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The Einstein-Smoluchowski promeasure for (quasiparticle) gases
Physica A 180 (1992) North-Holland
309-335
The Einstein-Smoluchowski (quasiparticle) gases
promeasure
for
Zbigniew Banach Insliture of Fundamental Technological Research, Department of Fluid Mechanics, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland
Siawomir Piekarski Institute of Fundamental Technological Research, Department of Theory of Continuous Media, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland
Received
25 April
1991
Considering an ideal Bose-Einstein gas composed of quasiparticles (phonons, magnons, rotons, etc.), the object of this paper is the development of a tractable generalization of Einstein’s theory of equilibrium fluctuations. It is assumed that the state of the gaseous system may be characterized by the distribution function f(k) which determines the number of quasiparticles with quasimomentum hk. It allows one to embark on a program in which the inversion of Boltzmann’s relation connecting entropy and probability can be used as an organizing principle for the development of the theory dealing with the equilibrium fluctuations of f(k). In the present approach, Einstein’s distribution law in the Gaussian approximation is identified with the equilibrium promeasure CL_defined on the suitably chosen, infinite-dimensional Hilbert space H. This promeasure gives the possibility of calculating the fluctuations in the value of f(k) and its moments. Although pL, cannot be extended to a measure on H, what one discovers is that, in this case, one obtains a manifestly infinitedimensional analog of Einstein’s theory of equilibrium fluctuations. The generalization of the method to the case of classical and relativistic gases is straightforward.
1. Introduction
In this paper we study in detail some aspects of the theory of equilibrium fluctuations. Precisely speaking, our more concrete objective is to formulate and motivate a statement which says that a relatively simple method based on Einstein’s interpretation [l, 21 of Boltzmann’s principle leads to a tractable theory of equilibrium fluctuations occurring in gaseous systems composed of quasiparticles (phonons, magnons, rotons, etc.) [3-51. Beginning from the Boltzmann(-Peierls) equation [3-61, a second paper [7] will emphasize the essential similarities and differences between the static and 0378-4371/92/$05.00
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1992 - Elsevier
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Publishers
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time-dependent
theories
of fluctuations.
formalism
of our two papers
its almost
immediate
Thus.
for example,
simple which
gases
that
the
and will. for the most part,
find
application
in the theory
one
like to consider
would
[S, 9). Similarly.
the basic constituents
In 1910. Einstein
It should
is very universal
one would
[ 1] developed
might
also
of classical
be stressed and relativistic
such objects
like to discuss
be photons
or neutrinos
a time-independent
gases.
as the rarefied
relativistic
gases
in
[ 10, 111.
approach
to fluctuations
by introducing statistical concepts into thermodynamics and without making any explicit reference to the detailed microscopic description of the system. As we know,
his basic idea was the inversion of Boltzmann’s relation connecting that one can obtain the entropy and probability. This implies, in particular, relative probabilities of fluctuation states in terms of differences of entropy. In the Einstein thermodynamic theory of equilibrium fluctuations attention is turned toward the macroscopic system, the state of which may be characterized by a finite set of extensive variables. Although Einstein’s theory is absolutely insensitive to the specific choice of a thermodynamic phase space spanned by a finite number of extensive variables. a particular phase space associated with energy, underline
volume, and mole numbers plays a privileged role. In order its importance, we shall refer to it as Gibbs space 121.
to
According to Einstein’s interpretation [I] of Boltzmann’s principle. the only acceptable function that dcscribcs the relation among probability and entropy is the exponential. Then the key simplification is the fact that the entropy S of the (isolated) system need not be considered as some abstract (unknown) function of extensive variables U, but may instead be expanded around equilibrium with respect to the variations of (Y. The physical content of this expansion should be clear: one is assuming that equilibrium is characterized by the maximum of the entropy. Since the very existence of thermodynamics is based on the observation that the fluctuations in the extensive variables arc small compared to the mean values of these variables. the expansion of the entropy
can be broken
off directly
after
the second
term.
Consequently.
on
using the inversion of Boltzmann’s relation. one obtains the probability distribution function in the Gaussian approximation. thcrcby simplifying the theory of equilibrium fluctuations enormously. The basic viewpoint adopted here is that the physics of a gas of quasiparticles may be adequately described in terms of the one-point distribution function f(k) [3-6, 121. A.4 will be emphasized in section 2.1, this function determines the number of quasiparticles with quasimomentum hk. Since flk) is an extensive variable, in the present paper we embark on a program in which the inversion of Boltzmann’s relation can be used as an organizing principle for the development of the theory dealing with the equilibrium fluctuations of f(k).
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
311
Clearly this means that we let the distribution function f(k) become a random variable. Although the general ideas which we have here in mind are very similar to those of Einstein [l], our main contribution is to arrive at the theory of equilibrium fluctuations for f(k) in a novel fashion. The basic concept is in fact straightforward. In order to obtain the inversion of Boltzmann’s relation, one must express the total number of microstates corresponding to the coarse-grained distribution function f(k) in terms of another object - the entropy S of the system identified with some complicated functional S(f) off. As demonstrated in great detail by a number of authors (cf., e.g., refs. [3, 13, 14]), this idea can be motivated in a natural way at least in the context of the theory of ideal gases. Now, if one supposes that the gaseous system is close enough to equilibrium, one is led to approximate the deviation of S(f) from its value for the Bose-Einstein distribution function f,, as being a very simple functional of the nonequilibrium part cpoff (f - fo - cp). Precisely speaking, one has in mind a viewpoint in which the approximate expression for S(f) - S( fi,) is proportional to the square of the norm 1p lH induced by the scalar product of some suitably chosen, infinite-dimensional Hilbert space H. Thus, the paper will address the problem of how to compute the probability for such macrostates f where the entropy can uniquely be determined from the norm of cp. To give this program a precise meaning in a straightforward way, we should deal with, and investigate the mathematical features of, a very specific Bore1 probability promeasure [1_5-171, namely, the Einstein-Smoluchowski probability promeasure )(L~of scale parameter E, defined on the real separable Hilbert space H which we call Grad’s space [18]. Here we anticipate that E-’ is of the order of the volume V of the system. Among other things, the construction or derivation of pF is based on a combined application of the inversion of Boltzmann’s relation and the approximate expression for S(f) S(&,). Consequently, what one discovers is that, in this case, one obtains a manifestly infinite-dimensional analog of Einstein’s thermodynamic theory of equilibrium fluctuations. Indeed, one knows full well that a knowledge of the “continuous” distribution function f(k) is formally equivalent to a knowledge of all the moments of f(k). Therefore, the most rigorous approach would be to enlarge the basic thermodynamic space both by adding the higher order moments of f(k) and by treating them as the fundamental fluctuating variables on the same footing as the traditional Gibbs variables [2]. Due to the existence of CL,, this new approach can be formulated self-consistently in the context of the measure-theoretic treatment of probability theory [U-17]. Also, in order to evaluate the fluctuations in the value of f(k) and its moments (the Ikenberry-
distribution Tchebychef coefficients (6, 121). one need only identify Einstein’s law in the Gaussian approximation with pLcand then proceed in the discussion as if one were working Now,
given
investigations cannot
in the traditional
such a general
program,
is an observation
be extended
to a measure
consequences
Gibbs
space.
the important
result
that the Einstein-Smoluchowski [ 15-171 defined of this new result,
on Grad’s
of our preliminary promeasure
p8
space H [ 181. The
simplest
physical
attempt degrees
to deal with continuous systems described by an infinite of freedom, will be recognized and analyzed elsewhere.
which are essential
when we number
of
Here we proceed as follows. Since in section 2 only the vibrations of the lattice (crystal) are concerned, we treat it there as if it were a quasiparticle gas. It is only for pedagogical purposes that we begin with an ideal phonon gas. Then, making use of a sort of microcanonical prescription based principle of equal a priori probabilities [13]. and appealing to some
on the kind of
passage to the continuum limit, we offer in section 3 the precise definition of Grad’s space H. In addition, section 3 discusses the ideas, systematically investigated in section 4, standing behind the Einstein-Smoluchowski promeasure pc of scale parameter F. Next, applying the Einstein-Smoluchowski promeasure ~~ (which, by its very nature [15-171, cannot be extended to a Bore1 probability measure on H), in section 4.3 we calculate explicitly the so-called second moments [19,2(I) of the random energy %, of the random heat flux q, and of the random quasimomentum P of the macroscopic system. Clearly, to set a proper perspective upon such second moments, it is necessary to understand first the general method for calculating the expected values of the random variables determined from the Ikenberry-Tchebychef expansion coefficients of f(k) [6,12]. Thus we make a few remarks on this method in section 4.2. Some auxiliary technical material, being in fact a supplement to sections 3.2, 4.2 and 4.3, is included as appendices A and B. Section 5 is for the discussion and final remarks. Here we adopt the notation of our two previous papers. Thus the reader interested in arriving at information about it is kindly referred to ref. [6] and section
2 of ref. (211.
One final word regarding our approach. As a matter of fact, it should be said that we choose to think of the (kinetic) theory of gaseous systems and the Einstein thermodynamic theory of equilibrium fluctuations as a unified structure. However, since the actual unification of various deductive schemes is always a very difficult enterprise, we do not adopt a very detailed description of a gas of quasiparticles (phonons, magnons, rotons. etc.) [6,7, 121. Instead, we use the simplest model that still retains the crude qualitative physical properties that might be important. Of course, from the point of view of solid state physics, a more complicated model would be of interest.
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
313
2. Physical motion
2.1. Microcanonical
ensemble
As explained already in the introduction, the fundamental quantity that describes the macrostate of a gas of quasiparticles is the coarse-grained distribution function f(k). The objective here in section 2.1 is to define this function and to calculate the total number JV~ of microstates corresponding to it. Since we focus our attention on a microcanonical ensemble [9] and further restrict it to obey the principle of equal a priori probabilities [3, 13, 141, the calculations leading to f(k) and JV~ are not difficult. For concreteness sake, let us consider an ideal phonon gas which inhabits an insulating crystal composed of only one type of atom. Since we simplify our picture of the gas by neglecting the polarization effects, i.e., the effects depending on the direction of atomic vibrations, it is easy to prove that if a crystal (lattice) of volume V has N atoms, then there will be N normal modes, each characterized by its own “wave number” k, with the characteristic frequencies O(k). Denoting by E the elementary (unit) cell of the reciprocal lattice, and treating k as a vector in the corresponding reciprocal space, one can see that distinct values of k all fall within E. It is of interest to introduce the quantity A(k) which signifies how many phonons exist in the mode k. Note that the set {A(k); k E E} of numbers, the occupation numbers of N normal modes, completely specifies the microstate of the system. Let % be a total energy of the phonon gas, and suppose that 27rfi is Planck’s constant. Then the condition of the form
simply represents the fact that the entire phonon gas is an isolated system, and hence has some fixed energy 8. This is clearly an idealization because we never deal with truly isolated systems in the laboratory. For N z lo’“, it would be either impossible or impractical to decide which of the order of 10N microstates consistent with the energy constraint (2.1) the system is in. Alternatively, one could cut up the unit cell E into a large number of small but finite subcells E’,, IZ= 1, . . . , M, and, instead of exactly specifying the values of A(k), characterize the state of the gas approximately by giving the number F, of phonons in every subcell E’,. Hence we arrive at @=GEn, n=l
Fn:=
c kEE,
A(k);
(2.2)
the E,, can be constructed this way seems
a precise.
important
equilibrium
to show that
fluctuations
special,
definition
our analysis
which is essentially
provides
Of course,
of a particular
the distribution kE@,
function
f(k).
i.e.. the average
Proceeding number
in
and
a basis for the theory
independent
k E E,, if and only if n = n,.
one subcell:
we suggest
of a macrostate,
decomposition of C:. To each k E E corresponds determined from the observation by n,, uniquely
of the denoted only
such that they are disjoint.
if excessively
it of
choice
a positive integer, that k belongs to further,
of phonons
we define in the mode
by
f’(k) := F,,(N,,) ’ .
(2.31)
n-n,.
where N,,:=
c
I.
(2.3b)
!,i C), Suppose now that the characteristic frequency R(k) be (practically) a constant whenever k E E,, , tl = 1. . M. If we are to regard J1 as being a function fl on the set of subcells, then from the energy constraint (2.1) WC easily conclude that
(2.41)
where
i&C&J= f](k) The number number normal
modes
of distributing
of vibration
=(N,,+F,,-I)!
1.
,/’
UV,,of microstates
of ways
Il
(N,,
-
(2.43)
if II = 11~
l)!F,,!
corresponding
to F,, can be identified
F,, indistinguishable
phonons
among
with the the
N,,
of the crystal:
(2.5)
Now. let us set 4 : = {F,,; n = 1. , M}. Then, since interchanging phonons in different subcells does not lead to a new microstate of the system, the total number of microstates compatible with 9, denoted by .Vi. is immediately calculated from
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
31.5
for gases
(2.6) In the special case of equilibrium, if we know nothing about our system, apart from its energy 8, we have no reason to consider any one of the microstates {A(k); k E @} consistent with (2.1) to be more important than any other. Thus we should treat each of them equally, i.e., we should adopt the principle of equal a priori probabilities. Specifically, according to this principle, the probability of finding the system in the macrostate 9 is given by
(2.7) where the sum in the denominator extends over all possible sets 9* of integers F,, . . . , F,, satisfying the energy constraint (2.4a). Now, in view of the definition (2.3a), we can use 9 and f interchangeably to characterize the macrostate of the system. Of course, the total number of microstates corresponding to S is given by (2.6).
