Chaos, Dissipation, Arrow of Time,

in Quantum Physics i
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NIST Technical Note 1403

Chaos, Dissipation, Arrow of Time,

in Quantum Physics

Michael Danos

Physics Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899

May 1993

« WW -6

e.

^rts o* *

U.S. Department of Commerce

Ronald H. Brown, Secretary

National Institute of Standards and Technology

Raymond G. Kammer, Acting Director
National Institute of Standards U.S. Government Printing Office For sale by the Superintendent

and Technology Washington: 1993 of Documents

Technical Note 1403 U.S. Government Printing Office

Natl. Inst. Stand. Technol. Washington, DC 20402

Tech. Note 1403

12 pages (May 1993)

CODEN: NTNOEF
CHAOS, DISSIPATION, ARROW OF TIME,

IN QUANTUM PHYSICS

Michael Danos*

Physics Laboratory

National Institute of Standards and Technology

Technology Administration

U.S. Department of Commerce

Gaithersburg, MD

Abstract

A compact description of the evolution of a many-body system, e.g., a dilute gas, is provided by the generalization of the usual reaction S-matrix or U-matrix to a system S- or U-matrix. Using this tool it is demonstrated, that (i) the characterization of quantum chaos

turns out to be very transparent: already exceedingly simple systems, including time-reversal

invariant states, are capable of exhibiting quantum chaos; (ii) the time-reversal invariance of

the Hamiltonian leads to relaxation of arbitrary non-equilibrium states of chaotic quantum

systems, i.e., to dissipation, which thus allows the definition of a quantum arrow of time;

(iii) the second law of thermodynamics, and hence the complete field of thermodynamics, is a consequence of quantum physics.

""Visiting Scholar, Enrico Fermi Institute, University of Chicago, Chicago, Illinois.

111

1 Introduction

The question of chaos, in particular also of quantum chaos, over the last few years has attracted

considerable attention. Surprising as its recognition was, the existence of mathematical chaos,

or as it also is called, deterministic chaos, does not offer conceptual paradoxes. This is not

immediately so for quantum chaos, and for classical chaos - which, being the limit for large

quantum numbers of quantum chaos, should be distinguished from mathematical chaos which

has inherently infinite precision. To wit, the term "chaos" implies in some sense unpredictability

and irreversibility of the motion of the system; sometimes it is expressed as "high sensitivity

to initial conditions." A conceptual difficulty in searching for chaos in quantum physics arises

from the fact that the motion of the system is governed by a Hamiltonian which is time- reversal

invariant, and that the spectrum of the system is discrete, or actually must be discretized in

order to achieve localizability of the system components. Time-reversal invariance and chaos on

the face of it seem contradictory. Because of the time-reversal invariance of the Hamiltonian

non-chaotic time-reversal invariant states must actually exist. One of our tasks will be to identify

those states. The aim of this short note is to clear up these points and to briefly indicate their

consequences. To that end, in section 2, by studying the time evolution of a quantum system we

will give a description of the essence of quantum chaos, and a simple mathematical criterion for

its identification. Furthermore, we will demonstrate that in quantum physics only exceptional

systems do not exhibit the character of chaos, and that it is essentially impossible, i.e., only in an un-real, idealized manner possible, to construct such states. This will set the stage for the

understanding of the mechanism of dissipation in quantum physics, which is the subject of the

following section.

One of the important points we shall demonstrate is that a chaotic quantum system undergoes

relaxation, i.e., it exhibits the effect of dissipation, simply as the consequence of its evolution being

governed by quantum dynamics. This will be done in section 3. Of course, being part of quantum

physics, the description of the evolution allows the treatment of arbitrarily off-equilibrium situations. More precisely, as we will see, the system evolves towards equal occupation probability

for the accessible phase space cells. This way the basic axiom of statistical thermodynamics,

viz., the assumption of equal a priori probability for the occupation of the phase space cells at thermal equilibrium, turns out to be a consequence of the quantum physics evolution. Hence,

all results of statistical thermodynamics turn out to be correct; in particular, the second law of

thermodynamics is fulfilled. More broadly, the whole field of thermodynamics thus turns out to be a consequence of the dynamics in quantum physics. At the same time the entropy can be used

to define the quantum physics arrow of time in non-equilibrium systems.

The broader question of the thermodynamic variables will be addressed in a separate paper.

