley-Interscience, New York, 1975). A. Einstein, Ann. Phys. (Leipzig) 33 (1910) 1275. L. Tisza, Generalized Thermodynamics (MIT Press, Cambridge, MA, 1977). K. Huang, Statistical Mechanics (Wiley-Interscience, New York, 1963). D.A. McQuarrie, Statistical Mechanics (Harper and Row, New York, 1976). J. Szczepanski, Physica A 157 (1989) 955. C. Truesdell and R.G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York; 1980). H. Grad, Principles of the kinetic theory of gases, in: Handbuch der Physik, vol. 12, S. Flugge, ed. (Springer, Berlin, 1958) p. 205. U. Weinert, S.L. Lin and E.A. Mason, Phys. Rev. A 22 (1980) 2262.
3%
2.
Banuch,
S. Piekarski
I Einsrein-Smoluchowski
promeasure
L’S Boltzmann
equaiiorl
[IS] H. Beck, P.F. Meicr and A. Thelhmg. Phys. Stat. Sol. A 23 (1974) II. 116) K. Maurin, Methods of Hilbert Spaces (Polish Sci. Publ.. Warsaw. 1972). [ 171 Y. Choquet-Bruhat. C. Dewitt-Morette and M. Dillard-Bleick. Analysis. Manifolds. and Physics (North-Holland. Amsterdam. 1977). [IX] H.-H. Kuo, Gaussian measures in Banach spaces. in: Lecture Notes in Mathematics. vol. 463. A. Dold and E. Eckmann, eds. (Springer. Berlin, lY75). [ 191 C.W. Burrill. Measure. Integration, and Probability (McGraw-Hill, New, York. 1972). [ZO] M.S. Green. J. Chem. Phys. 20 (1952) 1281. [2l] R. Kubo, J. Phys. Sot. Jpn. 12 (lYS7) 570. [22] Z. Banach and S. Piekarski. J. Math. Phys. 30 (1YXY) 1x26. [23) G. Sansone. Orthogonal Functions (Interscience. New York. IYSY). 1241 P.L. Bhatnagar. E.P. Gross and M. Krook. Phys. Rev. Y3 (lYS4) 511. (251 G. Szego, Orthogonal Polynomials (Am. Math. Sot.. Providence. RI. IS3Y). [2hl K.S. Thorne, Mon. Not. R. Astron. Sot. 191 (IYXl) 430. 127) N. Udey and W. Israel, Mon. Not. R. Astron. Sot. 19’4 (19X2) 1137.
A 180 (1992)
336-358
North-Holland
The Einstein-Smoluchowski promeasure the Boltzmann(-Peierls) equation
Received
25 April
Beginning function
from
f(k.
inversion
of
1991
the specific model of the Boltzm~~nn~Peicrl~ the
I).
time-dependent
Boltrmann’t,
Boltzmann-Pcierls
relation.
equation
values of flk.
i\ convcnicnt
distribution study
(phonons.
carefully
in time
a kinetic
studied.
The
Ructuationh
distribution
equation
for the distribution
IS developed
function
f(k.
I)
magnons.
rotons.
etc.).
that the randomness
from
that
Einstein‘s
satisfica
The
hy the Einstein-Smoluchowski
approximation. of Ructuations
version
The
of the Green-K&o
exact formula
for
in this paper. cmcrgcs as a portion
equilibrium
in the moments
the thermal
assumption distribution promcasul-c
analog of Einstein‘s
promeasure of fl/,.
t).
CL* is used to
Some
approach to transport conductivity
the
the macro-
crucial
m the statistical
01 p, as being the infinite-dimensional
law in the Gaussian with
of
tw charactcrizcd
I) C:III
to think
the correlations
associated derived
The
into the present approach is simply
of the initial I_L,. It
theory
is to he vicwcd as ;I random variable characterizing
state of a gas of quasiparticles cntcring
versus
cocfticicnt
problems
proccsscs K,.
which
a~-e i\
of the general formalism
1. Introduction A convenient way of describing the gas composed of quasiparticles (phonons, magnons, rotons, etc.) [l-3] is to specify a one-point distribution function f(k, t). Since this function determines the number of quasiparticles with quasimomentum hk which is found in a system of volume V at time t (cf. the text before eq. (2.3a) in ref. [4]), one may suppose that f’(k, t) satisfies the space-independent, integro-differential equation, the so-called BoltzmannPeierls equation. Regardless of the program of obtaining such a kinetic equation for a one-point distribution function, in the previous paper (41 we set up a general 037%4371/92/$05.00
0
1992 - Elsevier
Science Publishers
B.V. All
rights
rcservcd
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
337
framework for a systematic theory of fluctuations occurring in quasiparticle gases “kept” in an overall equilibrium state. Thinking of the kinetic theory of gases [l, 2,5,6] and the Einstein thermodynamic theory of equilibrium fluctuations [7,8] as a unified structure, the physical picture which we have here in mind is not of course very similar to what one normally considers in the traditional description of both classical and quasiparticle gases: the distribution function f(k, t) is to be viewed as a random variable characterizing the macrostate of a system [4,9, lo]. In order to make this picture precise, we need prescribe the initial value f(k, 0) of f(k, t) and observe that, because every datum f(k, 0) evolves in time deterministically according to the Boltzmann (-Peierls) equation involved, any randomness will arise, then, only from the randomness in the statistical distribution of the initial values of f(k, t) in the ensemble of similarly prepared systems. In addition to these remarks, there remains the interesting problem of specifying in a sensible way what one means by a “statistical distribution” of f(k, 0). Although the initial experimental preparation of an ensemble may be quite arbitrary in principle [ll], the crucial assumption that enters into the present discussion is simply that the fluctuations of the initial distribution probability function f(k, 0) are characterized by the Einstein-Smoluchowski promeasure pue of scale parameter F. (As noted already in ref. [4], the reciprocal of the scale parameter E is proportional to the volume V of a system.) The first paper in this series [4] constructed p, and developed a manifestly infinite-dimensional analog of Einstein’s thermodynamic theory of equilibrium fluctuations [7,8]. The purpose of the second paper is to demonstrate how that approach may be used to formulate and answer the concrete physical problems in the context of Maxwell’s equations of transfer [12]. Before considering them in detail, we must now explain the notion of Maxwell’s equations of transfer and describe the general features of the moment method of solving the Boltzmann-Peierls equation [13]. If one multiplies both sides of the Boltzmann-Peierls equation by the IkenberryTchebychef functions forming a complete set [3,4] and then integrates the resulting expressions with respect to k, one obtains a system of infinitely many relations to be satisfied by the moments of f(k, t). Due to the completeness of the Ikenberry-Tchebychef functions [3,4], this system of relations, Maxwell’s equations of transfer [12], is formally equivalent to the Boltzmann-Peierls equation. At first sight, the equations of transfer seem to be a likely tool, but any attempt to study the time-behavior of the moments off(k, t) runs straight onto the nasty problem of evaluating the complicated collision integrals appearing on the rhs of those equations. And, even if one invents an exact method for the evaluation of the collision integrals, one may conclude very little a priori.