2.2.
Transition
to
the subsystem which is not isolated
The derivation of (2.7) is valid insofar as the isolated systems are concerned. However, we may suspect that a slight generalization of the method of section 2.1, namely, the method of equal a priori probabilities, would enable us to discuss also the equilibrium fluctuations of the energy ‘8. To this end, we consider an isolated system made up of two subsystems whose energies are, respectively, Z? and E - 8 (E = const.). We assume, as usual, that 58< E and that both subsystems are macroscopically large. Since there is the redistribution of the quantity ZYamong the two subsystems, the probability of finding the system in the macrostate B is now given, up to a proportionality constant, by P(9)
- N,J-(E
- %)
(2.8)
Here T(E - 8) denotes the total number of microstates available to subsystem with energy E - 8 if these microstates have equal weight. T(E - Z?) must be evaluated subject to the condition that the energy determined from (2.4a). Because of this condition, it is natural to regard state of a composite system as being represented by a single “point” Alternatively, we may describe this state by _r?
the The 8 is the 9.
316
Z. Banach, S. Pieknrski
I Einsteirz-Smolu~hoMski
3. Definition
of Grad’s space H
3.1.
to the continuum
Passage
We shall discuss for a very
large
(but
still finite)
the connection
the entropy
Vh of the isolated
following
limit
now the asymptotic
establishing
is to consider
formula
system.
between system.
for the rhs of (2.6)
This formula
the total
number
As a matter
the case in which
both to obtain the functional terms of the entropy Vh( f)
pmmeusure for gases
dependence corresponding
Stirling’s
plays
of interest
a central
JY, of microstates
role in and
of fact, what we do in the method
113) can be used
of Vh upon f and to calculate to J
.h”? in
If both N,, and F,, arc very large numbers for each n, as is almost always the case. then we may evaluate the quantity (2.6) easily; indeed. by use of (2.5) and Stirling’s approximation”’ to In J;, we conclude that
where .,+; := exp[k,‘Vh(
f’)] ,
h(f)
’ 1 dk F[f(k)]
:= k&&f’)
(3. lb)
,
EIf(k>l:= [I + f(k)] ln[l + f(k)] -f(k)
(3. lc)
In f(k) ;
here and henceforth k,, denotes Boltzmann’s constant. (3.1) we have utilized the fact that. for macroscopic M. can be replaced by an integration n,n=l,..., following manner (3, 141: N,, 3 V&T-‘)-
j dk .
(3. Id) In going from (2.6) to systems, a sum over over k. k E (2. in the
(3.2a)
(3.2b)
we use the letter L!Zto denote the function on the set { F,,/N,,; n = 1,. , M}. In this paper we do not hesitate to call Vh(f) the entropy that corresponds to
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
317
f, because the tradition of the (kinetic) theory of quasiparticle gases gives a persuasive reason for that name [3,14]. If we allow ourselves much freedom, we may set aside all direct reference to the detailed atomic structure of the crystal and seek the frequency 0 and the definition of @ most advantageous for rigorous arguments. Thus arise the following two conditions: (i) the frequency R depends upon the magnitude of k through the transformation rule of the form fin(k) 3 a(y), where y := R,‘/kj and k, := (6~r*N/1/)“‘; (ii) the elementary cell @ is given either by @!, := {k E iE:O 0}, [E being a three-dimensional Euclidean vector space. It is unlikely there be any crystal such as to make the resulting atomic structure compatible with these two conditions. Nevertheless, the assumptions (i) and (ii) have been adopted frequently in analytical studies. We may say that they characterize the isotropic one-branch Debye model. For this model, one derives the equality V(&r3)-’
1
if@=Q
dk = A’
(3.3)
e eq. (3.3) motivates, in fact, our definition of 4,. The case CF= cF_, while of physical interest, is not typical and in ref. [7] serves mainly the purpose of establishing a broader agreement between the theory of Boltzmann [8,9], which provides no upper bound for the magnitude of the velocity of a particle, and the theory of Peierls [22]. Now, let us introduce the Bose-Einstein distribution function fo that has the same principal moment#’ as f: f”(w) : = (em - 1))’ ,
w(y; T):=
hL?(y)lk,T,
(3.4a) (3.4b)
Comparison
of (3.4) with the gross condition of the form
hV(Sd-’
1
dk O(y) f(k) = 26’
(35)
c shows that from (3.ld),
T
is the kinetic-theory we arrive at
X2 We call I, dk f2( y) f(k)
the principal
temperature
moment
off.
corresponding
to %‘.Beginning
ff(f’)-ff(J,)-f;,0/-‘=1n(l+e
“‘p)-e +(I
+./‘)e
‘“p
‘“[PI1 - ( 1 + p,,) In( 1 + p,,)] s 0 ,
(.3.&i)
where +=f’,,‘(f-fJ>-1,
PI, .=
de” - ’ ) > _ 1 p + e”’
Clearly f;,~ is a function whose principal moment condition holds for every .f’ that satisfies (3.4b):
(3.6b) is 0: thus
the
following
(3.7)
We may integrate (3.6a) over t? and then by appeal to (3. Ic), (3.7). and (3.6b) conclude that /I( f’) - Cz(_/;,) 5 0 and that h(f) - .f’(f;,) = 0 if and only if p = 0 almost everywhere on L5. Summing up these properties of h( .f’), WC SW that among all distribution functions having the same principal moment, the Bose-Einstein distribution function J, gives /I( 1’) its greatest value. Since in the approach so far developed the entropy density h( J‘) depends in a very complicated way upon f, the general expression for .:1;, as defined by (3.lb)-( 3. Id), and the corresponding theory of equilibrium fluctuations would he almost totally impenetrable. The fact that we can choose V at will, however, allows us to simplify the problem considerably. Indeed, observing that both h( f‘) and ,,1’;have a maximum value when f’= J;,. the spread of .+; about .I;,, can be made arbitrarily narrow by taking the volume V to be arbitrarily large, and we are led to expect that the probability of finding the system in the macrostates f other than those for which If,, ‘(f’ - !;,)I 4 1 is practically equal to zero. We may regard this somewhat vague statement as expressing the content
of Einstein’s
Whether principles,
there is not
basic postulate
origin and significance of simplest consequences and of section 4.1 is to derive, of cqs. (2.7) and (2.X) for of degrees of freedom. 3.2.
The
upproximare
The entropy the objective
in his theory
of equilibrium
fluctuations.
he a precise derivation of Einstein‘s postulate from first presently known. In section 3.2 we shall discuss. not the the inequality I]‘,, applications of that using the results of quasiparticlc gases
expression
for
‘( ./‘- !;,)I 4 1. but some of the inequality. The primary object sections 3.2 and 3.3, the analog described by an infinite number
V[h( f) - h( A,)]
functional (3.1~) is too complicated to use as it stands. Thus, of this section is to derive the approximate expression for
2. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
319
s
VIh(f) - h(.h)l. o as to estimate the deviation of h(f) from its value for the Bose-Einstein distribution function f,, we define the nonequilibrium part of f(k)/&,(~), denoted by A(o) cp(z, g; T), as follows: f(k) = h,(w)tl + NW>cp(z> g; VI ;
(3.8a)
here 2 = qy;
o’(y;
T):=
4Yi {
wI(y;
T) := aw(y;
7,)
if fl( y) ly = const , if a(y) ly f const ,
T)
T)ldy
)
g:=Ik(-‘kED6:={gEIE:lg(=l}.
(3.8b) (3.8~) (3.8d)
we have made the change of variables k 3 (2, g). Interpreting (3.8b)-(3.8d), Of course, in view of the precise definition of w’(y; T), we may expect that considerations based upon (3.8) may restrict the possible choices of n(y), i.e., restrict the class of frequencies for which the method itself can be formulated definitely. Some of such restrictions are suggested in appendix A. For the purposes of the precise theory of equilibrium fluctuations, we must state our position somewhat more abstractly by introducing a condensed notation: 1
Yl *-
i co
if@= if@=
z,, .= hrlir+i(y;T), %+:=(o,y,),
1
(3.9a) zl:=;imf(Y;%
(3.9b)
%(p:=(Z”,Z1).
(3.9c)
Without going into details which are presented that
in appendix A, we postulate
(3.9d) be a solution for y of the equation i(y; T) = z, y E Pi2+. Let h(z; T), ze6Rt, Then, as the analysis of appendix A shows, it is straightforward to derive restrictions on 0(y) which ensure that the function A( 0 ; T) both exists and is continuously differentiable. In fact, we consider the function h(o ; T) such that A(0 ; T) E C’(%!,f); thus A( 0 ; T) has a continuous derivative. Setting
320
Z.
h’(z;
T) := ah(z;
z E 2,;. properties inequality
l&much.
T)/az,
Whether
we shall assume
or not such
of R(y)
is another
A’(z; T) > 0 until
Let H denote
i Einstein-Smoluchowski
S. Piekarski
henceforth
an assumption and
matter,
appendix
the space of real-valued
promeasure
for gases
that h’(z; T) > 0 for each
be consistent we shall
defer
with the general our proof
of the
A. functions
which are square-integrable
on
92+0 x 06 with weight BV(z; 7’) := A2A’VV,1(~). The immediately following analysis shows that to render the method of this paper definite, we must specify not only A(w) but also VVJo).
Clearly (o,o)(, cp E H, denoted
If cpl and (p2 both
lie in H, we set
re p resents the scalar product in the space H, and the norm by I+Q(~, is the nonnegative square root of (cp, cp>r,: thus
of
(3. lob) Considering
a situation
in which the gas is only slightly
perturbed
an overall equilibrium, we wish to find the scalar functions such that if cp = (j&A-‘( f - A,) belongs to H, we obtain
A(o)
away from and V!/,,(w)
where ~:=471./3N
(N-V).
(3.1 lb)
To refer to this program, we first insert (3.8a) into (3.ld) and then replace logarithm ln( 1 + X), (2’1< 1, by its Taylor expansion#3 X - 4X’ + :
the
(3.12) Combining dk = RiA’A’ dg dz with Ri = 6rr’NlV and using and VW,,(w) satisfy the conditions of the form
(3.Ic),
we see
that A(w)
(3.13a) *’ To avoid being overwhelmed by unnecessary complications, given here. We note, however. that X is proportional to q~.
the precise definition of X is not
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
321
(3.13b) hence
A(w)= 41 +.Mw)l
9
W,(w)= &&J) [l + h,(w)1.
(3.13c)
Now we may say that with the choice made above for A(w) and VVO(o), the mathematical properties of LJ( y), which are listed in appendix A, imply that the space H just defined is a real separable Hilbert space. Since our method of introducing H renders systematic some of the ideas Grad [18] seems to have employed in his own studies regarding classical gases [8,9], we name H Grad’s space. We should note one more thing. To require cp to be a member of HO:= { qbE H: (1, #), = 0} is the abstract form of the energy constraint (3.4b) that we impose upon fin this section. Clearly HO is a closed subspace of H. The elementary problem before us, then, is to construct the Einstein-Smoluchowski promeasure p, on H,, by utilizing (3.lb) and the following approximate expression for k,‘V[h( f) - h( &)I: cp E H,) C H ,
(3.14)
As remarked already in the introduction, we should think of p, as being the infinite-dimensional analog of Einstein’s distribution law in the Gaussian approximation. The construction of CL,ensures the possibility of obtaining all the information about the fluctuations of f(k) and its moments. It is useful to summarize the ideas of this section as follows. While the procedure we have used in deriving (3.1) is based upon a particular choice of the decomposition of @ into disjoint subcells @,,, the definition of Grad’s space H is completely independent of it. Even more important is the fact that the same remark concerns the notion of equilibrium promeasure. 3.3. Comments on the composite system As noted already in section 2.2, in fluctuations for a system made up of further discussion and interpretation. the basic object of our theory a more entropy S(f) of a composite system, S(f) := k, ln[N$T(E
- S)] .
the corresponding theory of equilibrium two subsystems we clearly require some In this case, we are led to introduce as complicated functional off, namely, the
(3.15)
Then,
using k,,F
in which
the expression
‘(l,cp),,=T
(3.lha)
‘(X-f,,,),
(cf. cq. (3.5))
%,,:= hV(Xn’)
and appealing S(I)-
of the form
’j
to (3.la). V/2(1;,) + T
dk R(y)
.
i,(o)
(3.lb),
(3.16b)
and (3.11a),
‘(K - c?,:,,)- k,,(2~)
we arrive
at
‘lp15, + k,, In f(E
- 8)
(3.17)
Since only the values of ‘;Cnear g,, are expected to be important, and since 1ci - A,,( e E - v,,, we may perform the expansion of k,, In T(E - % ) around k,, In T(E - go). Neglecting in the expansion nonlinear terms which are normally very small compared to (-k,(i2~)(cplf, (F = 47ri3N; cf. eq. (3.1 lb)), we obtain
k,,lnI’(Ewhere
~)~k,,In
I(-
T,, : = d[k, In T(E - &)]/aE
clearly we set r,, = T. Thus, calculated is
In this equation
C,,)-
r,;‘(G
is the temperature
from eq. (3.17),
(3. IX)
- c,,),
of the larger subsystem;
we see that the quantity
to be
we have written
S( h,) := Vh( A,) + k, In I‘(E - g,,,)
(3.19b)
for brevity. Given the above approximate expression for k,,‘[S( f) - S( A,)] and the inversion of Boltzmann’s relation (3.15), one will therefore be justified in constructing the Gaussian promeasure pF not only on the proper subspace H,, of N but also on Grad’s space N itself. Such an approach will in fact be developed. However, since the precise definition of p+ leads to some mathematical problems, the objective in section 4.1 is to assemble the machinery necessary for that approach.