2 Time-Reversal Invariant States

Since the Hamiltonian (by assumption!) is time-reversal invariant then for any solution of the

equations of motion, say tl>(t), the time-reversed state, <f>(t) = tp(—t)*, is also a solution. Consequently
which is time-reversal invariant, is also a solution. Our first task is to identify the time- reversal

invariant states; then we will show that they do not exist in nature and why. To that end, we

shall first consider the evolution in time of a two-body collision, and then that of a dilute gas. The prototype elastic two-body scattering system at energy E is described as consisting in the

asymptotic region of an incoming plane wave and an outgoing spherical wave. This description

has two related difficulties: the waves fill all space and generate unphysical interference patterns;

and there is no before or after the collision, they collide all the time [1]. The system is delocalized

in space and time. To avoid these difficulties we shall employ the Weyl eigendifferential states which are defined as [2] fEk +e *w(Ek ) =Af dE *(E) (2) JEk -i

where € —> is understood and Af is the normalization constant. Furthermore, ^(E) is the

complete system wave function, i.e., it includes all channels and the actual collision region.

The Weyl states have several useful characteristics: (a) they are as close to being eigenstates of

the Hamiltonian as can be while being localized in space and time; for large negative t the colliding

systems in a "physical" scattering state are far apart and are approaching each other; at t = they

collide; and at positive t the reaction products fly apart; (b) the original continuous spectrum has

been discretized, i.e., the number of Weyl states is denumerably infinite and thus only summations

and not integrations are encountered: they are Kronecker orthonormal; (c) in contrast to the

Gaussian wave packets of the Wigner representation they have simple mathematical properties;

(d) they are minimum-uncertainty wave packets and can be directly used to define the phase

space cells: when replacing in the wave function $ the time t by t — //, then in the classification

of phase space the indices k and / of Eq (2) play the roles of the commonly used coordinates p

and q, respectively. When mentioning phase space cells it will be tacitly assumed that this has

been done. Also, we will be able to achieve all our results using the Weyl wave functions, which

represent pure states. We will not need the more general description by density matrices which

would be needed if impure states were required [3]. We will conduct our discussions by first considering the individual sharp energy components,

y(E) of Eq. (2), and then investigate the modifications resulting from the Weyl superposition

according to Eq. (2). Thus, for example, a time-reversal invariant physical Weyl scattering state,

constructed according to Eq. (1), therefore, at f < would have in addition to the original

incoming state in the "input channel," in which the colliding particles approach each other,

also the time-reversed original outgoing state, i.e., particles in the original "output channels,"

approaching each other, all of them to meet at t = at the origin. All these components must

be phase-related.

When using Weyl states the description of the evolution of the state in the interaction picture

presents no problems as at t = — oo the collision partners are infinitely far apart and do not

interact; i.e., the system can be described by asymptotic states. This way, for example, the

inner structure (e.g., the self-energy) of each of the colliding partners can be computed without

the interference by the other partner. Hence, given the initial state in terms of the asymptotic

states, i.e., the states which are the solutions of the channel Hamiltonians, then the final state,

again in terms of the asymptotic states, is computed exactly by multiplying the state vector of

the initial state by the S-matrix; or by the U-matrix if the starting point is at some finite time

[4]. The difference between these two forms is that the U-matrix contains the closed channels,
i.e., the channels which contain off-the-mass-shell particles, while the S-matrix ignores them.

Since the closed channels decrease in amplitude exponentially as the systems move apart, the

difference between the U- and the S-matrix is important only very close to the actual collision.

For brevity we shall always speak of the S-matrix, with the proviso that if need be one should

use the U-matrix.

We will conduct the discussion in the asymptotic state representation. We describe the evo- lution of the system in time by following the evolution of some given initial state; that state may

be known as the result of the system having been suitably prepared, or it may be a member

of the (unknown) density matrix of the system. In case the energy of the two-body collision is above the inelastic threshold, the S-matrix is N by N, with N > 1; when including the photon

channels, and describing the photon states by Weyl functions, N is denumerably infinite at any

energy. For simplicity we ignore the photon channels; then N is some finite number, depending on the energy E of the collision (dropping the subscript k of Eq. (2) for brevity). The asymptotic

states then can be classified according to the number of (elementary or composite) particles they

contain.

For the discussion it is convenient to describe the channel states in hyperspherical coordinates

[5] which contain only one radial coordinate. Consequently, omitting the angular parts, then for a given state, ^(E) all channels, c, asymptotically for rc — oo have the form

V>c = ac e

l'(^-^) _ bc e - ,

'( /:crc +£ct) . (3)

Here ac is the amplitude "before," and bc "after" the collision. In general these states have

\ac \ ^ |6e |, i.e., the in-current is not equal to the out-current; thus, they are not time- reversal

invariant. In a "physical state" ac exists in one channel, the incoming channel, bc in all channels.

