612
case one is led to just the same equat,ions already given by Gibbs
(even here, however, as was noted in the Brandeis lectures, but
not in this book, the information theory viewpoint leads to a greater
flexibility in our choice of the basic variables).
The only place where new results are possible is in the extension
to nonequilibrium problems, where previously no definite, un-
ambiguous theory existed. However, the author’s discussion of
small deviations from equilibrium (transport coefficients, etc.) in
Chapter 8, is conducted in a way that owes nothing at. all to the
information theory approach. It reproduces only the results which
had been found by Onsager, Casimir, Green, Kirkwood, Callen,
Krrbo, Mori, and others prior to 1958, and which made no use of
information theory. A more complete account of this development
can be found in R. Kubo’s lectures (1968 Boulder Lectzcrcs in, Z’hro-
retical
Physics, Vol. 1, W. Brittin and L. Dunham, Eds. New York:
Interscience 1959). At this point, let us turn back to that “tech-
nical problem of getting the answers.”
In all of this early work, one was troubled with the so-called
“induction t,ime” and ‘Lplateau” phenomena. It was necessary to
carry out some kind of coarse-graining, llsually a time or space-time
average, before equations of the desired phenomenological form
(diffusion current proportional to density gradient, etc.) emerged.
In this work, one first set up an ensemble usually called the “local
equilibrium” or “frozen-state” ensemble, which described spatial
variations of t,emperat,ure, particle densit,y, etc. But when we try
to calculate the resulting fluxes (heat flow, etc.) or rates of relaxation
to equilibrium from this ensemble, we find to
our
dismay that the
result is identically zero.
Mathematically, it was found necessary to integrate the eqllations
of motion forward for a short “induction time” before the irrevers-
ible process gets going; and uncertainties about just how long we
must integrate before reaching the conjectured “plateau” vallles
led to ambiguities in the final formulas for transport coefficients.
Perhaps even worse from a practical standpoint, the inbroduction
of coarse-graining has the consequence that one cannot treat gen-
eral problems; but is limited to the quasi-stationary or long-wave-
lengt,h limit. This has been a famous difficulty in the theory of
irreversible processes, well recognized for over twenty years.
It is precisely at this point that t,he information theory approach
can make one of its most important contributions to the “technical
problem of getting the answers.” The need for carrying out this
forward integration and coarse-graiuing is, of course, merely a
symptom of the fact that the initially chosen ensemble did not cor-
rectly represent the physical situation (or, more accurately stated,
the correct range of microstates) in which we have the irreversible
process. These operations are corrective measures which in some
way compensate for the error in the initial ensemble.
But, if the theory were correctly set up in the first place, sllch
artificial devices would not be necessary; the flllxes and transport
coefficients coldd be calculated by direct qrladratrlres over the
initial ensemble. Furthermore, instead of ml&ilating the formalism
by coarse-graining in order to obtain equations of a preconceived
phenomenological form, we could tell from the t.heory the exact
conditions under which this form is correct. However, before the
introduction of the information theory viewpoint, nobody was
able to understand the exact nature of this error, much less how to
correct it; and it is not recognized in this book.
Here is the secret. Recall that in the information theory approach,
we are given certain information and use the principle of maximum
entropy to construct an ensemble corresponding to that information.
But, we can equally well reason in the opposite direction; the en-
tirely new insight into this problem is the realization that, given
any proposed ensemble, it makes perfectly definite sense to ask,
“What
is the &formation contained in this ensemble?”
It is the same
as asking, “With respect to
which
constraints does this ensemble
have maximum entropy?” To me, it is one of the most beautiful
aspects of this theory that the mathematics works out so that the
answer can be read off immediately, by mere inspection of any
ensemble.
IEEF, TRAXSACTIOKS ON IKFORMATION THEORY, JULY,
1968
As soon as the problem was looked at in this way, the answer
was obvious at a glance; a very essential piece of information was
missing in the “frozen-state”
ensemble. To restore it, one needs
to incorporate information not only over a space region at one ill-
stant of time, but over a space-time region; and the partition fnnctiotl
gets generalized to a partition functional over functions defillrtl
in this region. This insight into how to correct the “frozen-stat.e”
ensemble had the effect of opening the flood gates; immediatel!.,
both old and new formulas for transport coefficients pollred fort tr
as fast as one could write. Because the forward integration w:l.,
done with, one could specify the conditions under which Kubo’:
formulas were exact; and give corrections for other conditions
Because the coarse-graining was done with, we were no longer re
stricted to quasi-stationary processes like diffusion; wit,h equal east
the formalism yielded general formulas for rapid processes sllch :I
ultrasonic att,enuation. With further mathematical developrnellt
I believe it will be possible to find equally general treatments (
highly nonlinear phenomena, such as shock waves.
It is in the areas just indicated that the real power of the info
mation theory approach is found. Until the full theory has bet
written up, and a few more applications worked out, an objecti
assessment of its usefulness and limitations will not be possibl
In summary, this book is a useful introduction; but the read
should be forewarned that the real justification of this approa
appears only in further developments that begin where this bo
leaves off. Except, for a few apparently original proofs, it cont,ai
no actual results that had not been published elsewhere; in m(
cases prior to 1958 and in much more complete form. The seric
student of the subject will, therefore, be astonished, dismayed, a
handicapped by the failure to provide even a single reference. TII
is not even any acknowledgement of the work-or indeed,
existence-of Gibbs and Shannon, who created the two streams
thought here fruitfully merged.
EDWIK
T. Jnr
Dept. of Ph>
Washington Univel
St. Louis, MO. 6:
Detection, Estimation, and Modulation Theory, Part I, Harry L.
Trees (John Wiley and Sons, Inc., New York, 1968, xiv +
697
$20.00).
Van Trees has succeeded, where many have failed, in w1
a textbook for a
course
in detection and estimation theory. De
from course notes, this text inchtdes many of the periphern’
tures which make a book suitable for class option. Among
are numerous examples, over 120 pages of problems (in small I
and a la-page glossary of notation. The text is well refers
and is indexed by both subject and author.
The core material consists of a lucid treatment of basic to]
detection aud estimation theory with applications to commulli
problems. Most of the rnaterial is fundamental, having al)]
previously in one text or another, although not under one
Treatment, of the subject is classical. Signal representations
ou the Karhunen-Loeve expansion lead to eventual involverr
the properties of integral eqrlations. The following chap
chapter sketch will indicate the specific topics covered.
After an introductory chapter, Van Trees reviews the b:
pects of decision theory from the Bares and Neyman-Pearsol
points. These ideas are then extended to the estimation probl
discussion of various estimation criteria-the Bayes risk, m
mean-sqllare error, maximum likelihood, maximum a pc
probabilit.y-together with the conditions under which tl
be applied, and relationships among the resultant estimate
iations of the Chernoff bound and the Cram&-Rae boluld
rived for use when performances in the decision and es1
problems are not easily computed. This situation certain1
when an estimator cannot be represented explicitly as a
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