Z. Banach, S. Piekarski I Einstein-Smoluchowskipromeasure for gases
4. The Einstein-Smoluchowski
323
promeasure p,
4.1. Basic concepts If one were working with the general entropy functional S(f) as given by (3.15), this would be a very complicated problem. Fortunately, in the usual way one can instead choose to focus upon the approximate expression (3.19a) and to provide thereby the axiomatic definition of pc that will arise naturally in terms of it. However, it is worthwhile to introduce first the notion of promeasure which is essentially independent of (3.19a). This will, for example, demonstrate that one should have been led to the theory of fluctuations even if one had never discussed the ideas of Einstein. Let @j(H) be a collection of (closed) subspaces of H of finite dimension together with the partial ordering relation C . It will be convenient to use the symbol d(H) to denote the set of finite-dimensional orthogonal projections of H. For R E a(H), define QR E d(H) by QR(H) = R; thus the image of H by QR is R. When @(H) 3 R C S E Q(H), one can introduce a mapping from S onto R, denoted by QsR, through the relation of the form QR = Q,s,Q,T. Elementary inspection shows that QSR is a restriction of Q, to S. A Bore1 probability promeasure [15] on H is a family Al.:= { pR; R E Q(H)} defined as follows: (i) pu,, R E Q(H), 1s . a Bore1 probability measure on R; (ii) if &5(H) 3 R C S E B(H), then pR(G) = pu,(D) for every Bore1 subset G of R, where D is a (Bore]) cylinder in S induced by G and QsR; thus D := Q.;;(G) := {p E S: QsR’p E G}. Since p is a projective system of measures on the projective system {R, QsR; R, S E@(H)} of finite-dimensional subspaces of H, we are justified in using for p here the name “promeasure” [15]. The notion of Bore1 probability promeasure is altogether natural. Indeed, in formulating a mathematical model of a random experiment, one must assign probabilities to the various events under consideration. That is, given R E 4(H) and a Bore1 subset G of R, to the (Borel) cylinder D of H defined by D := Q,‘(G) := {cp E H: QR(p E G} one must assign a probability b(D) : = p,(G) of the event D taking place. At times we shall use the symbol ad, to signify the set of all Bore1 cylinders of H. Using the approximate equation (3.19a), we are now ready to provide the precise definition of F~, establishing thereby an important connection of Einstein’s ideas [l] with the ideas of Choquet-Bruhat et al. [15] discussed above. For each R E Q(H), let %‘Rbe a Bore1 o-field in R, and consider the function pu,, : B3, + [0, l] defined by
324
Z. Bunuch.
S. Piekurski
I Einstein-Smoluchowski
promeasure
for guse.s
(4.1)
G lies in % R ’ n is equal to dim R, and p,_ denotes the Lcbesgue measure on R. By far the most important promeasure on Grad’s space H is that which is where
generated
by pFR. We call this the Einstein-Smoluchowski
parameter
F and hereafter reserve the symbol pF to denote it. For any R E Q(H)
and
GE
any
933,, we
set
&(,!I)
: = pLF,((G)
whenever
promeasure D E P,,
of scale and
D =
Q&3. At this stage, it is useful to introduce the notion of a cylindrical function on H. Let fE be a three-dimensional Euclidean vector space, and consider for each cy. cy=o.1,2 . . . . . x. the cvth tensorial power iE” := 8 “iE of [E. Since IE” is a real separable Hilbert space with the inner (tensor) product”J M’; 0 M’,‘, M’,’ and MI: being elements of fE”, one can introduce the Bore1 a-field :dCVin E”. A mapping X from H into [E” is called a cylindrical function on H if there exists R E Sj( H) such that X = XQ,3. We say that a cylindrical function X : H + IE”is measurable, or a random tensor of degree cy. if the inverse image of D E :#I<,by the restriction of X = XQ, to R E G(H) is a Bore1 subset of R. In addition. the function X just mentioned is said to be integrable on H with respect to the promeasure p : = { P,~; S E o(H)} if the restriction of X = XQ, to R E $T(H) is on R with respect
integrable
I A+)
X(P) :=
The
expected
I ,ddv)
writes
Wcp>.
(4.2)
I(
II
function
to P,<; one then
value,
X. written
or expectation or mean, of an integrable cylindrical E[X; ~1, is defined to be the integral of X; thus
(4.3)
Eq. (4.1) is of course independent of the choice of a basis in H, and, for that reason, particularly useful; and, if it be so desired (cf. section 4.2), it may be “projected” into any convenient chart. This expression, namely, eq. (4.1), resembles quite closely a result first established by Einstein [l], differing, however, in that Einstein “excluded” the possibility of considering the infinite number of degrees of freedom. The next stage in the analysis is to obtain the explicit form of pF for a specific basis of interest. And, therefore, it is natural ” Concerning of ref. 121).
the definitwn
of the inner
(tensor)
product.
see rqs.
(A.h)-(A.9)
in appendix
A
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
325
to ask: Is there a connection between the theory of Einstein [l] and the theory of Grad [Ml? Could one at least capture, in some suitable way [6], a quasiparticle analog of Grad’s moment procedure? To these problems the remaining sections of this work are primarily devoted. 4.2. Fluctuations of the Ikenberry-Tchebychef
coeficients
aaxP
The class of possible orthonormal sequences in H is extremely broad, far too broad for us to be able to expect to get from any of its members the effective method for calculating the expected values of cylindrical functions of physical interest. To proceed further, we must be somewhat more specific about the choice of orthonormal sequences in H, and in this section we shall define, and present some of the basic properties of, the so-called Ikenberry-Tchebychef sequence [6]. We begin from the definition of the Tchebychef functions Ap(z; T). As the discussion in section 4.3 and appendix B suggests, fields intended to correspond with those common in the kinetic theory of classical gases are generated by the functions SP( y; T) of the form L~(Y; VI"
[yIo(y;
T)][i(y;
T)]”
for p = 0,2,4, . . . , w , for /3 = 1,3,5,. . . ,m,
(4.4)
where Y is the greatest integer =Sp/2 and y E 3 + (cf. eq. (3.9~)). By (3.8b) the effect of substituting w = o( y; T) into (4.4) leads to a family {S,( y; T); co} which is either linearly independent or linearly dependent; in /?=O,l,..., case of need, we must reject all unnecessary elements? Let [L’(!%,‘) be a set of all real-valued functions which are square-integrable on %!i with weight VV(z; T). Then it is possible to prove that S,[h(o; T); T] are elements of k2($?i). We relegate to appendix A the steps involved in arriving at this conclusion. Consider the scalar product { 0 1o} for k2(2i,+) as given by (A.2). If we orthogonalize the set {Sp[ h(z; T); T]; p = 0, 1, . . . , co}, using the method described on p. 1820 in ref. [6], we obtain a sequence of Tchebychef functions Ap(z; T), which, apart from the sign, is uniquely determined by the following conditions:
{A0tAY>= appv >
p,y=0, 1,. . . ) cc.
(4.5)
Here and henceforth the symbol S,, indicates a Kronecker delta. To understand the definition of Ikenberry’s tensorial harmonics Y”(g), one final word regarding notation. The image space ZIIE”, (Y3 2, of the symmetrizer *’ See appendix B
II in E” will be denoted and LY= 0 by setting kernel
of the trace operator
tensors the
by ET. We extend
elements write (kc’)
Tr in Ey. denoted
cy = 1 and
LY= 0 by setting
of KcrCVTr will be called
(M” ) for := M”).
Fix a single [23,21] Y”(g), Y”(g):=
the
symmetric
unit vector cy = 0. 1,.
traceless
the definition := E and
[(2u + 1)!!/4ntu!)“(
of
of Ker,?Tr
Ker,,Tr := 8.
tensors
M” E [E” (( M’ ) : = A4 ‘,
of
of degree
to The
part
tensorial
(Y. We
harmonics
(4.6)
@‘“g)
projection
The
traceless
g = (k] ‘k. Then Ikenberry’s , x, are defined by
If E((Y(u) E E’” is a natural (4.6)
Ker,Tr
symmetric
a set of real numbers.
by KerCVTr, (Y2 2. is a subset
M” E Ey such that Tr M” = 0. We extend
cases
of E: to the cases cy = I
the definition
(E: := E and Ez := 9., 9 being
121) of IE” onto
KerCkTr, we see from
that
J
dg Y”(g)@
(4.7a)
Y”(g) = ~,J(c+).
W being the set of unit vectors in E. Let (g,,(y); Y = I.. .2cu + I} be an orthonormal basis in KerCVTr: i?,,(r) 0 c!,,(s) = 6,, Then the natural projection E(cy ]a) can uniquely be written as a sum ?
Hence,
using
(4.7a),
we arrive
4s Y:‘(g) Y:‘(g) = L4,
at the equation (4.7b)
.
in which YF( g) := ;,“(~)a Y”(g) E 3. The orthonormal sequence (Yl’A,]r= 1,. .2~ + I: U, @ =O. 1.. . _. x} in H is referred to as the Ikcnberry-Tchebychef sequence. That this sequence forms a basis in N for many common isotropic dispersion relations R = O(y), is ensured by the supplementary analysis, as initiated, for example. at the end of appendix A. If cp belongs
to H, then
cp has a unique
expansion
(4.9)
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
327
and this expansion converges in the mean to the function cp. The tensorial expansion coefficients aasP : = (cp, Y”A,),, E Ker,Tr, as well as their components given by UT” := (cp, Y,“A,),, will be called the Ikenberry-Tchebychef expansion coefficients of cp. Elementary inspection shows that cpE H is a member of H,, if and only if a”.” = 0. Looking back at section 4.1, we may describe the tensorial quantities lo, Y”A,), and Co, Y”A,),@J(~, Y”AyjH as representing two simple and important examples of measurable cylindrical functions on H. Adopting the definitions (4.1) and (4.2), we conclude that the random tensors just mentioned are integrable on H with respect to the Einstein-Smoluchowski promeasure p”,. Indeed, if we set EF[X] := E[X; ~~1, we arrive at E,[LI”.~] = 0 , EF[cPP @a”“]
(4.10a) = ~iS&,E(ala).
(4. lob)
The significance of Tchebychef functions Ap(z; T) is not hard to see: these functions represent the quasiparticle analog of Laguerre polynomials [24] which one normally introduces in the context of the (kinetic) theory of classical gases [S, 91. The possibility of constructing Ap(z; T) through a deeper analysis of Einstein’s ideas should not be surprising, reflecting, as it does, the different way in which one has chosen to view the generalization [6] of Grad’s method [18]. In any event, the starting point is the implementation of very physical and universal assumptions that are intended to provide both the precise definition of Grad’s space H and the approximate expressions for V/z(f) and S(f) (cf. eqs. (3.14) and (3.19a)). What one must do now is to show that the “physical” moments of the distribution function f(k) can be related to the Ikenberry-Tchebychef expansion coefficients uaXPof cp. Then, given eqs. (4.10), it should be clear how one may calculate the fluctuations of those moments. The calculation will be easy if one observes that the aa,P are independent random variables and that the expression for )cp1; is given by
I&=
cc
a=0 p=o
PBoPB.
(4.11)
In this case, one may proceed in evaluating the effects of equilibrium fluctuations as if the “thermodynamic space” spanned by a”,p were not infinitedimensional [19,20]. In summary, if one chooses to work in terms of aa.‘, one may then hope to infer constructive information. It is not possible for other formal counterparts to Grad’s method.
328
4.3.
Z.
Banuch,
S. Piekarski
/
Einstein-Smoluchou,~ki
promrrtsure
for gu.ses
A simple example
For immediate present
notation
use we introduce may be written
% := hV(8n3)-’
] dk c
the quantities
in the
8, P, and 4, which
as
Of,
(4.12a)
P:=hV(Xn’)-‘1dkkf,
(3.12b)
where f =fi,( 1+ Aq) and cp E H. To apply (4.12) in the context of section 4.2, WC are to interpret the cylindrical functions 6 and P as, respectively, the random energy and the random quasimomentum of a subsystem with volume V, and the cylindrical function q as the random heat flux vector for this subsystem. Turning our attention back to the definition of SO( y; T) and the general method of constructing u”.’ (c f. section 4.2 and appendices A and B), WC note that since %, P and 4 are linear combinations of finitely many IkenberryTchebychef expansion coefficients a”.’ of p E H [6, 121. it is possible to adopt eqs. (4.10b) to calculate the expected values of F”. P 8 P, and q @ q. If R(y) iy # const, the calculations. though elementary and straightforward, are lengthy, so we shall only set down the final results for fI( y)/y = c, c being a strictly positive constant, omitting but not disregarding other important dispersion relations. The isotropic and dispersionless
(phonon)
formulas that the equations
O(y)
one obtains
when
V ‘( % - W,,) = a,, (T) a”.” ,
model.
