Table Of ContentHeavy Ion
Collisions
Cargese 1984
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Volume 128-Techniques and Concepts of High-Energy Physics III
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Volume 129-Transport in Nonstoichiometric Compounds
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Volume 130-Heavy Ion Collisions: Cargese 1984
edited by Paul Bonche, Maurice Levy, Philippe Quentin, and
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Series 8: Physics
Heavy Ion
Collisions
Cargese 1984
Edited by
Paul Sanche
CEN Saclay
Gif-sur-Yvette, France
Maurice Levy
Universite Pierre et Marie Curie
Paris, France
Philippe Quentin
Universite de Bordeaux
Bordeaux, France
and
Dominique Vautherin
Institut de Physique Nucliare
Universite de Paris XI
Orsay, France
Plenum Press
New York and London
Published in cooperation with NATO Scientific Affairs Division
Proceedings of a NATO Advanced Study Institute entitled
Heavy Ion Collisions: From Collective Motions to Quarks,
held September 2-15, 1984,
in Cargese, Corsica, France
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute (1984: Cargese, Corsica)
Heavy ion collisions.
(NATO ASI series. Series B, Physics; v. 130)
"Proceedings of a NATO Advanced Study Institute entitled Heavy Ion Colli
sions: From Collective Motions to Quarks, held September 2-15, 1984, in
Cargese, Corsica, France"-T.p. verso.
"Published in cooperation with NATO Scientific Affairs Division."
Bibliography: p.
Includes index.
1. Heavy ion collisions-Congresses. I. Bonche, P. II. North Atlantic Treaty
Organization. Scientific Affair Division. III. Title. IV. Series.
QC794.6.C6N376 1984 539.7'54 85-24403
ISBN-13: 978-1-4684-5017-0 e-ISBN-13: 978-1-4684-5015-6
DOl: 10.1007/978-1-4684-5015-6
© 1986 Plenum Press, New York
Softcover reprint of the hardcover 1s t edition 1986
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved. No part of this book may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying,
microfilming, recording, or otherwise, without written permission from the Publisher
PREFACE
The 1984 Cargese Advanced Study Institute was devoted to the
study of nuclear heavy ion collisions at medium and ultrarelativis
tic energies. The origin of this meeting goes back to 1982 when
the organizers met at the GANIL laboratory in Caen, France which
had just started accelerating argon ions at 44 MeV per nucleon.
We then realized that 1984 should be the appropriate time to review
the first results obtained with such new kinds of facilities. The
material contained in this volume, presenting many beautiful re
sults on nuclei at high excitation, fully confirms this point.
Many stimulating exchanges between experts in rather diffe
rent fields already took place during the school and we hope that
this cross fertilization will lead to further developments.
About half of the present volume is also devoted to the
field of relativistic heavy ion collisions, which is now expanding
rapidly. As an illustration, let us recall that the construction
of a 30 on 30 GeV per nucleon collider at Brookhaven has been
recognized last year as one cf the major priorities by the US
Nuclear Science Advisory Committee.
We would like to express our gratitude to NATO for its ge
nerous financial support which made this institute possible.
We also wish to thank the Institut de Physique Nucleaire et
a
de Physique des Particules (France), the Commissariat l'energie
atomique (France) and The National Science Foundation (USA) for
the attribution of travel grants.
Special thanks are due to our colleagues Gordon Baym (Urba
na), S.E. Koonin (Caltec) and W.J. Swiatecki for valuable sugges
tions regarding the program of the school.
It is also a pleasure to express our gratitude to the
Universite de Nice for making available the facilities of the
Institut d'Etudes Scientifiques de Cargese. Last but not least we
wish to thank Ms Marie-France Hanseler for her outstanding orga
nizational work, and the Staff of the Cargese Institute for making
our stay in Cargese a very enjoyable one.
M. Levy
P. Bonche
P. Quentin
D. Vautherin
v
CONTENTS
Collective variables and dissipation
R. Balian
Energy dissipation in nucleus-nucleus collisions
around 40 MeV per nucleon: New phenomena
or transition trends ? 33
M. Lefort, B. Borderie, D. Jacquet, and the
Nuclear Chemistry Group
Angular momentum dynamics in damped nuclear reactions, 69
J. Randrup* and T. Dossing
Nuclear structure at high spin 99
B. Herskind
Chaos in nuclei or statistical mechanics of small systems
fluctuations 119
H.A. Weidenmuller
Spectral fluctuations and chaotic motion 145
o. Bohigas, M.J. Giannoni*, and C. Schmit
Time-dependent variational principle for the expectation
value of an observable: Mean-field applications 165
M. Veneroni
Dynamics of the 'relativistic heavy ion collisions 209
J. Cugnon
Skyrmions, dense matter and nuclear forces 283
C.J. Pethick
Nucleus-nucleus collisions at high energies
~5
M.A. Faessler
vii
Comments on multiplicity in proton nucleus collisions 361
A.K. Kerman*, T. Matsui, and B. Svetitsky
An introduction to lattice QCD 367
G. Martinelli
Index 407
Note in the case of several authors, the name of the
speaker is followed by an *
viii
COLLECTIVE VARIABLES AND DISSIPATION
Roger Balian
Service de Physique Theorique, CEA, Centre de Saclay
91191 Gif-sur-Yvette Cedex, France
ABSTRACT
This course is an introduction to some basic concepts of non
equilibrium statistical mechanics. We emphasize in particular the
relevant entropy relative to a given set of collective variables,
the meaning of the projection method in the Liouville space, its
use to establish the generalized transport equations for these
variables, and the interpretation of dissipation in the framework
of information theory.