Of course, owing to the time-reversal invariance of the Hamiltonian, the time-reversed state of

any of those solutions is also a solution. There do exist time-reversal invariant states, viz., the

eigenchannel states [6], which are the eigenstates of the S- matrix. There exist N such states. Since the eigenchannel states form a complete set of states, any state, for example, a physical

scattering state, can be written as a superposition of eigenchannel states. In general, all N

eigenchannel states are needed for this superposition. A system which is in a pure eigenchannel

state "before" the scattering event remains unscathed, i.e., is unchanged by the scattering process.

For these states there holds

bc ,n = ac , n e

2ir>» . (4)

(The S-matrix being unitary, it must have eigenvalues of absolute value 1; hence the eigenvalues,

e

2tVn

, as written in Eq. (4), must have real eigenphases rjn .) Thus, the channel functions of an

eigenchannel state are of the form (still in hyperspherical coordinates)

V'cn = ac<n e

-»( £ct+T,„) sin(fcc rc - Vn ) ; (5)

they have standing waves in all channels c, which evidently is necessary for time-reversal invari- ance, and which are phase-related. (Of course, a 1 by 1 S-matrix has only 1 channel, which is thus an eigenchannel.) The eigenchannel states are the only stationary time-reversal invariant states. In principle, even if not in practice, such states can be constructed. These states are delocalized

continuum states. The Weyl eigenchannel states for t <C have phase-related incoming particles

in all channels which collide at t = and fly apart for t > 0, maintaining the relative amplitudes

and the relative phases in the sense that rjn before the collision goes into —rjn after the collision.
A physical state - which has an incoming wave in one channel and outgoing waves in all channels - is not time-reversal invariant. Again, the time-reversed state with incoming waves in all channels, outgoing wave in one channel, is also a possible solution. This time-reversed state then

can be used to construct a time-reversal invariant state according to the prescription of Eq. (1).

However, see below.

In a certain, somewhat inaccurate way, there exists a system which in some sense is an

eigenchannel state. That case is given by the system which at the considered energy allows only

elastic s-wave scattering. Then indeed the s-wave part of the wave function is described by a

1 by 1 S-matrix and has the form (5), i.e., it is in the time-reversal invariant form. However,

the other multipolarities in the incoming plane wave remain in the time-reversal non-invariant

superposition of a modified plane wave having the original direction of propagation. A convergent

pure spherical s-wave has not yet been achieved in the laboratory.

Before continuing the discussion of the quantum physics system it may be informative to recall

the case of a classical system. There exist exceptional systems which have closed trajectories in

phase space. They almost by definition are in the non-chaotic regime; hence such systems must

be "small." In a chaotic system the closed trajectories have measure zero; this statement actually

defines a chaotic system. It is a different, but equivalent, form of the usual statement that chaotic

motion is exponentially sensitive to the initial conditions. For the exceptional closed trajectory

systems the entropy remains constant. All such systems are idealizations and not realizable in

nature.

We now return to the quantum system. There the observation, equivalent to the above measure zero, is that the eigenchannel states (and also the time-reversed physical states) are represented by solutions which are unstable in the mathematical sense. Consider a given physical

scattering state. Its exact time reversed state has an outgoing wave in exactly one channel. This

situation is, however, mathematically unstable, in that a change of the amplitude even for only one of the incoming waves will generate a solution containing outgoing waves in all channels.

(This change of the amplitude can be thought of as the addition of a physical state which has its incoming wave in that channel. This then results in outgoing waves in all channels.) In the same

way, changing the amplitude only in one channel in an eigenchannel solution will unbalance the

in- versus the out-currents in all channels.

This simple fact has dramatic consequences for a many-body system as we now will explain.

We take the case of a dilute ideal gas in a mathematically perfect box, where the molecules are described by Weyl states in the asymptotic state representation. To describe the evolution of this

system it is useful to introduce the system U-matrix; it is defined similarly as the usual U-matrix

in terms of the Tomonaga-Schwinger equation [4], where the interaction picture interaction is the

sum of the individual interactions between the collision partners. It is a generalization of the

familiar graph expansion methods of many-body theory [7]. A more detailed discussion will be

given in a separate paper. The restriction to a dilute gas leads to a substantial simplification and

to conceptual transparency of the problem since then the evolution of the system can be treated

in terms of the cluster expansion, which can be visualized as a sequence of 2- or 3-body collisions.