Here
we simply
= cy and ti ’ = (0,~).
V ‘P = a,,(T)
a’.” .
record
If we start
q = a’,(T)
the with
a’.” , (4.13a)
in which
(4.13b)
Z. Banach,
ap(T):’
(YJT) :=
S. Piekarski
I Einsteitdmoluchowski
!g (!g2 (Lg T
promeasure
)
32!
(4.13c:
($)“*(g5’*,
~~:=nV(s~‘)-‘Idk~~“=~
for gases
(4.13d) (F)‘V,
(4.13e)
E
we obtain
w-2@ - %)21 = 4@Y2 E(O(0)
(4.13f)
E,[V2(P@P)]
= F[Crp(T)]* E(l(1) )
(4.13g)
Wll) .
(4.13h)
1
E,[q @sl = +,(n12
We remark in passing that 4 = (6a2NIV)1’3 and E = 41~13N. Although, in a sense, the argument given here is not new, our approach, as being independent of all the axioms central to the theory that forms the mathematical foundation [25] of extended thermodynamics [12], seems to justify, condense, and rationalize some of the earlier propositions and arrangements, e.g., by Jou et al. [19,20]. The discussion in this paper differs from that in refs. (19,201 in one comparatively major respect. Rather than truncating the series on the rhs of (4. ll), we attempt to estimate the expected values of random variables of physical interest according to the pu,-method. One good reason for so doing is simply to emphasize that this can, in fact, be done. Another interesting question has already been alluded to in the introduction. As we know, the Einstein-Smoluchowski promeasure pE defines a mapping fi, from gd, into [0, l] (cf. the text after eq. (4.1)]; I;, is finitely additive on gnH. Let ~(9~) be a minimal o-field over gH. In view of the fact that p, is a canonical Gaussian promeasure determined by the positive quadratic form (2~))‘( 0 ]i on the infinite-dimensional Hilbert space H, we see that fi, cannot be extended to a countably additive set function on (+(gdH) [15-171. Thus the possibility of working with the promeasure p, may be viewed as a reflection of a profound change in perspective vis-a-vis the ordinary interpretation of Einstein’s theory for a system described by a finite number of degrees of freedom. In ref. [7] we point out the importance of H and pu, in any attempt to recover from the kinetic theory of gases results which resemble those found in the Green-Kubo method [13].
5. Final remarks The
main
fluctuations. theory
and discussion
problem based
of ideal
of this paper on Einstein’s
(quasiparticle)
stage in the discussion of which equal
the macrostate
probabilities
the theory
[I] of Boltzmann’s of the distribution of a system.
of equilibrium relation,
gases [3, 141 into a consistent
was the definition
to characterize
a priori
was to combine
inversion
scheme. function
Adopting
[131,the idea then was to obtain
and the The first f’in terms
the method
of
the explicit form of corresponding to f:
.‘\‘; of microstates the expression for the total number Provided that one was willing to make some kind of passage to the continuum limit, this entailed no real difficulties and led us to the notion of entropy. It should
be stressed
that,
given the entropy
S(f)
of a system,
one can determine
Following Einstein, the next stage in the analysis was the approximation of S(J’) - S( f;,) by cq. (3.19a). This was done. not by means of a formal expansion of S(f) - .~(A,) m \ome : mathematical quantity, but, instead, by means of the implementation of physical assumptions that were intended to exploit the universal ideas of Grad [18]. Indeed, one was able to conclude quite generally that. for a system only slightly perturbed away from an overall equilibrium. the entropy functional S(f) - S(f;,) could be expressed in terms of the norm / 0 /,! associated with Grad’s space N. The next order of business was then to view the approximate expression for S( f’) - S( .f;,) as being the useful starting point for constructing the equilibrium promeasure p> on H [15). If one so desires. this can all be done using the elementary techniques employed in the theory of measure and integration ]lS-17). Since one may think of the ideas structured around p, as representing the infinite-dimensional analog of Einstein’s inversion of Boltzmann’s relation, the promcasurc p*, is referred to as the Einstein-Smoluchowski promeasure of scale parameter F. (The scale parameter E is equal to 47~/3N.) Certainly the most significant new feature that arises in our generalization Einstein’s theory is the fact that /..L~cannot be extended to a measure, insofar Grad’s
space
H itself
is concerned.
Fortunately,
the formalism
of as
of this paper
does not rely upon the existence of an equilibrium probability measure on H, and, for this reason, has the distinct advantages of being much simpler in practice and of a more general applicability. For example, the basic physics may be formulated in terms of the cylindrical functions X on H [151. What requires more work to establish, but is of an even greater importance, is that the special basis in H, the so-called Ikenberry-Tchebychef basis [6], provides the effective method for calculating the expected values of cylindrical functions of physical interest. These functions include energy, quasimomentum, and heat
nux.
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
331
Given this state of affairs, one was led to the consideration of a very natural quasiparticle variant [6] of Grad’s method 1181 and to the description of a connection of the moment procedure of Grad’s type with the Einstein thermodynamic theory of equilibrium fluctuations. Consequently, the important point to observe was that the Ikenberry-Tchebychef expansion coefficients ua,p of cp, cp being the nonequilibrium part of f, could be viewed as representing a privileged class of random variables. If one then implements such an assumption, one arrives at eqs. (4.10). The theory of dynamic fluctuations is more complicated than the theory of static fluctuations but not different in kind. Indeed, the decisive new result gotten by considering and motivating the Einstein-Smoluchowski promeasure pu, on Grad’s space H, sketched in this preliminary paper, will allow us to set up an axiomatic structure for the systematic theory of time-dependent fluctuation phenomena occurring in gaseous systems described by the deterministic and stochastic Boltzmann(-Peierls) equations. For example, in ref. [7] we shall study the correlations in time of fluctuations in the Ikenberry-Tchebychef expansion coefficients ua.p of cp under the circumstances that the gaseous system is, on the average, in an equilibrium state.
Acknowledgements One of us (Z.B) acknowledges Prof. D. Jou’s invitation to Barcelona. We thank Profs. J. Casas-Vazquez and D. Jou for helpful discussions concerning the Einstein thermodynamic theory of equilibrium fluctuations.
Appendix A. Mathematical properties of n(y) In section 3.2 we invented for quite general quasiparticle gases a forma1 method of defining Grad’s space H. However, as the remarks after eqs. (3.8d), (3.9~) and (3.13~) indicate, if we wish to obtain from the kinetic-theory approach conditions ensuring the existence of pu,, we should turn our attention away from abstract dispersion relations (w( y; T) : = ?if2( y) lk, T) in favour of a broad class of special ones. Case 2. 0(y)
= cy, c = const > 0. Since in this instance z = (hclk,T)y, y E .C?? +, z E 92; = (0, z,), and the weight function
we arrive at h(z; T) = (k,Tlhc)z, MY(z; T) is given by
332
Z. Bunclch,
vv(2; T) = of course.
S. Piekarski
Einsteirl-Smoluchowski
promeasure
for
gase.,
($g (e:1’;)2 ;
z, = hclk,T
Cuse 2. w(y;
I
T)ly
with the following T)EC’(%
when
depends
.%’ = (0,l)
(A.11 and z, = ~0 when
on y. By definition,
~(0;
S2’ = (0, x).
T) is a function
on %! ’
properties: ‘),
(a)
~(0;
(b)
lim w( y; T) = 0, I”0 +
o’(Y;
w”(Y; T)>O
T) >O,
lim o’(y; \-I~
T)
(c) y/w(y; T) is square-integrable on (0,l) with weight y’. T) > 0. then yi From (a) and (b) we see that if 2,) := lim,_,,+w’(y; and property (c) is satisfied automatically. In o(y; T) 0. Establishing (a)-(c), we have said nothing about our choice of 3 ’ : .9X’ = (0,l) for CF= CFr: (0 s z~, < z, < =) and %! + = (0, m) for CF= LF_ (0 G z,, < z, = SC). If 3 + = (O,m), we add to (a)-(c) the following two properties as being sufficient to justify the formal steps proposed in this paper: (d) As z 3~ the rate of growth of z is no greater than that of w[ h(z; T); T]. (e) There are positive functions @, and Ccz of T and a nonnegative number p such that A'(z; T) [A(z;
T)]’ s
Example 1. The power const S 2).
C, + CJz (magnon)
- zo)” model:
for all z E (z,,, m) 0(y)
= cy’ (c = const > 0.
1 < 1=
Example 2. The isotropic ( phonon) models with dispersion: (i) 0(y) = cy( 1 + 6y’) (c = const > 0, 6 = const > 0, 0 < I= const G 3); (ii) a(y) = cy[l + 6 x y’ln(1 + y)] (c = const>O, ~Y=const>O, O
I4+f4
:= 1 dz w(z; .*:,
T) ‘p, (2) (P&I
64.2)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
333
With these elementary definitions by way of introduction, the above-mentioned properties of a(y) allow us to conclude that if p = 0, 1,2,. . . , ~0, then zP and (A(z; T)l@[A(z; T); TI) z’ are members of k2(.%?i). Moreover, since W(z; T) is nonnegative and measurable in Lebesgue’s sense, since { l]l} > 0, and since, for certain positive functions FU and C of T, W(z; T) s M exp(-d=z), we can prove that the collection of polynomials defined on 9.; forms a dense subspace of fL*($Z!i). (Cf. theorem 3.1.5 in Szegd’s book (ref. [24], p. 40) and the lemma of Dijkstra and van Leeuwen as formulated on p. 468 in ref. [26].)
Appendix
B. Comments
on S,[A(o;
T); T]
It is possible to show that
v-l@ - g,,) = ~‘3 v-‘p
=
J-; &
k,T(cp,
f
Y’S,,),
Y’&),
k,Th
(B.1)
,
03.2)
3
and q = E
$
for 0(y)/y
7-t
k,T(cp,
Y’S,),
03.3)
= c = const (S, = 1, R: = 6n*NIV),
V-‘(8
- i&,0> = I,‘% -$
k,T(cp,
Y’S,,),
and that ,
03.4)
and
q=Jz
3
&
(k,T)2b, Y’S,),
03.6)
for 0(y) ly # const (S,, = 1, Ri = 6n*NIV). Since 8, P and q are basic variables that we use in describing quasiparticle gases, eqs. (B.l)-(B.6) motivate our definition of SPIA(o ; T); T]; cf. also refs. [3-6,121. If some of the functions S, are linearly dependent, then we must reject all unnecessary elements, leaving the procedure of orthogonalization discussed in
section
4.2 otherwise
unchanged:
Case 1 in appendix A. The set {S,;
{SIP; p = 0, 1,2, . . . , x} = {z”;
its subset
Example
1 in appendix A. The set {S,;
byitssubset Example
{S,,,S,,S,,;p=1,2
linearly
IS,,, S,,
S,,;
p = 0, 1,2.
. . . . . =}={l,fz
/3 = 1,2,.
be replaced
. m} (in passing,
by ob-
for z, = x).
Z(i) in appendix A. Since f(fiicik,,T)
subset
, x} should
p = 0, 1.2,
that Ao(z; T) = ~‘%(2rr’))‘(hc/k,T)~‘~
serve
the
J3 = 0, 1,2,
,x}
. . , =} should be replaced ‘,z”;p=l,2 ,.... J;).
LO+, + SLlj +i = (I + l)S,,.
of the set {S,;
p =O, 1..
only ,%} is
independent.
Example
Z(ii)
in appendix
A. The
set
{S,;
/3 = 0, 1,
. ~3;) is linearly
in-
dependent.
Note added in proof We remark
that,
since
E+[(% - &)‘] = C&T’
E(O\O)= 1, eq. (4.13f)
may be written
as (N. la)
,
where
(N.lb) which
is a clear indication
that eq. (4.13f)
is a textbook
result.
References [I] (21 [3] 141 [5] [6] [7] [X]
A. Einstein. Ann. Phys. (Leipzig) 33 (1910) 1275. L. Tisza, Generalized Thermodynamics (MIT Preas, Cambridge. MA. 1977). V.L. Gurevich. Kinetika Fononnykh Sistem (Nauka, Moscow, 1980). R.A. Guyer and J.A. Krumhansl. Phys. Rev. 148 (1966) 766. H. Beck, P.F. Meier and A. Thellung, Phys. Stat. Sol. A 24 (1974) 11. 2. Banach and S. Piekarski, J. Math. Phys. 30 (1989) 1816. Z. Banach and S. Piekarski. Physica A 180 (1992) 336, this volume. P. Rcsiboi5 and M. De Leener, Classical Kinetic Theory of Fluids (Wiley-Interscience, York, 1977). [9] R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley-Interscience, York, 1975).