338
Z. Bunach,
Indeed,
S. Piekarski
as is well known,
say, the first II moments moments
alone
this should transfer
the equations
be true would
for the moment the “forward
promeasure
that govern
of f(k, t) do not become
until the remaining
m’ > m. Such a general called
I Einstein-Smoluchowski
higher
seem apparent of order
feature
coupling
order
the evolution
in general moments
m contains
the moments equations
of the equations
in time of.
equations
for these
are calculated.
once one realizes
of Maxwell’s
L’SBolrzmann eyuation
of order
of transfer
of moments”
That
that the equation
of
m’ where
is sometimes
[ 121.
Thus, what various authors have typically done (cf.. e.g., refs. [13, 141) has been to introduce some truncation scheme. and then to write down as a starting point of the discussion a closed system of equations for finitely many moments. Of course, this leaves the distribution function f‘(k, t) largely undetermined, since only the infinite set of moment equations can determine f(k, t). The assumptions based upon the truncation scheme, albeit drastic, are not unreasonable. And indeed, if nothing else, one may argue for their validity making use of the numerical experiments [14]. On the other hand. the exact solutions for Maxwell’s equations of transfer are possible, and it is very instructive to explore in some detail their properties in each nontrivial case. This paper will adopt the latter alternative. Precisely speaking, calling attention to the support and interpretation given by Gurevich [2] to a special case of the collision operator, we obtain exact equations describing detailed properties of the Ikenberry-Tchebychef expansion coefficients LI’“.’ of the distribution function [3,4]. Our equations are the quasiparticle analog [3] of those derived by Grad’s method of moments [13]. Then, since we do not consider the space-dependent effects”, the next stage in the analysis involves the further specification of the collision operator so as to arrive at the exact solution of Maxwell’s equations of transfer. This is of course nothing more than another way to look at the problem of solving the Boltzmann-Peierls equation. The above observations emphasize the essential similarities between the kinetic theories of quasiparticle and classical gases [12, 151. Now, the net upshot of the preceding paragraphs randomness in the statistical distribution of the initial
is as follows. values of f(k,
That the t) can be
described by the Einstein-Smoluchowski promeasure p+ is really a fact of crucial importance for, given only that f(k. t) satisfies the deterministic Boltzmann-Peierls equation, one may discuss the time evolution of I*, . In particular, observing that the collision operator is the infinitesimal generator of a strongly continuous semigroup {U(t); t 2 0} of operators [16] acting on the proper subspace N,, of Grad’s space H, one concludes that the reciprocal image of pF by U(t) is a promeasure on H,, [17-191. By the way, in the previous paper ” The distribution function f(k, I) represents directly before q. (2.31) in ref. [4]).
the global
property
of the system
(cf. the text
2. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
339
[4] we introduced the Hilbert spaces H and H, without making any explicit reference to the Boltzmanm-Peierls equation. Leaving aside these technical issues, in our approach the crucial physics will be associated with the Ikenberry-Tchebychef expansion coefficients ~“‘~(t) of the distribution function f(k, t). To be completely explicit, one particularly important question to be investigated concerns the calculation of the unequaltime correlation functions of the random variables ~“‘~(t). In fact, the macroscopic quantities describing transport processes can be written as linear combinations of finitely many integrals calculated from the suitably chosen timecorrelation functions of ~2~~’ (t). This will, for example, enable one to construct a kinetic version [lo] of the Green-Kubo theory [20,21]. Indeed, adopting the results established in ref. [4], the desired calculations can really be justified on a formal level, and the exact formula for the thermal conductivity coefficient K= which we derive in this paper emerges as a portion of the general formalism. And finally, because the proposed form of the collision operator is not very complicated, one can actually address more difficult problems than simply albeit somewhat obtaining the exact formula for K=. It is straightforward, tedious, to construct the stochastic Boltzmann-Peierls equation and to discuss its fundamental properties. This is a problem that deserves some more attention in future investigations. In summary, it should be stressed that, for our model of the BoltzmannPeierls equation, all the physics associated with Maxwell’s equations of transfer and the Einstein-Smoluchowski promeasure p, can be treated exactly; it is only the intermediate calculations leading to the notion of an equilibrium promeasure that are to be considered in an approximate way [4]. By focusing upon conceptual issues, this paper will emphasize the essential connections between the kinetic theory of quasiparticle gases [l, 21 and the Einstein thermodynamic theory of equilibrium fluctuations [7,8]. Here we proceed as follows, Sections 2 and 3 describe a simple collision model and construct exact solutions of the kinetic equation by means of solutions of the system of equations for the moments of the distribution function f(k, t). Given the Einstein-Smoluchowski promeasure CL, on the proper subspace H,, of Grad’s space H (cf. ref. [4]), section 4 derives a variant of the time-correlation function formalism originally thought limited to a gas composed of classical molecules [5,6]. In this derivation, the role of Maxwell’s equations of transfer introduced in section 3 will be clearly evident. Then, in section 5 we adopt the Green-Kubo approach to transport processes [20,21] and calculate the thermal conductivity coefficient K=. While most of the explicit calculations are carried out without first specifying the detailed form of the Tchebychef functions Ap(z; T) [4], appendix A takes note of this restriction and explains, if not directly then at least through some useful inequalities and
340
Z. Banach,
theorems, model tion
S. Piekarski
a way of removing
(O(y)
iy = const).
3.1 is included
section
I Einstein-Smoluchowski
it for the isotropic
The auxiliary
as appendix
vs Boltzmann
and dispersionless material
the work
associated
equution
(phonon) with sec-
with final
remarks
of
[4]),
distribution
6.
Terminology
As we know
equation
and basic definitions (cf. the
text
before
function f(k, t) determines the number system of volume V at time t. The evolution
technical
B. We end
2. A model of the Boltzmann-Peierls 2.1.
promeasure
of f(k. t) is given
d,S= J(f)
eq.
(2.3a)
in ref.
of quasiparticles kinetic equation
the
with wave vector k in a that governs the time
by (2.la)
’
off due to collisions. where J(f) represents the rate of “increase” the behavior of f(k, t), it is necessary to determine J(S) explicitly. [3], we assume that the collision operator and t into another one, is characterized
[J(f)1 (k. t>= -
&
J(o), by
which transforms
To discuss As before
a function
of k
lf(k, t>-.M~>l
+
i dk, {g(k> k) f(k. t) [f(k 0 + 11
-
g(k k,) f(k, f>[f(k,, f) + 11)
(2. lb)
3
where w(y; T):=hL?(y)(k,T)-’
,
y := k,; ‘Ikl ,
y, := k,;‘lkJ
g := (kl-‘k
g, := (k,(-‘k,
,
s(k,
k*):=
L(Y) x(s”s*)
4, := (6&V/V)“3
, .