This then can be used to factorize the system U- (or S-) matrix into reaction U- (or S-) matrices.

Of course, this simplification is not essential for the existence of a system U-matrix; it only allows

for easy visualization.
We now return to our problem. If there axe M particles in the box, the overall wave function

is contained in a 3M-dimensional finite configuration space; at fixed total energy the phase space

of the system is also finite and has, say, N cells. Then the overall system S-matrix is N by

N; it is sparse but not disconnected. Except for its size, it has all the same attributes as the

S-matrix of a two-body system discussed above. It allows the definition of system eigenchannel

states. Again, these states are time-reversal invariant, and are the only time-reversal invariant

states. The remarks that the time-reversal invariant solutions are mathematically unstable made

above for a two-body system thus apply also for a many-body system. (Since N is the volume of

the phase space it actually is ridiculously inconceivable to construct a state with phase-related

incoming waves in all N channels which would be needed for the time-reversal invariant state.)

The point of the mathematical instability of the time-reversal invariant solutions requires a

precise demonstration. In the eigenchannel representation the S-matrix is diagonal with diagonal

elements e-2"7 ". In this representation the state vector of the given eigenchannel, n, is 6n , m - It has non-vanishing amplitudes in all components of the asymptotic state representation. Each

action of the S-matrix generates the "outgoing" wave by multiplying each of the "incoming" wave components by the factor e _2n?n

, common to all these components. The ratio of the amplitudes

thus is not altered by the collision; the system is in a stationary state. If, however, only one more

eigenchannel solution, say m, is admixed then "after" the collision the amplitude in channel c will be

bc = ac , n e2lT) " + ac, m e~2irim

. (6)

Since in view of the orthogonality of the eigenchannels in general the ratio aCin /aC)Tn is dif- ferent for the different channels c, the state after the collision is not an eigenchannel state. Furthermore, as long as (r?n — Tjm)/-K is not a rational number, the system will never return

to the state it occupied "before the first collision." Therefore, the Poincare time then is infinite. Nonetheless, a time-reversed invariant state can be constructed by the prescription (1). Admixing further eigenchannel solutions does not change the situation in any qualitative manner. Of

course, in that case there exist more relative phases, generally irrational multiples of it, between

the contributing eigenchannel states; this then allows for greater variety of trajectories in the

asymptotic-representation Hilbert space. This discussion thus provides the precise definition of

quantum chaos: except for the case of being in a single eigenchannel state, every quantum system

is chaotic unless all ratios of the eigenphases are rational numbers. From this follows that already a "small" system with a 2 by 2 S-matrix, thus having 2 eigenchannel states, according to Eq. (6) can exhibit quantum chaos. Furthermore, time-reversal invariance does not imply that the state

is non-chaotic; on the other hand a non-chaotic state must exhibit time- reversal invariance. So,

for example, a physical state time-symmetrized according to Eq. (1), can be, and in general will

be, in a chaos state.

Concerning the rationality of the ratio (r]n — r)m )/Tr, this can happen in an N by N S-matrix

if at all, then for 2 eigenchannels, and only by some accidental degeneracy. We have not been

able to invent a non-trivial system exhibiting this characteristic in a non- accidental way, as we

are dealing with systems above some inelastic thresholds. Besides, even an approximate Poincare

return, defined in some suitable manner, is thwarted because the Weyl states do "flow apart" in a time 1/e, cf. Eq. (2), which is short compared with even any reasonable "quasi - Poincare"

time.
3 Dissipation

Having recognized that owing to the unstable nature of the time-reversal invariant states, in

particular their non-return character, there is nothing left to distinguish them from a generic

state. The question we now must address is what is the nature of the state the system evolves

into, starting from some arbitrary "initial state." To answer that question we investigate, again

for the case of a dilute gas, the change induced in the system by "the next collision." As explained

in the previous section, the time evolution is governed by the (N by N) system U- or S-matrix.

Assume that before the considered collision, i.e., after the last collision, the amplitudes of the

system wave function in the asymptotic representation are a,-, and, furthermore, we assume that

the state is not an eigenchannel state. Then the amplitudes after the next collision are given by

multiplication of the original amplitudes by the system S-matrix. Denoting the system S-matrix

elements by R{j, then the new amplitudes, dj, thus will be

ax = (l-Rn)ai + #12 a2 + #13 a3 + • •

• (7a) a 2 = #2101 + (1 - R22) a2 + #23 03 + ••• (7b)

dj = Rji a x + (l-Rjj)aj •• + Rjk <*k + ••• (7c)

dk = Rk i a x ••• + Rkj dj + ••• + (1 - Rkk) a>k + •• (7d)

Owing to the unitarity of the S-matrix there holds

E M

2 = £ N

2 = 1 (8)

The elements Rjk, j i

1 k, describe cross-flow of probability amplitudes between the channels

j and k, the elements Rjj reflect the net in-(or out-)flow into channel j.