New New
Z. Banach, S. Piekarski
[lo] [ll] [12] [13] [14] [15]
I Einstein-Smoluchowski
promeasure
for gases
335
K.S. Thorne, Mon. Not. R. Astr. Sot. 194 (1981) 439. N. Udey and W. Israel, Mon. Not. R. Astron. Sot. 199 (1982) 1137. Z. Banach and S. Piekarski, J. Math. Phys. 30 (1989) 1826. D.A. McQuarrie, Statistical Mechanics (Harper and Row, New York, 1976). K. Huang, Statistical Mechanics (Wiley-Interscience, New York, 1963). Y. Choquet-Bruhat, C. Dewitt-Morette and M. Dillard-Bleick, Analysis, Manifolds, and Physics (North-Holland, Amsterdam, 1977). [16] C.W. Burrill, Measure, Integration, and Probability (McGraw-Hill, New York, 1972). [17] H.-H. Kuo, Gaussian measures in Banach spaces, in: Lecture Notes in Mathematics, vol. 463, A. Dold and E. Eckmann, eds. (Springer, Berlin, 1975). [IE] H. Grad, Principles of the kinetic theory of gases, in: Handbuch der Physik, vol. 12, S. Flugge, ed. (Springer, Berlin, 1958), p. 205. [19] L.S. Garcia-Colin, M. Lopez de Haro, R.F. Rodriguez, J. Casas-Vazquez and D. Jou, J. Stat. Phys. 37 (1984) 465. 1201 D. Jou, Physica A 155 (1989) 221. [21] Z. Banach and S. Piekarski, J. Math. Phys. 30 (1989) 1804. [22] R.E. Peierls, Quantum Theory of Solids (Clarendon, Oxford, 1955). [23] C. Truesdell and R.G. Muncaster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas (Academic Press, New York, 1980). [24] G. Szego, Orthogonal Polynomials (Am. Math. Sot. Providence. RI, 1939). [25] Z. Banach, Physica A 159 (1989) 343. [26] J.J. Dijkstra and W.A. van Leeuwen, Physica A 90 (1978) 4.50.
309-335
The Einstein-Smoluchowski (quasiparticle) gases
promeasure
for
Zbigniew Banach Insliture of Fundamental Technological Research, Department of Fluid Mechanics, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland
Siawomir Piekarski Institute of Fundamental Technological Research, Department of Theory of Continuous Media, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland
Received
25 April
1991
Considering an ideal Bose-Einstein gas composed of quasiparticles (phonons, magnons, rotons, etc.), the object of this paper is the development of a tractable generalization of Einstein’s theory of equilibrium fluctuations. It is assumed that the state of the gaseous system may be characterized by the distribution function f(k) which determines the number of quasiparticles with quasimomentum hk. It allows one to embark on a program in which the inversion of Boltzmann’s relation connecting entropy and probability can be used as an organizing principle for the development of the theory dealing with the equilibrium fluctuations of f(k). In the present approach, Einstein’s distribution law in the Gaussian approximation is identified with the equilibrium promeasure CL_defined on the suitably chosen, infinite-dimensional Hilbert space H. This promeasure gives the possibility of calculating the fluctuations in the value of f(k) and its moments. Although pL, cannot be extended to a measure on H, what one discovers is that, in this case, one obtains a manifestly infinitedimensional analog of Einstein’s theory of equilibrium fluctuations. The generalization of the method to the case of classical and relativistic gases is straightforward.
1. Introduction
In this paper we study in detail some aspects of the theory of equilibrium fluctuations. Precisely speaking, our more concrete objective is to formulate and motivate a statement which says that a relatively simple method based on Einstein’s interpretation [l, 21 of Boltzmann’s principle leads to a tractable theory of equilibrium fluctuations occurring in gaseous systems composed of quasiparticles (phonons, magnons, rotons, etc.) [3-51. Beginning from the Boltzmann(-Peierls) equation [3-61, a second paper [7] will emphasize the essential similarities and differences between the static and 0378-4371/92/$05.00
0
1992 - Elsevier
Science
Publishers
B.V. All rights
reserved
time-dependent
theories
of fluctuations.
formalism
of our two papers
its almost
immediate
Thus.
for example,
simple which
gases
that
the
and will. for the most part,
find
application
in the theory
one
like to consider
would
[S, 9). Similarly.
the basic constituents
In 1910. Einstein
It should
is very universal
one would
[ 1] developed
might
also
of classical
be stressed and relativistic
such objects
like to discuss
be photons
or neutrinos
a time-independent
gases.
as the rarefied
relativistic
gases
in
[ 10, 111.
approach
to fluctuations
by introducing statistical concepts into thermodynamics and without making any explicit reference to the detailed microscopic description of the system. As we know,
his basic idea was the inversion of Boltzmann’s relation connecting that one can obtain the entropy and probability. This implies, in particular, relative probabilities of fluctuation states in terms of differences of entropy. In the Einstein thermodynamic theory of equilibrium fluctuations attention is turned toward the macroscopic system, the state of which may be characterized by a finite set of extensive variables. Although Einstein’s theory is absolutely insensitive to the specific choice of a thermodynamic phase space spanned by a finite number of extensive variables. a particular phase space associated with energy, underline
volume, and mole numbers plays a privileged role. In order its importance, we shall refer to it as Gibbs space 121.
to
According to Einstein’s interpretation [I] of Boltzmann’s principle. the only acceptable function that dcscribcs the relation among probability and entropy is the exponential. Then the key simplification is the fact that the entropy S of the (isolated) system need not be considered as some abstract (unknown) function of extensive variables U, but may instead be expanded around equilibrium with respect to the variations of (Y. The physical content of this expansion should be clear: one is assuming that equilibrium is characterized by the maximum of the entropy. Since the very existence of thermodynamics is based on the observation that the fluctuations in the extensive variables arc small compared to the mean values of these variables. the expansion of the entropy
can be broken
off directly
after
the second
term.
Consequently.
on
using the inversion of Boltzmann’s relation. one obtains the probability distribution function in the Gaussian approximation. thcrcby simplifying the theory of equilibrium fluctuations enormously. The basic viewpoint adopted here is that the physics of a gas of quasiparticles may be adequately described in terms of the one-point distribution function f(k) [3-6, 121. A.4 will be emphasized in section 2.1, this function determines the number of quasiparticles with quasimomentum hk. Since flk) is an extensive variable, in the present paper we embark on a program in which the inversion of Boltzmann’s relation can be used as an organizing principle for the development of the theory dealing with the equilibrium fluctuations of f(k).
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
311
Clearly this means that we let the distribution function f(k) become a random variable. Although the general ideas which we have here in mind are very similar to those of Einstein [l], our main contribution is to arrive at the theory of equilibrium fluctuations for f(k) in a novel fashion. The basic concept is in fact straightforward. In order to obtain the inversion of Boltzmann’s relation, one must express the total number of microstates corresponding to the coarse-grained distribution function f(k) in terms of another object - the entropy S of the system identified with some complicated functional S(f) off. As demonstrated in great detail by a number of authors (cf., e.g., refs. [3, 13, 14]), this idea can be motivated in a natural way at least in the context of the theory of ideal gases. Now, if one supposes that the gaseous system is close enough to equilibrium, one is led to approximate the deviation of S(f) from its value for the Bose-Einstein distribution function f,, as being a very simple functional of the nonequilibrium part cpoff (f - fo - cp). Precisely speaking, one has in mind a viewpoint in which the approximate expression for S(f) - S( fi,) is proportional to the square of the norm 1p lH induced by the scalar product of some suitably chosen, infinite-dimensional Hilbert space H. Thus, the paper will address the problem of how to compute the probability for such macrostates f where the entropy can uniquely be determined from the norm of cp. To give this program a precise meaning in a straightforward way, we should deal with, and investigate the mathematical features of, a very specific Bore1 probability promeasure [1_5-171, namely, the Einstein-Smoluchowski probability promeasure )(L~of scale parameter E, defined on the real separable Hilbert space H which we call Grad’s space [18]. Here we anticipate that E-’ is of the order of the volume V of the system. Among other things, the construction or derivation of pF is based on a combined application of the inversion of Boltzmann’s relation and the approximate expression for S(f) S(&,). Consequently, what one discovers is that, in this case, one obtains a manifestly infinite-dimensional analog of Einstein’s thermodynamic theory of equilibrium fluctuations. Indeed, one knows full well that a knowledge of the “continuous” distribution function f(k) is formally equivalent to a knowledge of all the moments of f(k). Therefore, the most rigorous approach would be to enlarge the basic thermodynamic space both by adding the higher order moments of f(k) and by treating them as the fundamental fluctuating variables on the same footing as the traditional Gibbs variables [2]. Due to the existence of CL,, this new approach can be formulated self-consistently in the context of the measure-theoretic treatment of probability theory [U-17]. Also, in order to evaluate the fluctuations in the value of f(k) and its moments (the Ikenberry-
distribution Tchebychef coefficients (6, 121). one need only identify Einstein’s law in the Gaussian approximation with pLcand then proceed in the discussion as if one were working Now,
given
investigations cannot
in the traditional
such a general
program,
is an observation
be extended
to a measure
consequences
Gibbs
space.
the important
result
that the Einstein-Smoluchowski [ 15-171 defined of this new result,
on Grad’s
of our preliminary promeasure
p8
space H [ 181. The
simplest
physical
attempt degrees
to deal with continuous systems described by an infinite of freedom, will be recognized and analyzed elsewhere.
which are essential
when we number
of
Here we proceed as follows. Since in section 2 only the vibrations of the lattice (crystal) are concerned, we treat it there as if it were a quasiparticle gas. It is only for pedagogical purposes that we begin with an ideal phonon gas. Then, making use of a sort of microcanonical prescription based principle of equal a priori probabilities [13]. and appealing to some
on the kind of
passage to the continuum limit, we offer in section 3 the precise definition of Grad’s space H. In addition, section 3 discusses the ideas, systematically investigated in section 4, standing behind the Einstein-Smoluchowski promeasure pc of scale parameter F. Next, applying the Einstein-Smoluchowski promeasure ~~ (which, by its very nature [15-171, cannot be extended to a Bore1 probability measure on H), in section 4.3 we calculate explicitly the so-called second moments [19,2(I) of the random energy %, of the random heat flux q, and of the random quasimomentum P of the macroscopic system. Clearly, to set a proper perspective upon such second moments, it is necessary to understand first the general method for calculating the expected values of the random variables determined from the Ikenberry-Tchebychef expansion coefficients of f(k) [6,12]. Thus we make a few remarks on this method in section 4.2. Some auxiliary technical material, being in fact a supplement to sections 3.2, 4.2 and 4.3, is included as appendices A and B. Section 5 is for the discussion and final remarks. Here we adopt the notation of our two previous papers. Thus the reader interested in arriving at information about it is kindly referred to ref. [6] and section
2 of ref. (211.
One final word regarding our approach. As a matter of fact, it should be said that we choose to think of the (kinetic) theory of gaseous systems and the Einstein thermodynamic theory of equilibrium fluctuations as a unified structure. However, since the actual unification of various deductive schemes is always a very difficult enterprise, we do not adopt a very detailed description of a gas of quasiparticles (phonons, magnons, rotons. etc.) [6,7, 121. Instead, we use the simplest model that still retains the crude qualitative physical properties that might be important. Of course, from the point of view of solid state physics, a more complicated model would be of interest.
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
313
2. Physical motion
2.1. Microcanonical
ensemble
As explained already in the introduction, the fundamental quantity that describes the macrostate of a gas of quasiparticles is the coarse-grained distribution function f(k). The objective here in section 2.1 is to define this function and to calculate the total number JV~ of microstates corresponding to it. Since we focus our attention on a microcanonical ensemble [9] and further restrict it to obey the principle of equal a priori probabilities [3, 13, 141, the calculations leading to f(k) and JV~ are not difficult. For concreteness sake, let us consider an ideal phonon gas which inhabits an insulating crystal composed of only one type of atom. Since we simplify our picture of the gas by neglecting the polarization effects, i.e., the effects depending on the direction of atomic vibrations, it is easy to prove that if a crystal (lattice) of volume V has N atoms, then there will be N normal modes, each characterized by its own “wave number” k, with the characteristic frequencies O(k). Denoting by E the elementary (unit) cell of the reciprocal lattice, and treating k as a vector in the corresponding reciprocal space, one can see that distinct values of k all fall within E. It is of interest to introduce the quantity A(k) which signifies how many phonons exist in the mode k. Note that the set {A(k); k E E} of numbers, the occupation numbers of N normal modes, completely specifies the microstate of the system. Let % be a total energy of the phonon gas, and suppose that 27rfi is Planck’s constant. Then the condition of the form
simply represents the fact that the entire phonon gas is an isolated system, and hence has some fixed energy 8. This is clearly an idealization because we never deal with truly isolated systems in the laboratory. For N z lo’“, it would be either impossible or impractical to decide which of the order of 10N microstates consistent with the energy constraint (2.1) the system is in. Alternatively, one could cut up the unit cell E into a large number of small but finite subcells E’,, IZ= 1, . . . , M, and, instead of exactly specifying the values of A(k), characterize the state of the gas approximately by giving the number F, of phonons in every subcell E’,. Hence we arrive at @=GEn, n=l
Fn:=
c kEE,
A(k);
(2.2)
the E,, can be constructed this way seems
a precise.
important
equilibrium
to show that
fluctuations
special,
definition
our analysis
which is essentially
provides
Of course,
of a particular
the distribution kE@,
function
f(k).
i.e.. the average
Proceeding number
in
and
a basis for the theory
independent
k E E,, if and only if n = n,.
one subcell:
we suggest
of a macrostate,
decomposition of C:. To each k E E corresponds determined from the observation by n,, uniquely
of the denoted only
such that they are disjoint.
if excessively
it of
choice
a positive integer, that k belongs to further,
of phonons
we define in the mode
by
f’(k) := F,,(N,,) ’ .
