We address the question of the microscopic origin of dissipation in collective motion of a quantum many--body system in the framework of a parametric random matrix approach to the intrinsic dynamics. We show that the fluctuation--dissipation theorem is generally violated and, moreover, energy diffusion has a markedly non--Gaussian character and the corresponding distribution has very long tails. Such features do not support a Langevin or Fokker--Planck approach to dissipation in collective nuclear motion

Bulgac, A

Dang, G D

Kusnezov, D

CERN Document Server

YCTP-N1-95
January, 1995
Quantum Dissipation
Aurel BULGACa;b;1, Giu DO DANGb;2 and Dimitri KUSNEZOVc;3
a Department of Physics, FM{15, University of Washington, Seattle, WA 98195, USA4
b Laboratoire de Physique Theorique et Hautes Energies,
Universite de Paris{Sud, Ba^t. 211, 91405 Orsay, FRANCE
c Center for Theoretical Physics, Sloane Physics Laboratory,
Yale University, New Haven, CT 06511-8167,USA
Abstract
We address the question of the microscopic origin of dissipation in collective
motion of a quantum many{body system in the framework of a parametric random
matrix approach to the intrinsic dynamics. We show that the fluctuation{dissipation
theorem is generally violated and, moreover, energy diusion has a markedly non{
Gaussian character and the corresponding distribution has very long tails. Such
features do not support a Langevin or Fokker{Planck approach to dissipation in
collective nuclear motion.
PACS numbers: 24.10.Cn, 24.60.-k, 05.60+w, 05.40.+j
1Internet: bulgac@phys.washington.edu
2Internet: dodang@psisun.u-psud.fr
3Internet: dimitri@nst.physics.yale.edu.
4permanent address
While theory and experiment have gone a long way in the study of the collective
nature of large amplitude nuclear motion, the theoretical understanding of the coupling
between the collective and the intrinsic degrees of freedom is still in its infancy. Most of
the approaches are more or less phenomenological in nature. Hill and Wheeler [1] sug-
gested almost forty years ago that Landau{Zener transitions are at the origin of nuclear
dissipation. Over the years there have been a relatively large number of studies of this
particular mechanism [2] and the range of results is equally diverse. Implicit in this inter-
pretation is the presumption that irreversibility is of quantum origin. Even though there
exist quantum approaches (the linear response model [3], the hopping model [4], path in-
tegral method [5] and others), many formulations are basically classical , e.g. the so called
\wall formula" [6]. These include the more pragmatic phenomenological models, such as
the Langevin equation or Fokker{Planck equation [7], Maxwell’s model for friction with
memory eects [8], and to a certain extent kinetic approaches, e.g. two{body dissipation
mechanism [6]. In an analysis of a generic problem of coupled slow and fast degrees of
freedom, Berry and Robbins [9] obtained friction for the slow subsystem only by treating
the entire system classically, attributing it to \a clash between the essence of quantization,
namely the discrete spectrum of frequencies, and the essence of chaos, namely mixing and
a continuous spectrum extending to zero frequency" [9]. The present status of our under-
standing of the microscopic origin of dissipation in many{body quantum systems is thus
rather unsatisfactory. In this letter we explore the nature of dissipation in a many-body
system, using parametric random matrix theory, which allows for direct solution in many
cases of the quantum dynamics of the system.
The physical systems we explore are many-body systems which exhibit excitations
on two distinct time scales, described by collective (slow) and intrinsic (fast) degrees of
freedom. In order to address how energy is transferred from the slow (X;P ) to the fast
(x; p) modes, we will assume that the slow modes evolve classically at constant velocity
V0 according to X(t) = V0t. (Although we do not consider it here, this restriction can
be lifted, and the more general problem solved using the results and methods presented
here.) As a consequence of this assumption, we can solve for the quantum dynamics of
the fast subsystem, and even obtain analytic results for situations which are analogous
to the conventional adiabatic and diabatic limits. The intrinsic system is dened by its
matrix elements, and is taken as complex, described by its average level density, (E),
and its spectral fluctuations, in the form
H(X) = H0 +H1(X): (1)
Here H0 is chosen to be a diagonal N N matrix, dening the average density of states,
with hkjH0jli = [H0]kl = "kkl. In the basis of the eigenstates of H0, we dene H1(X) as a
parameter dependent, N N real Gaussian random matrix, which is completely specied
by its rst two moments
[H1(X)]ij [H1(Y )]kl = [ikjl + iljk]Fij(X − Y );
[H1(X)]kl = 0: (2)
2
Fij(X−Y ) is a \bell{shaped" correlation function with a characteristic width X0, and the
overline stands for the ensemble average. The dependence on i; j allows for the description
of banded matrices, where an eective number of states N0  N can be coupled by H1(X).
Such a parametrization implies that correlations between dierent instantaneous spectra
corresponding to dierent ‘shapes’ X are eective only within a distance  X0. The
average level density for each xed shape X is given by H0, while its spectral fluctuation
properties (in this case GOE, but all our formalism applies to GUE as well) are determined
by H1(X). We use a convenient parametrization of the correlator Fij(X) introduced in
Refs. [5]
Fij(X) =
W0q
("i)("j)
exp
"
−
("i − "j)2
220
#
F