Take as the measure of pair- wise non-uniformity the expression

Ajk = Kf - K|2. (9)

From (7c) and (7d) we compute the change in the non-uniformity (A^: after the collision)

Ajk - Ajk = (\d

3 \ 2 - \dk \ 2 ) - (la/ - M2 ) = -\Rjk \ 2 Ajk + [F]. . (10)

Because of the negative sign on the right-hand side, Eq. (10) has the character of a damping term.

Hence we have the important result that Ajk relaxes towards zero (see remark below). This is true

for every pair, j, k. The other terms, denoted by [F] in Eq. (10), are of the form a*j ak — aj a*k and

thus are phase-sensitive. These terms induce fluctuations in Ajk , which are damped by Eq. (10).

Both the fluctuations and the relaxation can be followed by repeated matrix multiplication.

(Note that the form of the damping term, including it being real, results from the time-reversal

invariance of the S-matrix.) Furthermore, the terms [F] are responsible for maintaining (unstably)

the amplitude and phase relations between the channels in the eigenchannel solutions.

6
An important remark here is in order. Namely, it is at this point that the results of the

previous section, in particular that the Poincare time for chaotic quantum systems is infinite, are

decisive. Without those results the statements of the above paragraph would be unsupported. One

would have to rely on some assumptions like random phases to eliminate, or at least to decrease

the influence of, the fluctuation inducing terms [F]; they otherwise could in effect provide the

dominant influence, e.g., leading to macroscopically observable fluctuations.

We now discuss our result. Recall that the channels of the system S-matrix are the cells of

the overall system phase space. Thus the amplitude a,j is directly the occupation amplitude for phase space cell j, and thus the occupation probability for that cell is j <z^ | 2

. Hence, even though

the Hamiltonian is time-reversal invariant, the dynamics of the system tends to achieve equal

a posteriori probability for the occupation of the energetically accessible phase space. This is simply the consequence of the "no return" character of the reaction, and of the non-zero value

of some Rjk, together with the connectedness of the S-matrix. If the S-matrix should turn out

to be disconnected, then the equalization would take place within each connected piece of the

S-matrix. A fully disconnected S-matrix actually does not occur in nature; what does occur is that by virtue of the smallness of the relevant Rjk, some parts of the S-matrix are only weakly

connected to the rest of the S-matrix. In that case the overall equalization of the occupation

probabilities would be slowed down; the overall system very well might, and actually usually

does, have a set of different relaxation time constants.

In summary: the central result of this discussion is that evidently quantum dynamics achieves

directly a uniform occupation of the accessible phase space. Now recall that all results of statistical

thermodynamics are arrived at by assuming axiomatically equal a priori occupation probability

of all accessible phase space cells, and evaluating the resulting probability by counting the phase

space cells. Our result shows that we have proven the correctness of that axiom for quantum

dynamics. (Note that Eq. (10) concerns only the occupation probabilities, |aj| 2

; the amplitudes

still may contain phase relations, which, however, has no importance for our result. A detailed

discussion of this point will be given in a subsequent paper.) Hence, not only are the results of

statistical thermodynamics correct, even more strongly: we have proven that statistical thermodynamics is a direct consequence of quantum physics. In particular, one of these consequences

is the validity of the second law of thermodynamics; it arises as a direct consequence of the

time-reversal invariance of the quantum physics Hamiltonian and the equations of motion derived from this Hamiltonian, which underlie the form and even the reality of the right-hand side

of Eq. (10). At the same time this allows the definition of quantum entropy (which, however, we

will not address here), and with it the definition of the arrow of time for non-equilibrium systems

in quantum physics. No arrow of time can be defined for systems at equilibrium. In view of

the fact that the first law of thermodynamics, being only the expression of energy conservation,

is contained in quantum physics, the overall conclusion is that the field of thermodynamics is simply part of quantum physics.

Note that in this whole discussion the question of measurement did not arise and did not

have to be addressed. Measurement evidently is irrelevant to the existence of dissipation. In particular, the well-known collapse of the wave function plays no role here. This is not to say

that an analysis of the measurement process for a many-body system is of no interest; to the

contrary, it is very worthwhile to give a careful discussion of that problem, in particular, in the

context of the definition of the observables representing the microscopic thermodynamic variables.