,
S(Y -Y.>
3 (2. Ic) (2. Id) (2.le)
Here T(T), L(y) and x(t), (51~1, are certain positive functions of their arguments, and S(y - y*) stands for Dirac’s delta. Without further comment, we shall use those symbols that either are reasonably standard or appear for the first time in ref. [4]. Here let us only recall that T is the kinetic-theory
Z. Banach,
S. Piekarski
/ Einstein-Smoluchowski
promeasure
vs Boltzmann equation
341
temperature corresponding to f and that f,(w) is the Bose-Einstein distribution function which has the same principal moment as f. The quantity 7 denotes in turn the effective (k-independent) relaxation time. In general, this relaxation time is a function of the temperature T: 7 = T(T). Of course, 2& and k, are the constants of Planck and Boltzmann, respectively, and fifl( y) represents the energy of a single quasiparticle with quasimomentum hk. The specification of the meaning of the symbols N and V is not necessary here, but can be found in section 2.1 of ref. [4]. Insofar as the aggregate of phonons in an insulating crystal at low temperature is concerned, the rate of change of f(k, t) that arises from the collisions associated with anharmonic contributions to the potential energy has already been calculated [2]. Unfortunately, the resulting nonlinear expression for this rate of change seems to be extremely complicated, and the idea to replace it by -Y’( f-h,), or by the (more realistic) proposition of Callaway#*, rests implicitly on the hypothesis that a large amount of detail contained in the definition of the collision operator J( 0) is not likely to influence significantly the values of many experimentally measured quantities. Obviously, after having introduced 7, the mean (k-independent) relaxation time, as a tool to solve analytically some problems standing behind (2.la), we might as well try to give the first term on the rhs of (2.lb) quite another interpretation. (In passing, it should be stressed that, in the present case, T is a constant.) With respect to the integral appearing on the rhs of (2.lb), the collision operator has such a structure if one considers, for example, the elastic scattering of phonons by different defects (imperfections) of a dielectric crystal. But we have explained already these questions, and of general interest in this regard is our discussion in section 3 of ref. [3]; cf. also Gurevich’s monograph [2]. Before going further, however, we would like to point out that S(k,, k) is nothing else than the transition rate for an elementary process which destroys a single quasiparticle in the mode k, and creates one characterized by the wave vector k. Of course, we assume that %(k,, k) = %(k, k,). Given (2.lc), the S-function allows only transitions satisfying the energy conservation law. In order to describe %(k, k,) completely, we must specify the scalar functions L(Y) and X(5), /5]~1, and to this aspect of the definition of %(k, k,) section 3.1 is primarily devoted. It is a simple matter to prove that the operator J(o) as given by (2.lb)-(2.le) reproduces correctly the essential features of the collision operators for quasiparticle gases [3].
X2 Cf., for example,
section 6.1 in ref. [22].
2. Banach,
342
2.2.
General properties
If the distribution (Z.la)
i Einstein-Smoluchowski
S. Pirkarski
is written
promeasure
d,q
=
function
f(k,
t) obeying
the Boltzmann-Peierls
I 7
equation
as
(2.2) -
equation
of the collision operator
f(k,f) =h(w> [I + A(w) cp(z,g. t; 731substituting
L’S Boltzmann
k
into
(2.la)
CFE
4, .
(2.2)
yields (2.3a)
VT [ 161 dense subspace
where L is a linear mapping from a suitably chosen Grad’s space H,, onto H,, (H,, C H). If we set
D( 8) of
o_:=I+;j, I being
(2.3b)
the identity
(;b)(z,
operator,
then
we have
g, t; T) := -7k:)[A(z; T)]‘L[h(z: T)]
I
x dg. Wgog,) [dz, g., f; T)
- dz, g, t; T)].
k.
(2.3c)
To arrive at (2.2), we have replaced k by (z, g). This is legitimate because, as noted already in ref. [4], z may be viewed as a function of y and T; in section 3.2 of ref. [4] we denoted this function by i(y; T). And thus, one can interpret the quantity A(z; T) which appears on the rhs of (2.3~) by saying that h(z; T) is a solution for y of the equation f( y; T) = z. Granted these preliminary remarks, belongs addition, space
we should also remember [4] that A(o) : = w[ 1 + h,(w)] and that 9 to H,, if and only if ( 1, cp) ,{ = 0 (in this context, cf. eq. (2.2)). In I6 is a set of unit vectors in a three-dimensional Euclidean vector
IE. The verification
of eq. (2.3~) for (2.1)
and (2.2)
is immediate;
and,
indeed, one can prove that, regardless of the apparently nonlinear form of J(f), the 3 IS a linear operator on H,,: $9 E H,, for cp E D(L). To see why, it is sufficient to adopt for (2.lb) the symmetry properties of S(k, k_). We may use now the definition (2.3~) of ;‘, to deduce some of the basic properties of O_.By a standard manipulation. we arrive at (2.4a)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzrnann equation
343
provided that both cp and $ lie in the domain D(lL) of the operator II; thus ILis self-adjoint and strictly positive. Since L(y) > 0 and X( 6) > 0, the equality sign in (2.4b) holds if and only if $(z, g*) - $(z, g) = 0 almost everywhere on#3 9?2,‘xKxK. If there exist positive constants C1 and CZ such that y2L( y) c C, for y E .!?i!+ and#3 X([)sC,for[E[-l,l], thenfromCF=@‘,:={kE[E: O<(lc]<&} and the choice made above for ,3 it follows that L is a bounded operator on H,. When#” @!= @, := {k E [E: ]k) > 0}, the problem becomes much more complicated, since in this instance we have to extend the analysis to a broader class of operators, namely, one whose elements are either bounded or unbounded. Applying the Yosida-Hille theorem (cf. ref. [16], p. 219, we easily show that -T-‘IL is the infinitesimal generator of a strongly continuous semigroup {U(t); t B 0} of operators acting on a Hilbert space H,. In addition, if we choose to write ]]U(t)l],, for the norm of U(t), we obtain the following inequality:
In order to understand more completely some fundamental problems in the theory of stochastic kinetic equations, it is very useful to restrict attention to isotropic one-branch Debye models such that @ = EF [4]. However, these particular models constitute only a subclass of all possible ones (@ = @,, CF= @,) for which (2.5) holds, and so it is reasonable to obtain some definite results from (2.5), as well as from the additional assumptions concerning both L(Y) and x(5) (c f . section 3.1), without first having to specify @. In ref. [4] the Hilbert space H and its proper subspace H,, were introduced without explicitly invoking the Boltzmann-Peierls equation. We recall that the basic concepts of our previous paper [4] should be traced to those of Einstein [7]. Thus the fact that the collision operator U. is self-adjoint and strictly positive on the proper subspace H, of Grad’s space H appears as a very natural prerequisite for substantiating the eventual conclusions concerning the link between the kinetic theory of (quasiparticle) gases and the Einstein thermodynamic theory of equilibrium fluctuations. Clearly, since a,( 1, cp(o, t; T)), = 0 for cp(o, 0; T) E D(L) C Ho, we must seek a physically meaningful formalism for “isolated” systems (cf. sections 2.1 and 3.1 in ref. [4]). Indeed, the property of cp just mentioned is equivalent to the statement that the energy % of the gas is a constant. xi For the precise definitions of the segments 92 + := (0, y,) and 9?: := (z,, z,) in (0, m), see section 3.2 in ref. [4]. The definitions of the elementary cells EF and Q, are originally given in section 3.1 of ref. [4].