X
X0

: (3)
Here F (x) = F (−x) = F (x)  1, F (0) = 1 and W0; 0 (N0  0(")) and X0 are
characteristic to the given system. The instantaneous spectra of a Hamiltonian H(X)
with constant average level density, [H0]kl = kkl, is shown in Fig. 1 for gaussian F (x) =
exp(−x2=2) (top) and exponential F (x) = exp(−jxj) (bottom) correlations. Notice that
a conventional adiabatic limit does not exist for the exponential, as the individual energy
levels undergo Brownian motion, and are not smooth on any time scale.
The time evolution of this system is found by solving the time{dependent Schro¨dinger
equation:
 (t) = T exp

−
i
h
Z t
0
dsH(X(s))

 (0) = U(t) (0): (4)
where T is the time-ordering operator, and U(t) the propagator. (We assume that the
initial state  (0) is uncorrelated with the Hamiltonian H(X(t)) at later times.) The
average propagator U(t) = U(t), found by using Eqs. (2) and resumming all leading order
diagrams in perturbation expansion of U(t) in the limit N0  1, satises the system of
integral equations [11]:
Uk(t) = U0 k(t)−
1
h2
Z t
0
ds1
Z s1
0
ds2U0 k(s2)Uk(t− s1)

NX
n=1
Fkn(X(s1)−X(s2))Un(s1 − s2); (5)
where U0 k( ) = exp(−i"kt=h). In order to compute averages of observables, we introduce
the set of generalized occupation number probabilities
Nk(t1; t2) = h (t1)jkihkj (t2)i
=
X
l
hljU y(t1)jkihkjU(t2)jlinl(0); (6)
where nl(t)  Nl(t; t) is the occupation probability of the state jli. Nk(t1; t2) satisfy the
following set of integral equations
Nk(t1; t2) = U

k (t1)Uk(t2)nk(0) +
1
h2
Z t1
0
ds1
Z t2
0
ds2
X
l
Nl(s1; s2)Flk(s1 − s2)
Uk (t1 − s1)Uk(t2 − s2): (7)
3
These equations specify the time evolution of the system, and we will consider (i) the
numerical solutions of (5)-(7) and the velocity dependence of the diusion constant, and
(ii) the extension of the formalism to the regime 1  N0 < N = 1, where we nd
analytic limits and a great simplication of the formalism as well.
The rst situation we study is that of constant average level density ([H0]kl = kkl),
as in a stadium billiard. Eqs. (5)-(7) have been solved numerically for N = 101 levels, a
bandwidth N0 = 21, a gaussian correlation F (x) = exp(−x2=2) with X0 = 1, and initial
conditions nk(0) = k;51. The resulting occupation numbers nk(t) are shown in Fig. 2
for the cases of fast (V0 = 4, top) and slow (V0 = 1=16, bottom) driving velocities. As
the results are symmetric with respect to the index k, nk(t) = n102−k(t), only k = 1− 51
are shown, counting from the bottom of the gure. One might expect that even a small
driving velocity would result in a complicated time evolution, as the Hamiltonian is time{
dependent, and has many small gaps in the instantaneous spectrum, where Landau{
Zener transitions might occur and thus induce \irreversibility" [1]. Actually, as we have
discussed at length in Ref. [10], this mechanism, which has been advocated in many
previous treatments [2], is valid only for isolated level crossings and thus is unrealistic
when there are many non{isolated ones as shown in Fig.1.
In Fig. 2, one can clearly distinguish two time scales: a relatively rapid initial transient
evolution, followed by a much slower one. While the initial transient behaviour is almost
identical in both cases, governed by the same spreading width Γ#, the long time behaviour
is strikingly dierent. For small driving velocities, the time evolution rapidly equilibrates,
and can be understood in terms of the V0 ! 0 limit, corresponding to constant random
matrix theory [11]. For large velocities there is a steady evolution to a dierent probability
distribution. The initial transient behaviour arises only because our initial occupation
probabilities nk(0) did not originate from an instantaneous eigenstate of H(0) (detailed
discussion on initial conditions will be presented elsewhere[10]). The subsequent long time
behaviour is due to the explicit time dependence of the Hamiltonian H(t) and would be
absent for a time independent one.
The diusion process associated with these time evolutions can be characterized by
the energy variance, E(t), and the energy diusion constant D(V 20 ), dened by
E(t) = h (t)j[H(t)− E(t)]2j (t)i  const + 2D(V 20 )t: (8)
t!1
In Fig. 3, D(V 20 ) can be seen to exhibit quadratic velocity dependence, in contradiction to
previous claims [2]. In the case we consider here of a symmetrical initial distribution nk(0)
and constant level density, the average energy E(t) = h (t)jH(t)j (t)i is constant, hence
the reaction force on the slow system, in particular the friction force, exactly vanishes.
This is consistent with a fluctuation{dissipation theorem in the following sense. Expressed
as γ = D, where γ is the \friction" coecient and  = 1=T = d ln (e)=de  0 is the
inverse thermodynamic temperature, we have the expected result dE(t)=dt = γ  0.
4
We shall outline briefly a further extension and simplication of the formalism in the
limit N ! 1, i.e. in the spirit of the standard constant random matrix theory, by
introducing the characteristic functional [11]
N (t1; t2;  ) = h (t1)j exp
"
iH0( − t1 + t2)
h
#
j (t2)i;
Nk(t1; t2) =
1
2h("k)
Z
dN (t1; t2;  )
 exp
"
−
i"k( − t1 + t2)
h
#
; (9)
N (t; t;  ) = exp
"X
n
hh (t)jHn0 j (t)ii
(i )n
hnn!
#
; (10)
where hh (t)jHn0 j (t)ii are cumulants. N (t1; t2;  ) satises the evolution equation
N (t1; t2;  ) = 
(t1)(t2) +
p
20W0
h2
Z t1
0
ds1
Z t2
0
ds2
N (s1; s2;  ) exp
2420
2
 