Simply, it lies outside of the frame of the present note.
To apply our results to nature, first note that no questions of general relativity, e.g., the

expansion of the universe, have been addressed. Ignoring that aspect, we recall that the eigenchannels belong to isolated points in Hilbert space. Since one knows that some parts of nature

are not on the eigenchannel points, no part of nature can escape equilibration. Finally, since for a (Weyl discretized!) N by N S-matrix (N is the total volume of the phase space of the universe)

the number of eigenchannel states is N, while the total number of states is TV!, the chances that a "New Universe" will land in an eigenchannel point, i.e., (1/(N — 1)!), seem to be negligibly small.

Acknowledgments

I would like to acknowledge many useful discussions about the concepts of the present paper

with many colleagues, including H. Bers, S. Chandrasekhar, U. Fano, H. Fritsche, D. Mueller,

Y. Nambu, D. Oxtoby, and R. Wald. I am particularly grateful for very helpful and stimulating

discussions with P. Carruthers and especially with J. Rafelski, who in addition not only con- tributed essentially in clarifying my thinking but actually provided the initial stimulus which led to this work.
References

* Visiting Scholar, Enrico Fermi Institute, University of Chicago, Chicago, Illinois.

[1] E. Merzbacher, Quantum Mechanics, 2nd edition, Chapter 12; John Wiley and Sons, New

York, London, Sydney, Toronto (1970).

[2] A. Sommerfeld, Partielle Differentialgleichungen der Physik, Chapter 5, Appendix 1; Akademische Verlagsgesellschaft, Leipzig (1948).

[3] J. von Neumann, Mathematische Grundlagen der Quantenmechanik; Springer (1932); trans- lation Mathematical Foundations of Quantum Mechanics; Princeton University Press, Princeton, NJ (1955).

[4] D. Lurie, Particles and Fields, Chapter 6; Interscience Publishers (1968); P. Roman, Intro- duction to Quantum Field Theory, Chapter 4; John Wiley and Sons, New York, London,

Sydney, Toronto (1969).

[5] P. H. Morse and H. Feshbach, Methods of Theoretical Physics, Chapter 11; McGraw-Hill

Book Co., New York, Toronto, London (1953).

[6] R. F. Barrett, et al, Rev. Mod. Phys. 45, 44 (1973).

[7] D. S. Koltun and J. M. Eisenberg, Quantum mechanics of many degrees of freedom; John

Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore (1988).

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Developed under a worldwide program coordinated by NIST under the authority of the National

Standard Data Act (Public Law 90-396). NOTE: The Journal of Physical and Chemical

Reference Data (JPCRD) is published bi-monthly for NIST by the American Chemical Society

(ACS) and the American Institute of Physics (AIP). Subscriptions, reprints, and supplements are

available from ACS, 1155 Sixteenth St., NW„ Washington, DC 20056.

Building Science Series -Disseminates technical information developed at the Institute on

building materials, components, systems, and whole structures. The series presents research

results, test methods, and performance criteria related to the structural and environmental

functions and the durability and safety characteristics of building elements and systems.

Technical Notes -Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or

definitive in treatment of the subject area. Often serve as a vehicle for final reports of work

performed at NIST under the sponsorship of other government agencies.

Voluntary Product Standards -Developed under procedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The standards establish

nationally recognized requirements for products, and provide all concerned interests with a

basis for common understanding of the characteristics of the products. NIST administers this program as a supplement to the activities of the private sector standardizing organizations.

Consumer Information Series -Practical information, based on NIST research and experience,

covering areas of interest to the consumer. Easily understandable language and illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order the above NIST publications from: Superintendent of Documents, Government Printing

Office, Washington, DC 20402.

Order the following NIST publications -FIPS and NISTIRs-from the National Technical

Information Service, Springfield, VA 22161.

Federal Information Processing Standards Publications (FIPS PUB) -Publications in this series collectively constitute the Federal Information Processing Standards Register. The

Register serves as the official source of information in the Federal Government regarding

standards issued by NIST pursuant to the Federal Property and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order

11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

NIST Interagency Reports (NISTIR)-A special series of interim or final reports on work

performed by NIST for outside sponsors (both government and non-government). In general,

initial distribution is handled by the sponsor; public distribution is by the National Technical

Information Service, Springfield, VA 22161, in paper copy or microfiche form.