344
2. Bunach.
3. Maxwell’s
S. Piekarski
equations
I Einstein-Smoluchowski
of transfer
promeasure
vs Rolfzmann
equutiort
(a( y)Iy # const)
3.1. The explicit form for the transition rate The objective the moments
here will be to obtain
of f - A,. In order
the irreducible
equations
of transfer
to do this, one must first specify
for
the functions
L(y) and X( 5) (cf. (2.1~)). Although the inequality (2.5) guarantees that as t+x: the distribution function f( 0 , t) = fo[l + AP( 0, t; T)], cpE H,. approaches that appropriate to kinetic equilibrium, (CPP 1 f; Vi,, the
details
30
of this
(3.1)
3 decay
are
not
known
explicitly.
While
the
problem
is
unsolved today for the genera1 functions L(y) and %( 0, some steps have been made toward solving it (cf., e.g., section 5 in ref. [3]), and our aim here and in section 3.2 is to present in outline the explicit solution in the special case when O(y) iy # const and when L(y) is given by
L(y) =
y-’ i
II 0
b,,[n’(y>]” ,
(3.2)
where O’(y):=dfl(y)/dy and 6,,, 0~ v G r, appear in the role of constant coefficients. Also, we assume that %( [), (51 G 1, can be expanded in a series of Legendre polynomials [23] P,( 5). v = 0, 1.2, . . . , x.
X(S) =
c d,Jy(S)
“=(I
since the analysis conclusion. The propositions
becomes (3.2)
(3.3)
3
simpler and (3.3)
thereby
yet without
are useful
loss of generality
for the reason
of the
they were useful
before [3], namely, that they allow us to calculate exactly the effect of the collision operator O_ on Y”( 0) S, [ A( 0 ; T); T], and thus yield definite and rigorous special results instead of the approximations with which we must remain content if we insist upon using any other propositions. (Concerning the precise definitions of Y”(g), S,(y; T) and A(z; T), see sections 4.2 and 3.2 in ref. [4]. However, to make this development selfsufficient, we recall that, for a( y)/y # const, &( y; T) := [w’(y; T)]’ and s Ifi+ ,(Y: T) := [Y/w(Y; T)l[o’(y; w’(y; T) being the derivative of VI”. w( y; T) := W(y) (k,T)-’ with respect to y. In addition, Y”(g) := [(2a + 1)!!/4ncu!]“‘( 0”s) are Ikenberry’s tensorial harmonics. Finally, h(z; T) is a solution for y of the equation w’(y; T) = z; y = A(z; T).)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
345
The symmetric traceless moments N*,P of f - f,, are defined as follows: N
I
a,’:= dk YYg) 4x T) S,(Y; 7’) (f-f,) CF
=
WP, yaqJm
(Y,p = 0, 1,2,. . . ) a,
T); T]), ,
(3.4a)
cPEf&.
(3.4b)
We introduce (3.4a) just so as to arrive at the equations of transfer for NaXPin which (3.2) and (3.3) cause no difficulty in the rigorous determination of ([Lq, Y”S,), from finitely many moments N*,O of f-f”. Directly from (2.2), (2.3), and (3.2)-(3.4) we find that
d
f
N”.P = _ _1 N”.P -4&c,
c, :=do-
7
i
b, (yj”
“=O
Nn,p+2v ,
No” = 0 ,
(3.5a)
1
-d 2a+*
0..
These particular equations of transfer are relations among moments that the kinetic theory is designed to deliver, and, if r 9 1, they suggest the way in which the essential phenomena of relaxation may occur approximately when a more realistic form for L(y) is used. As observed already in the introduction, the equation of transfer for N”,P necessarily connects the moment N”,’ of order p to the moments Na,p+2” of order /? + 2~ where v = 1, . . . , r. In one very drastic limiting case, namely, when the elastic scattering effects associated with different defects in a crystal were neglected entirely (3 = 0), one could of course obtain the quasiparticle analog of the classical model originally proposed by Bhatnagar, Gross, and Krook [24]. This simplified approach is not discussed here, however, because, irrespective of the forward coupling of our equations of moments, we can evaluate exactly the time evolution of N”,P by solving (3.5a). In (3.5a) we set Noa0 = 0; then the Bose-Einstein distribution function &(w) has the same principal moment as f(k, t): 1 dk fin(y) MO) = jdk C LF
O(Y) f(k, t) .
(3.6)
To go into the detail regarding the R(y) = cy case (c being a strictly positive constant) would needlessly duplicate the ideas already presented, but the discussion given in appendix B for Q(y) = cy satisfies the purpose of this
346
Z. Banach. S. Piekarski
paper:
to show that the method
suffices
to obtain
incorporating
a systematic
all results
The exact solution
3.2.
It is clear
that
/ Einsteirt-Smoluchocc?ski promeusure vs Boitzmcmrt equation
eqs.
of sections
3.1 and 3.2, applied
description
of the trend
in the theory
based
(3.5a)
D(5) C H,,), then elementary general solution of (3.5a) is =
=
toward
from the outset
# cy,
equilibrium,
upon
0(y)
= cy.
of Maxwell’s equations o,f transfer have
a unique
because they are ordinary linear differential If we denote by cp,,(z, g; T) the function
~“.~(r)
to R(y)
inspection
dk [:‘( y; T) Y”(g)
~:,(cp,,, j:‘[A(o;
for all r. r = 0, 1.2,.
solution
.
equations of first order in the time. cp(z. g, t: T) in which t = 0 (cp,,E
shows that for a general
w( I’; T) S,(y;
T) [f(k,
value
of r the
0) -.6,(w)]
T); T] SJA(o; T); T])f,.
(3.7a)
where i;(y;
T):=exp{
-
[ti~~~(y; T)l).
(3.7b)
~
(3.7)
[R’(y)]“S,(y;
from
T)=
(3.5a)
we have made
(Yj”
(3.7c)
‘. use of [4]
S,+,,,(y: T).
(3.8)
So as to exclude once and for all the r/rcV G 0 case. we shall assume henceforth. without restatement, that ccI 2 0; e.g., d,, > 0 and d, = 0 when I_Y> 0. The results (2.5) and (3.7) express the relaxation theorem. They show. if not directly then by inference, that as t 3x, the values of N”.“(t) all decrease “exponentially”
to the values
appropriate
to kinetic
equilibrium.
namely.
0.