(2.31)
n-n,.
where N,,:=
c
I.
(2.3b)
!,i C), Suppose now that the characteristic frequency R(k) be (practically) a constant whenever k E E,, , tl = 1. . M. If we are to regard J1 as being a function fl on the set of subcells, then from the energy constraint (2.1) WC easily conclude that
(2.41)
where
i&C&J= f](k) The number number normal
modes
of distributing
of vibration
=(N,,+F,,-I)!
1.
,/’
UV,,of microstates
of ways
Il
(N,,
-
(2.43)
if II = 11~
l)!F,,!
corresponding
to F,, can be identified
F,, indistinguishable
phonons
among
with the the
N,,
of the crystal:
(2.5)
Now. let us set 4 : = {F,,; n = 1. , M}. Then, since interchanging phonons in different subcells does not lead to a new microstate of the system, the total number of microstates compatible with 9, denoted by .Vi. is immediately calculated from
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
31.5
for gases
(2.6) In the special case of equilibrium, if we know nothing about our system, apart from its energy 8, we have no reason to consider any one of the microstates {A(k); k E @} consistent with (2.1) to be more important than any other. Thus we should treat each of them equally, i.e., we should adopt the principle of equal a priori probabilities. Specifically, according to this principle, the probability of finding the system in the macrostate 9 is given by
(2.7) where the sum in the denominator extends over all possible sets 9* of integers F,, . . . , F,, satisfying the energy constraint (2.4a). Now, in view of the definition (2.3a), we can use 9 and f interchangeably to characterize the macrostate of the system. Of course, the total number of microstates corresponding to S is given by (2.6).
2.2.
Transition
to
the subsystem which is not isolated
The derivation of (2.7) is valid insofar as the isolated systems are concerned. However, we may suspect that a slight generalization of the method of section 2.1, namely, the method of equal a priori probabilities, would enable us to discuss also the equilibrium fluctuations of the energy ‘8. To this end, we consider an isolated system made up of two subsystems whose energies are, respectively, Z? and E - 8 (E = const.). We assume, as usual, that 58< E and that both subsystems are macroscopically large. Since there is the redistribution of the quantity ZYamong the two subsystems, the probability of finding the system in the macrostate B is now given, up to a proportionality constant, by P(9)
- N,J-(E
- %)
(2.8)
Here T(E - 8) denotes the total number of microstates available to subsystem with energy E - 8 if these microstates have equal weight. T(E - Z?) must be evaluated subject to the condition that the energy determined from (2.4a). Because of this condition, it is natural to regard state of a composite system as being represented by a single “point” Alternatively, we may describe this state by _r?
the The 8 is the 9.
316
Z. Banach, S. Pieknrski
I Einsteirz-Smolu~hoMski
3. Definition
of Grad’s space H
3.1.
to the continuum
Passage
We shall discuss for a very
large
(but
still finite)
the connection
the entropy
Vh of the isolated
following
limit
now the asymptotic
establishing
is to consider
formula
system.
between system.
for the rhs of (2.6)
This formula
the total
number
As a matter
the case in which
both to obtain the functional terms of the entropy Vh( f)
pmmeusure for gases
dependence corresponding
Stirling’s
plays
of interest
a central
JY, of microstates
role in and
of fact, what we do in the method
113) can be used
of Vh upon f and to calculate to J
.h”? in
If both N,, and F,, arc very large numbers for each n, as is almost always the case. then we may evaluate the quantity (2.6) easily; indeed. by use of (2.5) and Stirling’s approximation”’ to In J;, we conclude that
where .,+; := exp[k,‘Vh(
f’)] ,
h(f)
’ 1 dk F[f(k)]
:= k&&f’)
(3. lb)
,
EIf(k>l:= [I + f(k)] ln[l + f(k)] -f(k)
(3. lc)
In f(k) ;
here and henceforth k,, denotes Boltzmann’s constant. (3.1) we have utilized the fact that. for macroscopic M. can be replaced by an integration n,n=l,..., following manner (3, 141: N,, 3 V&T-‘)-
j dk .
(3. Id) In going from (2.6) to systems, a sum over over k. k E (2. in the
(3.2a)
(3.2b)
we use the letter L!Zto denote the function on the set { F,,/N,,; n = 1,. , M}. In this paper we do not hesitate to call Vh(f) the entropy that corresponds to
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
317
f, because the tradition of the (kinetic) theory of quasiparticle gases gives a persuasive reason for that name [3,14]. If we allow ourselves much freedom, we may set aside all direct reference to the detailed atomic structure of the crystal and seek the frequency 0 and the definition of @ most advantageous for rigorous arguments. Thus arise the following two conditions: (i) the frequency R depends upon the magnitude of k through the transformation rule of the form fin(k) 3 a(y), where y := R,‘/kj and k, := (6~r*N/1/)“‘; (ii) the elementary cell @ is given either by @!, := {k E iE:O
1
if@=Q
dk = A’
(3.3)
e eq. (3.3) motivates, in fact, our definition of 4,. The case CF= cF_, while of physical interest, is not typical and in ref. [7] serves mainly the purpose of establishing a broader agreement between the theory of Boltzmann [8,9], which provides no upper bound for the magnitude of the velocity of a particle, and the theory of Peierls [22]. Now, let us introduce the Bose-Einstein distribution function fo that has the same principal moment#’ as f: f”(w) : = (em - 1))’ ,
w(y; T):=
hL?(y)lk,T,
(3.4a) (3.4b)
Comparison
of (3.4) with the gross condition of the form
hV(Sd-’
1
dk O(y) f(k) = 26’
(35)
c shows that from (3.ld),
T
is the kinetic-theory we arrive at
X2 We call I, dk f2( y) f(k)
the principal
temperature
moment
off.
corresponding
to %‘.Beginning
ff(f’)-ff(J,)-f;,0/-‘=1n(l+e
“‘p)-e +(I
+./‘)e
‘“p
‘“[PI1 - ( 1 + p,,) In( 1 + p,,)] s 0 ,
(.3.&i)
where +=f’,,‘(f-fJ>-1,
PI, .=
de” - ’ ) > _ 1 p + e”’
Clearly f;,~ is a function whose principal moment condition holds for every .f’ that satisfies (3.4b):
(3.6b) is 0: thus
the
following
(3.7)
We may integrate (3.6a) over t? and then by appeal to (3. Ic), (3.7). and (3.6b) conclude that /I( f’) - Cz(_/;,) 5 0 and that h(f) - .f’(f;,) = 0 if and only if p = 0 almost everywhere on L5. Summing up these properties of h( .f’), WC SW that among all distribution functions having the same principal moment, the Bose-Einstein distribution function J, gives /I( 1’) its greatest value. Since in the approach so far developed the entropy density h( J‘) depends in a very complicated way upon f, the general expression for .:1;, as defined by (3.lb)-( 3. Id), and the corresponding theory of equilibrium fluctuations would he almost totally impenetrable. The fact that we can choose V at will, however, allows us to simplify the problem considerably. Indeed, observing that both h( f‘) and ,,1’;have a maximum value when f’= J;,. the spread of .+; about .I;,, can be made arbitrarily narrow by taking the volume V to be arbitrarily large, and we are led to expect that the probability of finding the system in the macrostates f other than those for which If,, ‘(f’ - !;,)I 4 1 is practically equal to zero. We may regard this somewhat vague statement as expressing the content
of Einstein’s
Whether principles,
there is not
basic postulate
origin and significance of simplest consequences and of section 4.1 is to derive, of cqs. (2.7) and (2.X) for of degrees of freedom. 3.2.
The
upproximare
The entropy the objective
in his theory
of equilibrium
fluctuations.
he a precise derivation of Einstein‘s postulate from first presently known. In section 3.2 we shall discuss. not the the inequality I]‘,, applications of that using the results of quasiparticlc gases
expression
for
‘( ./‘- !;,)I 4 1. but some of the inequality. The primary object sections 3.2 and 3.3, the analog described by an infinite number
V[h( f) - h( A,)]
functional (3.1~) is too complicated to use as it stands. Thus, of this section is to derive the approximate expression for
2. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
319
s
VIh(f) - h(.h)l. o as to estimate the deviation of h(f) from its value for the Bose-Einstein distribution function f,, we define the nonequilibrium part of f(k)/&,(~), denoted by A(o) cp(z, g; T), as follows: f(k) = h,(w)tl + NW>cp(z> g; VI ;
(3.8a)
here 2 = qy;
o’(y;
T):=
4Yi {
wI(y;
T) := aw(y;
7,)
if fl( y) ly = const , if a(y) ly f const ,
T)
T)ldy
)
g:=Ik(-‘kED6:={gEIE:lg(=l}.
(3.8b) (3.8~) (3.8d)
we have made the change of variables k 3 (2, g). Interpreting (3.8b)-(3.8d), Of course, in view of the precise definition of w’(y; T), we may expect that considerations based upon (3.8) may restrict the possible choices of n(y), i.e., restrict the class of frequencies for which the method itself can be formulated definitely. Some of such restrictions are suggested in appendix A. For the purposes of the precise theory of equilibrium fluctuations, we must state our position somewhat more abstractly by introducing a condensed notation: 1
Yl *-
i co
if@= if@=
z,, .= hrlir+i(y;T), %+:=(o,y,),
1
(3.9a) zl:=;imf(Y;%
(3.9b)
%(p:=(Z”,Z1).
(3.9c)
Without going into details which are presented that
in appendix A, we postulate
(3.9d) be a solution for y of the equation i(y; T) = z, y E Pi2+. Let h(z; T), ze6Rt, Then, as the analysis of appendix A shows, it is straightforward to derive restrictions on 0(y) which ensure that the function A( 0 ; T) both exists and is continuously differentiable. In fact, we consider the function h(o ; T) such that A(0 ; T) E C’(%!,f); thus A( 0 ; T) has a continuous derivative. Setting
320
Z.
h’(z;
T) := ah(z;
z E 2,;. properties inequality
l&much.
T)/az,
Whether
we shall assume
or not such
of R(y)
is another
A’(z; T) > 0 until
Let H denote
i Einstein-Smoluchowski
S. Piekarski
henceforth
an assumption and
matter,
appendix
the space of real-valued
promeasure
for gases
that h’(z; T) > 0 for each
be consistent we shall
defer
with the general our proof
of the
A. functions
which are square-integrable
on
92+0 x 06 with weight BV(z; 7’) := A2A’VV,1(~). The immediately following analysis shows that to render the method of this paper definite, we must specify not only A(w) but also VVJo).
Clearly (o,o)(, cp E H, denoted
If cpl and (p2 both
lie in H, we set
re p resents the scalar product in the space H, and the norm by I+Q(~, is the nonnegative square root of (cp, cp>r,: thus
of
(3. lob) Considering
a situation
in which the gas is only slightly
perturbed
an overall equilibrium, we wish to find the scalar functions such that if cp = (j&A-‘( f - A,) belongs to H, we obtain
A(o)
away from and V!/,,(w)
where ~:=471./3N
(N-V).
(3.1 lb)
To refer to this program, we first insert (3.8a) into (3.ld) and then replace logarithm ln( 1 + X), (2’1< 1, by its Taylor expansion#3 X - 4X’ + :
the
(3.12) Combining dk = RiA’A’ dg dz with Ri = 6rr’NlV and using and VW,,(w) satisfy the conditions of the form
(3.Ic),
we see
that A(w)
(3.13a) *’ To avoid being overwhelmed by unnecessary complications, given here. We note, however. that X is proportional to q~.
the precise definition of X is not
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
321
(3.13b) hence
A(w)= 41 +.Mw)l
9
W,(w)= &&J) [l + h,(w)1.