TABLE
1 - Introduction
2 - The Liouvillean formulation of quantum mechanics
3 - Contracting the description
4 - Reduced densities
5 - Relevant entropy relative to a set of variables
6 - Metric structure of the spaces of states and observables
7 - Projection in the Liouville space
8 - Memory kernel
9 - Generalized transport equations
10 - Mean-field approximations
11 - Perturbation methods and short-memory approximations
12 - Conclusion
1. INTRODUCTION
During the last decade, phenomenological theories of heavy
nuclei have gradually been superseded by more microscopic approa
ches. It has become feasible to understand and even to compute
many dynamical processes, for instance to analyse collisions of
heavy ions, by starting from a many-body model of interacting nu
cleons without any adjustable parameter. This type of studies in
nuclear physics appears as a new branch of statistical mechanics,
and it takes advantage of many concepts and techniques drawn from
the study of other types of matter.
A specific feature of nuclear systems is their finite size,
which brings in more complications than simplifications. On the
one hand, properties associated with the thermodynamic limit of
statistical mechanics do not hold here : the extensivity of nuclear
matter can be considered only as an approximation. Surface effects
are essential, as well as size effects like shell structures. From
the dynamical viewpoint, the finite size of nuclear systems does
not allow them to relax towards thermal equilibrium as is usual in
statistical Mechanics, and the analysis of dissipative processes
requires special care, as we shall see. On the other hand, the num
ber of degrees of freedom is still much too large to allow complete
microscopic studies. Ive are therefore faced with a standard problem
in non-equilibrium statistical mechanics : is it possible to des
cribe adequately the system in terms of some small set of collec
tive variables which would replace the untractable set of positions
and momenta of all nucleons ?
We shall be mainly concerned in this course with the systema
tic construction of the equations of motion for a set of collective
variables, taking the microscopic dynamics as a start. The unified
framework and most of the techniques which we shall present have
been introduced long ago in the context of non-equilibrium statis
tical mechanics, but they are not so widely known and applied as
they deserve. The formalism is general enough to encompass many
existing theories and approximations currently used in nuclear
theory. In addition, the geometric interpretation which we shall
give may help to visualize the steps involved in the derivation of
the collective motion. Our scope will be merely formal, because
we wish to emphasize the conceptual rather than the technical as
pects. This should help the reader adapting the general theory to
such or such specific problem.
Several important qualitative differences exist between the
dynamics at the microscopic level and at the level of collective
variables, and the origin of these differences will be enlightened
by the general approach. As usual in statistical mechanics,-new
phenomena appear when the description is contracted, i.e., when
the non-collective variables are eliminated : the physics changes
2
with the level of description. Let us recall a few general features
of the type of contraction performed here. On the one hand, the
Schrodinger equation is linear ; however, isolating the collective
degrees of freedom leads to non-Zinearities which may have impor
tant consequences. In contrast to linear equations, which provide
nothing but superpositions of sinusoidal oscillations, a non
linear dynamics may lead to solitons, as well as to chaotic beha
viors (like in the Euler equations of hydrodynamics or in the He
non-Heiles model). On the other hand, the collective dynamics may
present a cZassicaZ structure in spite of the underlying quantum
nature of the problem. Actually, a nucleus is a quantum object,
and the quantities characterizing its state are therefore statisti
cal ; but they may follow classical equations of motion of the
Ehrenfest type. Finally, dissipation, which is absent at the micros
copic level, may take place in the collective equations of motion.
Our approach will introduce naturally most quantities and con
cepts of current use in the thermodynamics of irreversible proces
ses, such as ZocaZ temperatures, chemicaZ potentiaZs, hydrodynamic
veZocities, or transport coefficients. In particular, the whole
course will emphasize entropy, both as a measure of the missing
information and as a tool for eliminating the non-collective varia
bles considered as irrelevant. For instance, we shall characterize
the dissipation rate by using the concept of reZevant entropy in
troduced in § S. The more standard approach to this question in
nuclear physics relies on energetic considerations : the energy
is split, in a more or less natural way, into a "mechanical" part,
associated with the collective degrees of freedom, and a "thermal"
part accounting for the remainder, and the dissipation is defined
as an energy flow from the former to the latter part. However, from
the viewpoint of statistical mechanics and of thermodynamics, entro
py is a more basic quantity than energy, since energy is only one
among the conserved variables of the system. Having defined the
relevant entropy as the missing information relative to the set of
collective variables, we shall understand and measure the dissipa
tion as the increase of this relevant entropy, an extension of the
Carnot principle.
We shall make use of several techniques of statistical mecha
nics. We shall first introduce (§ 2) the LiouviZZean formuZation
of quantum mechanics. More general than the standard formulation
in Hilbert space, it encompasses useful formalisms such as the
polarization representation for a spin or the Wigner representa
tion. Working in the Liouville space is also essential for the
reduction of the number of degrees of freedom retained in the
dynamical description (§ 3 and 4). The concept of entropy will be
used to endow this Liouville space with a natural metric (§ 6).
We shall then interpret the elimination of the irrelevant varia
bles and the construction of the trajectory of the collective
3