The rates at which they do so are governed by the functions T<”( y; T) of y and T. We call these functions either the times of relaxation for N”‘“(t) or the P-independent relaxation times. (By way of digression, a very simple model based upon cu = 0 contains the most important features of the BoltzmannPeierls operator J(o), but it presents some shortcomings: this model does not allow the collision frequency to depend on the magnitude ,fQ of the wave vector k; JkJ = &y.) Of course. given (2.5) and (3.7), we have not proved anything so strong as the assertion of the ultra-narrow trend toward equilibrium, according to which in a grossly homogeneous condition if k E CF. then
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
347
lim,,,1f(k t) -&,(~)I= 0. CTruesdell and Muncaster (cf. ref. [12], p. 363) have purposely left unspecified the precise sense of the limit f 3 fO.) It is to be remarked that, although a knowledge of all the moments is theoretically equivalent to a knowledge of the distribution function f(k, t) = f,(w) ]I + A(o) cp(z, g, t; Ul, no direct expression of the latter is obtained from eq. (3.7a). This explains our interest in considering only such solutions (~(2, g, t; T) of (2.3a) as can be expanded in a series of the IkenberryTchebychef functions#4 Y”(g) A,(z; T). Thus, setting cp,(z, g; T) := (~(2, g, t; T), we obtain r
cPr =
x
c c a”,‘(t)0
Y”A, ,
(3.9a)
v=o y=o
(ao3”(t) - NO’O(t) )
A+‘.O(t)= 0)
;
(3.9b)
eq. (3.9a) is satisfied in the sense of the norm joIn in H [4]. The expansion coefficients a”,‘(t) have a simple representation in terms of q,. Indeed, if we “multiply” each side of (3.9a) by#” VV(z; T) Ya( g) A,@; T) and then integrate with respect to g E K and z E 3 (:, we find that P”(t)
=
(q~,,Y”A,), >
9,
E
W) .
(3.10)
Since Ap(z; T) is a linear combination of S,[A(z; T); T] with coefficients depending upon T [4], since 0 d y d p, and since [P( y; T) are independent of p, it follows from (3.4a), rp = q,, N”*P = N”.P(t), (3.7a) and (3.10) that
c”(t) = (~0, 5;Y”A,),
3
‘POE
W) .
(3.11)
Thus, while in general we are not able to calculate q, explicitly, for the very special circumstances of a model gas of quasiparticles and a function L(y) of the form (3.2) we need only specify the initial value ‘poE D(L) of q, and determine cp(y; T) from (3.7b) and (3.7~). It is this fact which makes the search for exact solutions of the type (3.9) and the exact, time-dependent theory of fluctuations feasible. We have presently no hint toward proving the corresponding relaxation L4If we orthogonalize the set {S,[h(z; T); T]; p =O, 1,. , m}, we obtain a sequence of Tchebychef functions A,(z: T) which, apart from the sign, is uniquely determined by the following conditions: {A, IA,} = S,,. Here the symbol SPYdenotes the Kronecker delta, and { 0 10) is a scalar product in the Hilbert space U.*(%~) of all real-valued functions which are square-integrable on 9,: C (0, m) with weight AZA’~‘fo(l +fO); A’(,?; T) := ah(z; T)/az. For more details, see sections 4.2 and 3.2 in ref. [4]. X5W(z;
T) := (A(z; T)]‘A’(z;
T)w*( A(z; T); T) fO[w(A(z; T); 7’)]{ 1 +h,[w(
A(r; T); T)]}.
snql fs~osua~ ruopue.~ asaql 30 sanlm palDadxa aql awlnw2 01 ww3!p lou s! l! ‘([p] ‘3al u! (E’P) .ba .33) [“rl :x]g =: [x]“g 30 uop!uyap aqi 01 y3eq 8u+a3ax .(%-I “‘f~) uo slosuai u~opuw p2~!.1puqIC:, 4u!aq SE papmh aq uw (~)~.,n@ (I&D pue (J)~.,,D wql as!ldms ou SC awo3 plnoqs I! ‘“)H uo ‘ri amseawo~d !ysMoq~nlo~s-u!alsu!~ aql 01 %up.~ome ch 30 uo!inq!.wp ~r?3!is!iels E [p] .3a1 u! pawnsse aheq am am!s ‘IahaMoH ‘x) aa.kiap 30 losual ssalaml z+lawurtCs pauyap-l[aM 3~~0s s! ‘0 4 3 ‘I1 ( d vn~ ;j ‘d, ) pz!l!u! ualz@ r! JO~ = (I)$/ 1,n ~l~luenb aq1 “‘fj 3 (a)0 3 h uo!i!puo~ (.maisi(s aql 30 i(%aua aql salouap g i(lgur?nb aql lr?ql 11waJ am .lsuon = 8 _pawos!., :uop!puo:, sno!Aqo ??u!Mo~lo3 aql u! paujeluo3 uo~1em~o3u~ aqi paldope amq aM ([l’~) put? (6’~) 8u!i\!lap u! ieqi 1~3 aqi sm~yuo3 ICldru!s 11 ‘paapul +u!s!ld_ms aq IOU plnoqs luawa3cldaI s!ql ‘am03 30 ‘(H # ‘)H ‘H 3 “H) “‘H 1(q H azzldal lsnur 3~ ‘0 = (~)~,.,,n a.mq ias am am!s ‘JaAaMoH ‘H a3eds sSpwt) uo amstmuo~d !ysMoq3nIours-u!alsu!~ aql pauyap amq aM [b] ‘3al u! ‘Zugeads iClas!aald) .wn!.tqg!nba lewaql u! suo!lenlmy luapuadap-aury aql %!qgsap suogmn3 uo!lt?laJlo3 3ql alelnDIe3 01 s! uogoas s!ql3o asodmd aql “d, 30 cm!l u! uognlom aql su.~ailo% ieql (c’z) uogmba ~egua~a33!p-o.Galu~ aqi 8u!sn .pau!umlap pur? pauyap aq hem salqc!.nzA uropuw alqw8alu!-almbs 30 suogmn3 UO!~~I~JIO~ lsaldw!s aql q3!qM LI! SiCBM 3UIOS alaq] UMOqS aAl?Lj 3M .[b] ‘331 30 t, Uql3,aS u! “JJ uo ‘ri amsealuold !ysMoq3nlours-u!alsu!~ aql Apnls 01 u@aq lsly ah
‘patomp
Aplew!-rd
am
YJOM s!ql 3o suog3as Zu!u!swai aql ‘smalqold asaql 0~ .ald!m!ld s,uumuz~~o~ 30 [L] uo!maldlaw! sGu!alsug uodn passq suogenlmg 30 hoaql luapuadapu! -am!1 aql %~!lap!suo~ alaM auo 31 se uo!ssnmp aql u! paaDold uaql pm
‘rl ampow! AIUO paau auo ‘(I)+,,D 30 suogenlmg uxn!lqq!nba aql alenpz,ta 01 laplo u! 'splo~ lay10 UI .([p] 'Jai u! 1‘p uogztas ‘33) “H uo uo!imn3 je~!.tpuqh u! pay!lsnI aq 111~ auo ‘uogenba LIB qsns (11’~) .ba’Jo suogmgdru! 1mgzmd aql ‘[p] I? %U!aq st? “( ua,@ ‘paapuI
“v~,A d_‘j “d)) 8u!leaJi .1ea13 al!nb aq plnoqs
“H uo ‘r/ amseauold um!lqg!nba asoqm put? (ii)7
suorlXIn3
!ysMoq3n~omS-u!aisu!~ aqi Icq paq!map air? suo!lenlmg uogmn3 P se ‘d, 30 ‘6, anlm ~E!J!u! aql 8ugaldlaluI ‘(E’E) pue (z*(;) icq ua@ aie (3)~ asoqm c lowado
ayl 103 alay
mop
aA2eq aM se ‘(L t;i)“~
mu!1 uo!lr?xe~al luapuadapu!-d aqi icq I! i!mgap 01 sn MOM 01 se aIdm!s OS aq 01 q3eoldde aql laadxa 01 aIqeuosea1 mu s! 11 Inq ‘ICpsca pawlsuowap aq um um!lqq!nba ‘@I%]3
.suopenba
01 puail 3!laug
aql30 waioaql leiaua% e ‘(s’z) iaq$o pm ,,suo!l~ialu! 30 s~el,,
&mbau! aql qi!M iaqlo 103 swaioaql
Z. Banach,
S. Piekarski
E,[a”‘P(t)] = 0
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
349
(4. la)
)
(4. lb) where
E := 4nl3N and#6 E(CZ(CZ)denotes, as we know [3,4], the natural projection of the ath tensorial power IE” of lE onto the irreducible subspace Ker,Tr of symmetric traceless tensors of degree (Y. The symbol { 0 ) 0 } represents the scalar product for L2(!ZZi) as determined by eq. (A.2) in appendix A of ref. [4]:
(4. lc) The result (4.lb) can be viewed in a slightly different way if one introduces a somewhat wider class of random variables Xi, (t, r+G) E [0, m) x HO, on (H,,, pL,), (4.2a) 9, :=
U(t) cp
9
clef&.