(3.13c)
Now we may say that with the choice made above for A(w) and VVO(o), the mathematical properties of LJ( y), which are listed in appendix A, imply that the space H just defined is a real separable Hilbert space. Since our method of introducing H renders systematic some of the ideas Grad [18] seems to have employed in his own studies regarding classical gases [8,9], we name H Grad’s space. We should note one more thing. To require cp to be a member of HO:= { qbE H: (1, #), = 0} is the abstract form of the energy constraint (3.4b) that we impose upon fin this section. Clearly HO is a closed subspace of H. The elementary problem before us, then, is to construct the Einstein-Smoluchowski promeasure p, on H,, by utilizing (3.lb) and the following approximate expression for k,‘V[h( f) - h( &)I: cp E H,) C H ,
(3.14)
As remarked already in the introduction, we should think of p, as being the infinite-dimensional analog of Einstein’s distribution law in the Gaussian approximation. The construction of CL,ensures the possibility of obtaining all the information about the fluctuations of f(k) and its moments. It is useful to summarize the ideas of this section as follows. While the procedure we have used in deriving (3.1) is based upon a particular choice of the decomposition of @ into disjoint subcells @,,, the definition of Grad’s space H is completely independent of it. Even more important is the fact that the same remark concerns the notion of equilibrium promeasure. 3.3. Comments on the composite system As noted already in section 2.2, in fluctuations for a system made up of further discussion and interpretation. the basic object of our theory a more entropy S(f) of a composite system, S(f) := k, ln[N$T(E
- S)] .
the corresponding theory of equilibrium two subsystems we clearly require some In this case, we are led to introduce as complicated functional off, namely, the
(3.15)
Then,
using k,,F
in which
the expression
‘(l,cp),,=T
(3.lha)
‘(X-f,,,),
(cf. cq. (3.5))
%,,:= hV(Xn’)
and appealing S(I)-
of the form
’j
to (3.la). V/2(1;,) + T
dk R(y)
.
i,(o)
(3.lb),
(3.16b)
and (3.11a),
‘(K - c?,:,,)- k,,(2~)
we arrive
at
‘lp15, + k,, In f(E
- 8)
(3.17)
Since only the values of ‘;Cnear g,, are expected to be important, and since 1ci - A,,( e E - v,,, we may perform the expansion of k,, In T(E - % ) around k,, In T(E - go). Neglecting in the expansion nonlinear terms which are normally very small compared to (-k,(i2~)(cplf, (F = 47ri3N; cf. eq. (3.1 lb)), we obtain
k,,lnI’(Ewhere
~)~k,,In
I(-
T,, : = d[k, In T(E - &)]/aE
clearly we set r,, = T. Thus, calculated is
In this equation
C,,)-
r,;‘(G
is the temperature
from eq. (3.17),
(3. IX)
- c,,),
of the larger subsystem;
we see that the quantity
to be
we have written
S( h,) := Vh( A,) + k, In I‘(E - g,,,)
(3.19b)
for brevity. Given the above approximate expression for k,,‘[S( f) - S( A,)] and the inversion of Boltzmann’s relation (3.15), one will therefore be justified in constructing the Gaussian promeasure pF not only on the proper subspace H,, of N but also on Grad’s space N itself. Such an approach will in fact be developed. However, since the precise definition of p+ leads to some mathematical problems, the objective in section 4.1 is to assemble the machinery necessary for that approach.
Z. Banach, S. Piekarski I Einstein-Smoluchowskipromeasure for gases
4. The Einstein-Smoluchowski
323
promeasure p,
4.1. Basic concepts If one were working with the general entropy functional S(f) as given by (3.15), this would be a very complicated problem. Fortunately, in the usual way one can instead choose to focus upon the approximate expression (3.19a) and to provide thereby the axiomatic definition of pc that will arise naturally in terms of it. However, it is worthwhile to introduce first the notion of promeasure which is essentially independent of (3.19a). This will, for example, demonstrate that one should have been led to the theory of fluctuations even if one had never discussed the ideas of Einstein. Let @j(H) be a collection of (closed) subspaces of H of finite dimension together with the partial ordering relation C . It will be convenient to use the symbol d(H) to denote the set of finite-dimensional orthogonal projections of H. For R E a(H), define QR E d(H) by QR(H) = R; thus the image of H by QR is R. When @(H) 3 R C S E Q(H), one can introduce a mapping from S onto R, denoted by QsR, through the relation of the form QR = Q,s,Q,T. Elementary inspection shows that QSR is a restriction of Q, to S. A Bore1 probability promeasure [15] on H is a family Al.:= { pR; R E Q(H)} defined as follows: (i) pu,, R E Q(H), 1s . a Bore1 probability measure on R; (ii) if &5(H) 3 R C S E B(H), then pR(G) = pu,(D) for every Bore1 subset G of R, where D is a (Bore]) cylinder in S induced by G and QsR; thus D := Q.;;(G) := {p E S: QsR’p E G}. Since p is a projective system of measures on the projective system {R, QsR; R, S E@(H)} of finite-dimensional subspaces of H, we are justified in using for p here the name “promeasure” [15]. The notion of Bore1 probability promeasure is altogether natural. Indeed, in formulating a mathematical model of a random experiment, one must assign probabilities to the various events under consideration. That is, given R E 4(H) and a Bore1 subset G of R, to the (Borel) cylinder D of H defined by D := Q,‘(G) := {cp E H: QR(p E G} one must assign a probability b(D) : = p,(G) of the event D taking place. At times we shall use the symbol ad, to signify the set of all Bore1 cylinders of H. Using the approximate equation (3.19a), we are now ready to provide the precise definition of F~, establishing thereby an important connection of Einstein’s ideas [l] with the ideas of Choquet-Bruhat et al. [15] discussed above. For each R E Q(H), let %‘Rbe a Bore1 o-field in R, and consider the function pu,, : B3, + [0, l] defined by
324
Z. Bunuch.
S. Piekurski
I Einstein-Smoluchowski
promeasure
for guse.s
(4.1)
G lies in % R ’ n is equal to dim R, and p,_ denotes the Lcbesgue measure on R. By far the most important promeasure on Grad’s space H is that which is where
generated
by pFR. We call this the Einstein-Smoluchowski
parameter
F and hereafter reserve the symbol pF to denote it. For any R E Q(H)
and
GE
any
933,, we
set
&(,!I)
: = pLF,((G)
whenever
promeasure D E P,,
of scale and
D =
Q&3. At this stage, it is useful to introduce the notion of a cylindrical function on H. Let fE be a three-dimensional Euclidean vector space, and consider for each cy. cy=o.1,2 . . . . . x. the cvth tensorial power iE” := 8 “iE of [E. Since IE” is a real separable Hilbert space with the inner (tensor) product”J M’; 0 M’,‘, M’,’ and MI: being elements of fE”, one can introduce the Bore1 a-field :dCVin E”. A mapping X from H into [E” is called a cylindrical function on H if there exists R E Sj( H) such that X = XQ,3. We say that a cylindrical function X : H + IE”is measurable, or a random tensor of degree cy. if the inverse image of D E :#I<,by the restriction of X = XQ, to R E G(H) is a Bore1 subset of R. In addition. the function X just mentioned is said to be integrable on H with respect to the promeasure p : = { P,~; S E o(H)} if the restriction of X = XQ, to R E $T(H) is on R with respect
integrable
I A+)
X(P) :=
The
expected
I ,ddv)
writes
Wcp>.
(4.2)
I(
II
function
to P,<; one then
value,
X. written
or expectation or mean, of an integrable cylindrical E[X; ~1, is defined to be the integral of X; thus
(4.3)
Eq. (4.1) is of course independent of the choice of a basis in H, and, for that reason, particularly useful; and, if it be so desired (cf. section 4.2), it may be “projected” into any convenient chart. This expression, namely, eq. (4.1), resembles quite closely a result first established by Einstein [l], differing, however, in that Einstein “excluded” the possibility of considering the infinite number of degrees of freedom. The next stage in the analysis is to obtain the explicit form of pF for a specific basis of interest. And, therefore, it is natural ” Concerning of ref. 121).
the definitwn
of the inner
(tensor)
product.
see rqs.
(A.h)-(A.9)
in appendix
A
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
325
to ask: Is there a connection between the theory of Einstein [l] and the theory of Grad [Ml? Could one at least capture, in some suitable way [6], a quasiparticle analog of Grad’s moment procedure? To these problems the remaining sections of this work are primarily devoted. 4.2. Fluctuations of the Ikenberry-Tchebychef
coeficients
aaxP
The class of possible orthonormal sequences in H is extremely broad, far too broad for us to be able to expect to get from any of its members the effective method for calculating the expected values of cylindrical functions of physical interest. To proceed further, we must be somewhat more specific about the choice of orthonormal sequences in H, and in this section we shall define, and present some of the basic properties of, the so-called Ikenberry-Tchebychef sequence [6]. We begin from the definition of the Tchebychef functions Ap(z; T). As the discussion in section 4.3 and appendix B suggests, fields intended to correspond with those common in the kinetic theory of classical gases are generated by the functions SP( y; T) of the form L~(Y; VI"
[yIo(y;
T)][i(y;
T)]”
for p = 0,2,4, . . . , w , for /3 = 1,3,5,. . . ,m,
(4.4)
where Y is the greatest integer =Sp/2 and y E 3 + (cf. eq. (3.9~)). By (3.8b) the effect of substituting w = o( y; T) into (4.4) leads to a family {S,( y; T); co} which is either linearly independent or linearly dependent; in /?=O,l,..., case of need, we must reject all unnecessary elements? Let [L’(!%,‘) be a set of all real-valued functions which are square-integrable on %!i with weight VV(z; T). Then it is possible to prove that S,[h(o; T); T] are elements of k2($?i). We relegate to appendix A the steps involved in arriving at this conclusion. Consider the scalar product { 0 1o} for k2(2i,+) as given by (A.2). If we orthogonalize the set {Sp[ h(z; T); T]; p = 0, 1, . . . , co}, using the method described on p. 1820 in ref. [6], we obtain a sequence of Tchebychef functions Ap(z; T), which, apart from the sign, is uniquely determined by the following conditions:
{A0tAY>= appv >
p,y=0, 1,. . . ) cc.
(4.5)
Here and henceforth the symbol S,, indicates a Kronecker delta. To understand the definition of Ikenberry’s tensorial harmonics Y”(g), one final word regarding notation. The image space ZIIE”, (Y3 2, of the symmetrizer *’ See appendix B
II in E” will be denoted and LY= 0 by setting kernel
of the trace operator
tensors the
by ET. We extend
elements write (kc’)
Tr in Ey. denoted
cy = 1 and
LY= 0 by setting
of KcrCVTr will be called
(M” ) for := M”).
Fix a single [23,21] Y”(g), Y”(g):=
the
symmetric
unit vector cy = 0. 1,.
traceless
the definition := E and
[(2u + 1)!!/4ntu!)“(
of
of Ker,?Tr
Ker,,Tr := 8.
tensors
M” E [E” (( M’ ) : = A4 ‘,
of
of degree
to The
part
tensorial
(Y. We
harmonics
(4.6)
@‘“g)
projection
The
traceless
g = (k] ‘k. Then Ikenberry’s , x, are defined by
If E((Y(u) E E’” is a natural (4.6)
Ker,Tr
symmetric
a set of real numbers.
by KerCVTr, (Y2 2. is a subset
M” E Ey such that Tr M” = 0. We extend
cases
of E: to the cases cy = I
the definition
(E: := E and Ez := 9., 9 being
121) of IE” onto
KerCkTr, we see from
that
J
dg Y”(g)@
(4.7a)
Y”(g) = ~,J(c+).
W being the set of unit vectors in E. Let (g,,(y); Y = I.. .2cu + I} be an orthonormal basis in KerCVTr: i?,,(r) 0 c!,,(s) = 6,, Then the natural projection E(cy ]a) can uniquely be written as a sum ?
Hence,
using
(4.7a),
we arrive
4s Y:‘(g) Y:‘(g) = L4,
at the equation (4.7b)
.
in which YF( g) := ;,“(~)a Y”(g) E 3. The orthonormal sequence (Yl’A,]r= 1,. .2~ + I: U, @ =O. 1.. . _. x} in H is referred to as the Ikcnberry-Tchebychef sequence. That this sequence forms a basis in N for many common isotropic dispersion relations R = O(y), is ensured by the supplementary analysis, as initiated, for example. at the end of appendix A. If cp belongs
to H, then
cp has a unique
expansion
(4.9)
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
327
and this expansion converges in the mean to the function cp. The tensorial expansion coefficients aasP : = (cp, Y”A,),, E Ker,Tr, as well as their components given by UT” := (cp, Y,“A,),, will be called the Ikenberry-Tchebychef expansion coefficients of cp. Elementary inspection shows that cpE H is a member of H,, if and only if a”.” = 0. Looking back at section 4.1, we may describe the tensorial quantities lo, Y”A,), and Co, Y”A,),@J(~, Y”AyjH as representing two simple and important examples of measurable cylindrical functions on H. Adopting the definitions (4.1) and (4.2), we conclude that the random tensors just mentioned are integrable on H with respect to the Einstein-Smoluchowski promeasure p”,. Indeed, if we set EF[X] := E[X; ~~1, we arrive at E,[LI”.~] = 0 , EF[cPP @a”“]
(4.10a) = ~iS&,E(ala).
(4. lob)
The significance of Tchebychef functions Ap(z; T) is not hard to see: these functions represent the quasiparticle analog of Laguerre polynomials [24] which one normally introduces in the context of the (kinetic) theory of classical gases [S, 91. The possibility of constructing Ap(z; T) through a deeper analysis of Einstein’s ideas should not be surprising, reflecting, as it does, the different way in which one has chosen to view the generalization [6] of Grad’s method [18]. In any event, the starting point is the implementation of very physical and universal assumptions that are intended to provide both the precise definition of Grad’s space H and the approximate expressions for V/z(f) and S(f) (cf. eqs. (3.14) and (3.19a)). What one must do now is to show that the “physical” moments of the distribution function f(k) can be related to the Ikenberry-Tchebychef expansion coefficients uaXPof cp. Then, given eqs. (4.10), it should be clear how one may calculate the fluctuations of those moments. The calculation will be easy if one observes that the aa,P are independent random variables and that the expression for )cp1; is given by
I&=
cc
a=0 p=o
PBoPB.