(4.2b)
Here {U(t); t 2 0) is a one-parameter semigroup on H, induced by -T-IL.. We shall refer to the collection of random variables {X’,; (t, $) E [0, m) X H,,} as a generalized random field over [0, ~0)x H,,. In view of (4.2), X$,X$? defines a cylindrical function on Ho [17-191, and since this function is integrable on Ho with respect to CL,, we obtain
-w&q
= (dJ,,U(t + $1+2>,.
(4.3)
Our interest in studying X’,(q) lies in the fact that pL,is not a measure on H,,. A theory based upon the fluctuating kinetic equation suggests in turn that the total time-correlation function of {Xi; (t, I/J)E [0, ccl)x Ho}, denoted by X>,X&, should be invariant under the distance-preserving transformation 3, defined by S,(XLIXS,,) := X~~“X>~“, where a E (0, to). Indeed, considering a situation in which the time evolution of cpr is governed by the stochastic Boltzmann-Peierls equation, one can prove that the equilibrium second moment of the generalized random field converges in the well-defined limit, not to EF[Xk,X$J as given by (4.3), but to (4.4) X6We assume that a crystal of volume V has N atoms (cf. section 2.1 in ref. [4]).
3.50
Z. Bonuch,
Eq. does
(4.4) not
However, the
is consistent
contradict
/ Einstein-Smoluchowski
space
positive,
promeusure
with the laws of statistical
the
principles
of large
since the Einstein-Smoluchowski
Hilbert
strictly
S. Piekurski
H,, for which the proof
the
of (4.4),
assemblies
although
possible,
Due to this fact, here we will not present
it. Rather, (4.4),
between
(4.3)
and
[5,6]
of discrete
equation
and thus particles.
~~ is not a measure
operator
important
difference
mechanics
promeasure collision
vs Bolrzrnattn
k is self-adjoint is extremely
in order
difficult.
to “remove”
we shall restrict
on and the
our attention
henceforth to the s = 0 case. Such an assumption is altogether natural, because in the calculations WC propose in section 5 the extra condition (t - s( = t + s plays a special role. In any event, it is reasonable to set down the following. H~pothesi.s.
According
to (4.3)
and (4.4).
we may note
that
if ]t - sl # I + s.
then the total time-correlation function is not governed by the linearized (linear) Bohzmann(-Peierls) equation. To calculate X:,,,Xi,, without going into detailed microscopic investigations, we need only discuss the mathematical properties of the stochastic Boltzmann(-Peierls) equation. Given eq. (4.3) for s = 0. the next stage in the analysis is to use this equation to estimate the thermal conductivity coefficient K., . Thus, in section 5 we shall discuss the Green-Kubo approach to transport processes (20,21]. Before doing that, we look at the problem concerning the evolution in time of the Einstein-Smoluchowski promeasure p, of scale parameter E. We recall that p, is a family {p,,,?: K E %\j(H,,)} defined as follows: (i) ,@(N,,) is a collection of (closed) subspaces of H,, of finite dimension together with the partial ordering relation C ; (ii) p?,,<. R E .\:1(H,,), is a Bore1 probability measure on R as given by eq. (4.1) in ref. (41; (iii) if $(H,,) 3 R C S E.\:,(H,,). then P,,~~(G) = I*+,,~(0) for every Bore1 subset G of R, where L) is a (Borel) cylinder in S induced by G and the orthogonal projection of S onto R. (For the precise definition of p,, see section 4.1 in ref. [4].) Let D, be a cylinder in U(t) H,, = H,, associated with U(t) G and the orthogonal projection of U(t) H,, onto U(r) R (G C R E;\:(H,,)). Moreover, we assume here that pi,r(,) ,? (U(r) G) is the probability of finding the system in the “macrostate” cp~E D, at time t where t 2 0. Clearly, observing that ‘p, = U(t) +I,,, we wish to regard this probability as unchanged in the course of time; thus we set
PL:, ,0(t) G) = PL,.dG) n(r)
Alternatively,
(4.5a)
we obtain (4.5b)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
351
A family & := {/-&:R~WU
is called the reciprocal image of p, by U(t). Since T(t) is a continuous linear mapping from H,, onto itself, one can prove that pLf,is a promeasure on H, (cf. the text at the top of p. 498 in ref. [17]). In rough summary of these results, we may say that the procedure we have used in deriving (4.5b) is meant to replace the theory based upon pL,and (2.3a) by the theory formulated in terms of &. In this context, it should be stressed that & cannot be extended to a measure on H,.
5. The thermal
conductivity
The actual calculation of a time-correlation function is a very difficult task in general. One of the appealing aspects of our model of the Boltzmann-Peierls equation is that it gives not only a formally exact representation of various time-correlation functions, but that it also does this in terms of integrals for which the thermal conductivity coefficient K~ can be evaluated explicitly. Notice that our model is not simply a mean free path model. In the Green-Kubo theory [20,21], the thermal conductivity coefficient K~ correlation function can be expressed in terms of the unequal-time E, t q(t) o q(O)1by
(5.1)
dt ~,MtPq(0)1,
V being the volume of a gaseous system and q(t), t 2 0, being the random heat flux vector. To illustrate how (5.1) may be applied, directly from (4.lb) and (3.7b) we see that
dt EJPB(t)
0
tPp&VI = @a+ l){A, h&J .