(4.11)
In this case, one may proceed in evaluating the effects of equilibrium fluctuations as if the “thermodynamic space” spanned by a”,p were not infinitedimensional [19,20]. In summary, if one chooses to work in terms of aa.‘, one may then hope to infer constructive information. It is not possible for other formal counterparts to Grad’s method.
328
4.3.
Z.
Banuch,
S. Piekarski
/
Einstein-Smoluchou,~ki
promrrtsure
for gu.ses
A simple example
For immediate present
notation
use we introduce may be written
% := hV(8n3)-’
] dk c
the quantities
in the
8, P, and 4, which
as
Of,
(4.12a)
P:=hV(Xn’)-‘1dkkf,
(3.12b)
where f =fi,( 1+ Aq) and cp E H. To apply (4.12) in the context of section 4.2, WC are to interpret the cylindrical functions 6 and P as, respectively, the random energy and the random quasimomentum of a subsystem with volume V, and the cylindrical function q as the random heat flux vector for this subsystem. Turning our attention back to the definition of SO( y; T) and the general method of constructing u”.’ (c f. section 4.2 and appendices A and B), WC note that since %, P and 4 are linear combinations of finitely many IkenberryTchebychef expansion coefficients a”.’ of p E H [6, 121. it is possible to adopt eqs. (4.10b) to calculate the expected values of F”. P 8 P, and q @ q. If R(y) iy # const, the calculations. though elementary and straightforward, are lengthy, so we shall only set down the final results for fI( y)/y = c, c being a strictly positive constant, omitting but not disregarding other important dispersion relations. The isotropic and dispersionless
(phonon)
formulas that the equations
O(y)
one obtains
when
V ‘( % - W,,) = a,, (T) a”.” ,
model.
Here
we simply
= cy and ti ’ = (0,~).
V ‘P = a,,(T)
a’.” .
record
If we start
q = a’,(T)
the with
a’.” , (4.13a)
in which
(4.13b)
Z. Banach,
ap(T):’
(YJT) :=
S. Piekarski
I Einsteitdmoluchowski
!g (!g2 (Lg T
promeasure
)
32!
(4.13c:
($)“*(g5’*,
~~:=nV(s~‘)-‘Idk~~“=~
for gases
(4.13d) (F)‘V,
(4.13e)
E
we obtain
w-2@ - %)21 = 4@Y2 E(O(0)
(4.13f)
E,[V2(P@P)]
= F[Crp(T)]* E(l(1) )
(4.13g)
Wll) .
(4.13h)
1
E,[q @sl = +,(n12
We remark in passing that 4 = (6a2NIV)1’3 and E = 41~13N. Although, in a sense, the argument given here is not new, our approach, as being independent of all the axioms central to the theory that forms the mathematical foundation [25] of extended thermodynamics [12], seems to justify, condense, and rationalize some of the earlier propositions and arrangements, e.g., by Jou et al. [19,20]. The discussion in this paper differs from that in refs. (19,201 in one comparatively major respect. Rather than truncating the series on the rhs of (4. ll), we attempt to estimate the expected values of random variables of physical interest according to the pu,-method. One good reason for so doing is simply to emphasize that this can, in fact, be done. Another interesting question has already been alluded to in the introduction. As we know, the Einstein-Smoluchowski promeasure pE defines a mapping fi, from gd, into [0, l] (cf. the text after eq. (4.1)]; I;, is finitely additive on gnH. Let ~(9~) be a minimal o-field over gH. In view of the fact that p, is a canonical Gaussian promeasure determined by the positive quadratic form (2~))‘( 0 ]i on the infinite-dimensional Hilbert space H, we see that fi, cannot be extended to a countably additive set function on (+(gdH) [15-171. Thus the possibility of working with the promeasure p, may be viewed as a reflection of a profound change in perspective vis-a-vis the ordinary interpretation of Einstein’s theory for a system described by a finite number of degrees of freedom. In ref. [7] we point out the importance of H and pu, in any attempt to recover from the kinetic theory of gases results which resemble those found in the Green-Kubo method [13].
5. Final remarks The
main
fluctuations. theory
and discussion
problem based
of ideal
of this paper on Einstein’s
(quasiparticle)
stage in the discussion of which equal
the macrostate
probabilities
the theory
[I] of Boltzmann’s of the distribution of a system.
of equilibrium relation,
gases [3, 141 into a consistent
was the definition
to characterize
a priori
was to combine
inversion
scheme. function
Adopting
[131,the idea then was to obtain
and the The first f’in terms
the method
of
the explicit form of corresponding to f:
.‘\‘; of microstates the expression for the total number Provided that one was willing to make some kind of passage to the continuum limit, this entailed no real difficulties and led us to the notion of entropy. It should
be stressed
that,
given the entropy
S(f)
of a system,
one can determine
Following Einstein, the next stage in the analysis was the approximation of S(J’) - S( f;,) by cq. (3.19a). This was done. not by means of a formal expansion of S(f) - .~(A,) m \ome : mathematical quantity, but, instead, by means of the implementation of physical assumptions that were intended to exploit the universal ideas of Grad [18]. Indeed, one was able to conclude quite generally that. for a system only slightly perturbed away from an overall equilibrium. the entropy functional S(f) - S(f;,) could be expressed in terms of the norm / 0 /,! associated with Grad’s space N. The next order of business was then to view the approximate expression for S( f’) - S( .f;,) as being the useful starting point for constructing the equilibrium promeasure p> on H [15). If one so desires. this can all be done using the elementary techniques employed in the theory of measure and integration ]lS-17). Since one may think of the ideas structured around p, as representing the infinite-dimensional analog of Einstein’s inversion of Boltzmann’s relation, the promcasurc p*, is referred to as the Einstein-Smoluchowski promeasure of scale parameter F. (The scale parameter E is equal to 47~/3N.) Certainly the most significant new feature that arises in our generalization Einstein’s theory is the fact that /..L~cannot be extended to a measure, insofar Grad’s
space
H itself
is concerned.
Fortunately,
the formalism
of as
of this paper
does not rely upon the existence of an equilibrium probability measure on H, and, for this reason, has the distinct advantages of being much simpler in practice and of a more general applicability. For example, the basic physics may be formulated in terms of the cylindrical functions X on H [151. What requires more work to establish, but is of an even greater importance, is that the special basis in H, the so-called Ikenberry-Tchebychef basis [6], provides the effective method for calculating the expected values of cylindrical functions of physical interest. These functions include energy, quasimomentum, and heat
nux.
Z. Banach, S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
331
Given this state of affairs, one was led to the consideration of a very natural quasiparticle variant [6] of Grad’s method 1181 and to the description of a connection of the moment procedure of Grad’s type with the Einstein thermodynamic theory of equilibrium fluctuations. Consequently, the important point to observe was that the Ikenberry-Tchebychef expansion coefficients ua,p of cp, cp being the nonequilibrium part of f, could be viewed as representing a privileged class of random variables. If one then implements such an assumption, one arrives at eqs. (4.10). The theory of dynamic fluctuations is more complicated than the theory of static fluctuations but not different in kind. Indeed, the decisive new result gotten by considering and motivating the Einstein-Smoluchowski promeasure pu, on Grad’s space H, sketched in this preliminary paper, will allow us to set up an axiomatic structure for the systematic theory of time-dependent fluctuation phenomena occurring in gaseous systems described by the deterministic and stochastic Boltzmann(-Peierls) equations. For example, in ref. [7] we shall study the correlations in time of fluctuations in the Ikenberry-Tchebychef expansion coefficients ua.p of cp under the circumstances that the gaseous system is, on the average, in an equilibrium state.
Acknowledgements One of us (Z.B) acknowledges Prof. D. Jou’s invitation to Barcelona. We thank Profs. J. Casas-Vazquez and D. Jou for helpful discussions concerning the Einstein thermodynamic theory of equilibrium fluctuations.
Appendix A. Mathematical properties of n(y) In section 3.2 we invented for quite general quasiparticle gases a forma1 method of defining Grad’s space H. However, as the remarks after eqs. (3.8d), (3.9~) and (3.13~) indicate, if we wish to obtain from the kinetic-theory approach conditions ensuring the existence of pu,, we should turn our attention away from abstract dispersion relations (w( y; T) : = ?if2( y) lk, T) in favour of a broad class of special ones. Case 2. 0(y)
= cy, c = const > 0. Since in this instance z = (hclk,T)y, y E .C?? +, z E 92; = (0, z,), and the weight function
we arrive at h(z; T) = (k,Tlhc)z, MY(z; T) is given by
332
Z. Bunclch,
vv(2; T) = of course.
S. Piekarski
Einsteirl-Smoluchowski
promeasure
for
gase.,
($g (e:1’;)2 ;
z, = hclk,T
Cuse 2. w(y;
I
T)ly
with the following T)EC’(%
when
depends
.%’ = (0,l)
(A.11 and z, = ~0 when
on y. By definition,
~(0;
S2’ = (0, x).
T) is a function
on %! ’
properties: ‘),
(a)
~(0;
(b)
lim w( y; T) = 0, I”0 +
o’(Y;
w”(Y; T)>O
T) >O,
lim o’(y; \-I~
T)
(c) y/w(y; T) is square-integrable on (0,l) with weight y’. T) > 0. then yi From (a) and (b) we see that if 2,) := lim,_,,+w’(y; and property (c) is satisfied automatically. In o(y; T)
T)]’ s
Example 1. The power const S 2).
C, + CJz (magnon)
- zo)” model:
for all z E (z,,, m) 0(y)
= cy’ (c = const > 0.
1 < 1=
Example 2. The isotropic ( phonon) models with dispersion: (i) 0(y) = cy( 1 + 6y’) (c = const > 0, 6 = const > 0, 0 < I= const G 3); (ii) a(y) = cy[l + 6 x y’ln(1 + y)] (c = const>O, ~Y=const>O, O
I4+f4
:= 1 dz w(z; .*:,
T) ‘p, (2) (P&I
64.2)
Z. Banach,
S. Piekarski
I Einstein-Smoluchowski
promeasure
for gases
333
With these elementary definitions by way of introduction, the above-mentioned properties of a(y) allow us to conclude that if p = 0, 1,2,. . . , ~0, then zP and (A(z; T)l@[A(z; T); TI) z’ are members of k2(.%?i). Moreover, since W(z; T) is nonnegative and measurable in Lebesgue’s sense, since { l]l} > 0, and since, for certain positive functions FU and C of T, W(z; T) s M exp(-d=z), we can prove that the collection of polynomials defined on 9.; forms a dense subspace of fL*($Z!i). (Cf. theorem 3.1.5 in Szegd’s book (ref. [24], p. 40) and the lemma of Dijkstra and van Leeuwen as formulated on p. 468 in ref. [26].)
Appendix
B. Comments
on S,[A(o;
T); T]
It is possible to show that
v-l@ - g,,) = ~‘3 v-‘p
=
J-; &
k,T(cp,
f
Y’S,,),
Y’&),
k,Th
(B.1)
,
03.2)
3
and q = E
$
for 0(y)/y
7-t
k,T(cp,
Y’S,),
03.3)
= c = const (S, = 1, R: = 6n*NIV),
V-‘(8
- i&,0> = I,‘% -$
k,T(cp,
Y’S,,),
and that ,
03.4)
and
q=Jz
3
&
(k,T)2b, Y’S,),
03.6)
for 0(y) ly # const (S,, = 1, Ri = 6n*NIV). Since 8, P and q are basic variables that we use in describing quasiparticle gases, eqs. (B.l)-(B.6) motivate our definition of SPIA(o ; T); T]; cf. also refs. [3-6,121. If some of the functions S, are linearly dependent, then we must reject all unnecessary elements, leaving the procedure of orthogonalization discussed in
section
4.2 otherwise
unchanged:
Case 1 in appendix A. The set {S,;
{SIP; p = 0, 1,2, . . . , x} = {z”;
its subset
Example
1 in appendix A. The set {S,;
byitssubset Example
{S,,,S,,S,,;p=1,2
linearly
IS,,, S,,
S,,;
p = 0, 1,2.
. . . . . =}={l,fz
/3 = 1,2,.
be replaced
. m} (in passing,
by ob-
for z, = x).
Z(i) in appendix A. Since f(fiicik,,T)
subset
, x} should
p = 0, 1.2,
that Ao(z; T) = ~‘%(2rr’))‘(hc/k,T)~‘~
serve
the
J3 = 0, 1,2,
,x}
. . , =} should be replaced ‘,z”;p=l,2 ,.... J;).
LO+, + SLlj +i = (I + l)S,,.
of the set {S,;
p =O, 1..
only ,%} is
independent.
Example
Z(ii)
in appendix
A. The
set
{S,;
/3 = 0, 1,
. ~3;) is linearly
in-
dependent.
Note added in proof We remark
that,
since
E+[(% - &)‘] = C&T’
E(O\O)= 1, eq. (4.13f)
may be written
as (N. la)
,
where
(N.lb) which
is a clear indication
that eq. (4.13f)
is a textbook
result.
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