(5.2)
So as to obtain the explicit formula for K-,., we consider now only the case in which ?A?’ = (0, W) and R(y) = cy (the isotropic and dispersionless (phonon) model). Then the dependence of q(t) upon ~“~(t) is characterized by the following particularly simple expression (cf. also eqs. (4.13) in ref. [4]):
q(t) =
2
(g)“* (k3g2
Given (5.1)-(5.3),
the calculations,
al’o(t).
though elementary
(5.3) and straightforward,
352
2. Bunach,
are lengthy, technical
S. Piekarski
I Einstein-Smoluchowski
promeasure
so we shall only set down the final result for
steps involved
in arriving
KS Boltzmunn equation
K., ,
not describing
the
at it:
(5.4a)
where
7.J‘?):= 7(T) [ 1 + V (J:=
-,
N
F n%(T) c, .s,, I,,,(+$z)“]’.
c, = cl,,-
$d,
(5.4b)
(5.4C)
The real difficulty lies in evaluating the integral on the rhs of (5.4a). However, the successful analysis of this integral now seems to be possible, for eqs. (3.2) and (3.3) have removed all the problems that dogged the GreenKubo approach to transport coefficients, and machine calculation should be able to deliver for the integral (5.4a) approximations as accurate as desired. Certainly the more serious problem is that of obtaining the accurate expression for T(T). Indeed, it should be evident that the presence of the defects and imperfections in a crystal does not contribute very much directly to the total thermal conductivity coefficient K~, because the ;{-contribution is generally much smaller than the contribution arising from the mean free path approximation. In other words, so long as the total number of defects and imperfections in a crystal is small enough for the relaxation time ~~(2) to remain independent of z, we can replace TV by T(T) in eq. (5.4a). Information about the way the coefficient T(T) depends upon T can then be found by analyzing the results described in detail in the experimental papers (cf., e.g., section 2.6 in the review paper by Beck et al. [15]). Nevertheless. it is important to remark once more that. using the GreenKubo method, we derived, for our blurred (but nontrivial) image of the “true” collision operator J(a). a formally exact representation of K~ which is identical with that found in the first Chapman-Enskog solution [2] of the BoltzmannPeierls equation. This then gives a justification for basing the computation of transport coefficients peculiar to the kinetic theory of quasiparticle gases on the time-correlation functions.
6. Final remarks We went beyond the Einstein theory of fluctuations (71. which deals explicitly only with a finite number of degrees of freedom, inasmuch as we could
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
353
consider gaseous systems described by continuous distribution functions f(k, t). Since a complete knowledge of the distribution function f(k, t) is formally equivalent to a knowledge of all its moments, there would appear to be no reason a priori to expect that the system in question should be represented in terms of a finite set of state variables. One would instead be led to anticipate the possibility of a much more complicated (and interesting) description. That these sorts of complications actually do arise has in fact been demonstrated in ref. [4] for the special case of the quasiparticle gas. The connection between the thermodynamic theory of equilibrium fluctuations and the kinetic theory of gases was stipulated by means of a postulate, the essential ingredient of which was the precise definition of the EinsteinSmoluchowski promeasure EL,on the proper subspace H, of Grad’s space H [4]. This promeasure paved the natural way for the calculation of the mean-square fluctuation in the value of the random variable a”,‘(t) determined from f(k, 2). At the same time, our construction of pu, was largely self-contained and did not rely on specialized statistical knowledge. This means, of course, that we decided to detach the infinite-dimensional analog of Einstein’s theory from its connections with statistical mechanics. Given this state of affairs, we are aware of the fact that the quantitative discrepancies between the approach taken by the present paper and statistical mechanics vanish asymptotically for large (macroscopic) systems, and that the two formalisms can be used interchangeably. Grad’s method of moments [13] has a much wider applicability than the most widespread approach in the literature would suggest. The reason for this is that in the kinetic theory of gases it is imperative to use Grad’s ideas as a point of departure, since these ideas are of interest in obtaining the generalized continuum-like equations#‘. However, in addition to the problem of allowing for the continuum models of less traditional form, there exist a number of other theoretical issues that one might raise. Thus, for example, of great interest for our purpose here is the relevance of Grad’s procedure [ 13,3] to the Einstein-Smoluchowski promeasure pL, on H,, [4]. Indeed, Grad’s method of moments implies that one can introduce in a sensible way a particularly important class of cylindrical functions on (H,,, pE), thereby simplifying the calculations enormously. To summarize, in the conventional presentation there is a clean-cut distinction between the Einstein theory of equilibrium fluctuations and the kinetic theory of gases. Actually, however, the application of pu, and Grad’s method of moments bridges the gap between the two points of view. Thus, it would also seem natural to apply the techniques developed in our two papers to the #’ These
equations
are not based
on constitutive
relations
of the Euler
or Navier-Stokes
type.
3.51
Z. Bunuch.
investigation
S. Piekarski
of fluctuation
/ Einstein-Smoluchowski problems
promeusure
related
to both
vs Bolrzmann
classical
eyuution
and relativistic
[ 26,271.
gases
Appendix
A. Comments
on the Tchebychef
polynomials
When R(y) = cy, c being a strictly positive constant. the Tchebychef function A,(z; T) becomes a polynomial in .z E .% i of precise degree p. We shall consider here only the still more special case in which !8 (r = (0, x). As we have shown in ref. (41, the system {A,; p = 0, 1.2,. . . x} is orthonormal, that is. % {A,(Ay)
the weight
vv(2;
If
dz N’(z; I-) A&;
=
function
T) A&;
VV(z; T) being
given
T) = 6,, 3
(A.1)
by
gj‘ (ey):
T):=
(A.21
denotes the highest coefficient of ACc(z; T), we may assume that has been so chosen that K,(T) >O. Although the isotropic and dispersionless (phonon) model (R(y) = cy, 8 ,; = (0, x)) is so simple as to be almost K~(
T)
K,(T)
degenerate, our treatment of it illustrates problems associated with the mathematical and solved.
the usual way in which specific properties of A,(z; T) are set up
If we write (A.3a) w,,(z) := -74”(z; T) [‘NJ’(z; T)]
’ ,
(A.3b)
w,(z) := VV(z; T) [%“(z; T)]
’
(A.3c)
9
we obtain
w,,(z)c 1
3
w,(z)S5+3z’.
Given the Laguerre polynomials functions Z@(z; T) defined by
(A.3d) L:)(z)
[23], one
can easily
show
that
the
(A.4)
Z. Banach,
S. Piekarski
are orthonormal
W&l
I Einstein-Smoluchowski
promeasure
vs Boltzmann equation
355
with respect to ‘W(z; 7’):
= 1 dz W(z; T) ~,(z;
T) .Z,,(z; T) = SPY.
(A.9
0
For each integer p, p 3 0. we define -yPas follows: -rP := ([AplApl)“2
(-4.6)
;
thus rP > 0 and rp G ({ApjAp})l’Z = 1. If we replace now Ap(z; T) by a linear combination of {L&(z; T); 0 =Sv s p} with constant coefficients cpV, eq. (A.6) becomes ri = c& + csr + . * . + c&, and from Cauchy’s inequality, -yPs 1, and (A.4) we conclude that#’ [23]
where, for convenience, we set L?/(z) := 0. Using the inequality (3) as formulated on p. 334 in ref. [23], property (A.7) shows us immediately that for all z in the finite interval (0, ,I, l> 0, there is a positive constant C, not depending upon z E (0, I), such that
(‘4.8) Regardless of (A.7) and (A.8), we also know [25] that the following relation holds for any three consecutive orthonormal polynomials A,, y = p - 1, p, p + 1: zA,
= ‘Q(‘$+,)
-1 Ap+, + npAp +
K~-,(K~)-‘A~_,
;
(A.9a)
here np is a constant not depending upon T. Hence, applying the definition of { 0 1o} (cf. eq. (A.2) in appendix A of ref. [4]), we obtain in view of (A.l) and (A.9a)
Kp(‘Q+~)-’ = b$h$+J
9
no = {zAplAs>.
xXFor the definition and properties of L:‘(z), on p. 334 in ref. [23].
(A.9b)
see pp. 295-302 of ref. [23] and the inequality (3)
Z. Banuch,
356
S. Piekarski
We now record
I Einstein-Smoluchowski
two easy but important
promeasure
consequences
vs Boltzmann equation
of (A.3d),
(A.7)
and
(A.9b). Theorem.
There
exists a positive
“/AK@ +I ) ‘a(p+q”,
C such that
na sC(j3
Proof. To prove (A.lO), ([Lw,(AZ])‘~‘([ZW,(A~])‘~~. that the rate of growth
constant
+ 1)“.
(A. 10)
it is sufficient to observe that ({ZAP (A y } j s The theorem then follows from (A.7) and the fact
of w,(z)
is no greater
than
n
that of z’; cf. (A.3d).
In section 3.2 we defined a solution of the Boltzmann-Peierls equation (2.3) as one that is determined completely by (3.9) and (3.11). The inequality (2.5) supports the existence of a broad class of such solutions, and so it is permissible to conclude that, for each choice of the initial value ‘p(,E D(L) C H,, of ‘p,, t E [O, r), the series (3.9a) converges in the sense of the norm ) 0 It, in H to an element ‘p, E D(L) C H,,; in addition, if ‘poE D(L), then lim,__, j(p,j,, = 0. Aside from these observations, it is natural to ask whether the series (3.9a) converges pointwise for (z, g) E !Z ,T x IK, and, if so, whether it converges to a classical solution of (2.3). We shall not present the details of these problems here. because such problems belong to mathematical physics rather than to statistical mechanics. All we shall intend to state is that the elementary properties of Ap(z; T) listed above permit us to establish quite easily sufficient conditions for the pointwise convergence of the series (3.9a). By way of digression, our method of dealing with A,@: T) can be immediately generalized to the case of relativistic gases [26,27].
Appendix
B. Comments
on the isotropic
and dispersionless
(phonon)
model:
($3: = (0, m), L&(y)ly = c = const) It
was
= h(z;
shown
in
T) = (k,T/kc)z
VW@; T) = (%I3 Now we define
appendix
A
of
ref.
[4]
that
y = (kL3T/Z1c)w(y; T)
and that (ezy;,2
the function
L(y)
(B.1) as follows:
L(y)=y-2 b ,, , 0 CVSY,
U3.2) being
some fixed nonnegative
coefficients.
We next introduce
the
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
symmetric traceless moments N”*@E Ker,Tr N a-0 .-.-
i
dk
vs Boltzmann equation
357
or f - f,,:
Y”(iL4.y; T)lB+l(f-fo)=
$(cp,Yap%,.
(B.3)
LF Starting with (2.3), by use of (B.2) and (3.3) we may derive the equations of transfer for N*.‘:
The above result also shows that eqs. (3.7) remain valid for L?(y) = cy if in their statement S,(y; T) = Sp[h(z; T); T] are replaced by [o(y; T)]” = 2’: N”%)
=
I dk iP(y; T) ya(g) l! = Gh,? 47WH 9
[w(Y;
T)lp+‘[fW) -.M~)l
cpE W) .
(B.5)
Since in the special case of an isotropic and dispersionless (phonon) model the Tchebychef function AB(z; T) is a polynomial with respect to z of precise degree /3 (in which the highest coefficient K~(T) is positive), we easily arrive at
a”.‘@)= boo,, lFY”A,), >
‘POEW).
03.6)
References [t] [2] [3] [4] [5] [6] [7] [8] [9] [lo] Ill] [12] [13] [14]
R.E. Peierls, Quantum Theory of Solids (Clarendon, Oxford, 1955). V.L. Gurevich, Kinetika Fononnykh Sistem (Nauka, Moscow, 1980). Z. Banach and S. Piekarski, J. Math. Phys. 30 (1989) 1816. Z. Banach and S. Piekarski, Physica A 180 (1992) 309, this volume. P. Resibois and M. De Leener, Classical Kinetic Theory of Fluids (Wiley-Interscience, New York, 1977). R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley-Interscience, New York, 1975). A. Einstein, Ann. Phys. (Leipzig) 33 (1910) 1275. L. Tisza, Generalized Thermodynamics (MIT Press, Cambridge, MA, 1977). K. Huang, Statistical Mechanics (Wiley-Interscience, New York, 1963). D.A. McQuarrie, Statistical Mechanics (Harper and Row, New York, 1976). J. Szczepanski, Physica A 157 (1989) 955. C. Truesdell and R.G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York; 1980). H. Grad, Principles of the kinetic theory of gases, in: Handbuch der Physik, vol. 12, S. Flugge, ed. (Springer, Berlin, 1958) p. 205. U. Weinert, S.L. Lin and E.A. Mason, Phys. Rev. A 22 (1980) 2262.
3%
2.
Banuch,
S. Piekarski
I Einsrein-Smoluchowski
promeasure
L’S Boltzmann
equaiiorl
[IS] H. Beck, P.F. Meicr and A. Thelhmg. Phys. Stat. Sol. A 23 (1974) II. 116) K. Maurin, Methods of Hilbert Spaces (Polish Sci. Publ.. Warsaw. 1972). [ 171 Y. Choquet-Bruhat. C. Dewitt-Morette and M. Dillard-Bleick. Analysis. Manifolds. and Physics (North-Holland. Amsterdam. 1977). [IX] H.-H. Kuo, Gaussian measures in Banach spaces. in: Lecture Notes in Mathematics. vol. 463. A. Dold and E. Eckmann, eds. (Springer. Berlin, lY75). [ 191 C.W. Burrill. Measure. Integration, and Probability (McGraw-Hill, New, York. 1972). [ZO] M.S. Green. J. Chem. Phys. 20 (1952) 1281. [2l] R. Kubo, J. Phys. Sot. Jpn. 12 (lYS7) 570. [22] Z. Banach and S. Piekarski. J. Math. Phys. 30 (1YXY) 1x26. [23) G. Sansone. Orthogonal Functions (Interscience. New York. IYSY). 1241 P.L. Bhatnagar. E.P. Gross and M. Krook. Phys. Rev. Y3 (lYS4) 511. (251 G. Szego, Orthogonal Polynomials (Am. Math. Sot.. Providence. RI. IS3Y). [2hl K.S. Thorne, Mon. Not. R. Astron. Sot. 191 (IYXl) 430. 127) N. Udey and W. Israel, Mon. Not. R. Astron. Sot. 19’4 (19X2) 1137.