2
+ i
s1 − s2 − 
h
!235
F
 
(s1 − s2)V0
X0
!
(t1 − s1)(t2 − s2); (11)
where (t) = exp(i"kt=h)Uk(t) (note (t) is state independent), for which an equation
similar to Eq. (5) can be derived. In this case, there is only one equation to be solved,
instead of N{coupled equations (cf. Eq. (7)), which results in a signicant simplication
of the entire formalism. Moreover, various analytic solutions can be obtained, as we
exemplify below, by analyzing the adiabatic and the diabatic evolutions of the occupation
numbers for a system with a realistic level density of the form (") = 0 exp(").  = 0
corresponds to the case we have just described, of constant average level density, while
the case of nite  approximates fairly well a many{fermion system.
The adiabatic limit corresponds to 0X0=hV0  1 (and also 0  1), from which we
nd
N (t; t;  ) = exp
(
2W0
h

F

V0
X0

− 1

t−
2W0j j
h
)
: (12)
All odd moments of H0 vanish identically (since F (x) = F (−x)), and in the limit t!1,
all even cumulants of H0 increase linearly in time. If F (x) = exp(−x2=2) (we shall use
this form hereafter for illustrative purposes) then
hh (t)jH2n0 j (t)ii =
2W0t
h
 
hV0
X0
!2n
(2n)!
2nn!
;
D(V 20 ) =
hW0V 20
X20
=
hΓ#V 20
2X20
; (13)
(Γ# = 2W0) resulting in a non-Gaussian distribution.
5
In the diabatic limit, 0X0=hV0  1, we nd
N (t; t;  ) = exp
8<:2X0W00h2V0
24exp
0@20
2
 

2
+
i
h
!21A− exp 202
8
!35 t
9=; : (14)
In this case again all the cumulants of H0 increase linearly in time
hh (t)jHn0 j (t)ii =
"
2X0W00
h2V0
exp
 
220
8
! 
i0p
2
!n
Hn
 
−
i0
2
p
2
!#
t; (15)
where Hn(x) are Hermite polynomials. From the explicit expressions for the rst and
second cumulants we thus obtain that
D = γ
 
1 +
220
4
!
; (16)
which shows that the Einstein fluctuation{dissipation theorem is generally violated.
The most salient feature of the solutions (12) and (14) becomes evident when one
considers the asymptotic behaviour of the cummulants. Since cumulants of higher than
second order are nonvanishing, Gaussian processes are not obtained in any of these lim-
iting cases for the energy diusion. hh (t)jHn0 j (t)ii either increase indenitely with n
or increase subsequently after an initial decrease, depending on the values of parameters.
As a result the energy distribution has very long tails. In particular for V0  0 the dis-
tribution corresponding to Eq. (12) has a Lorentzian shape. These features imply that
a Langevin or Fokker{Planck approach to energy dissipation is at least questionable. As
we have discussed in Ref. [10] the present results apply equally to the GUE case.
The present approach treats the fast subsystem quantum mechanically and the slow
subsystem classically, as has been done often in the past [2]. The energy diusion process
is described in terms of some intrinsic characteristics of the many{body system (thermo-
dynamic temperature, spreading width Γ#, 0 and X0) and V0. It is not clear yet whether
these characteristics have a meaningful classical limit separately or only in a given com-
bination, and this seemingly points to an apparent lack of a classical limit for the fast
degrees of freedom (h ! 0) of the solutions (13), (15). In Ref. [9], friction was obtained
only in a classical treatment of both fast and slow system, while in Ref. [5], dissipation
and friction appear only in a explicit quantum treatment (path integral) of the entire
system and the presence of quantum fluctuations in the slow subsystem was essential.
The wall formula [6] leads to a diusion constant D / V 20 as we have obtained here (see
Fig. 3 and Eq. (13)), but is essentially a classical result, which does not depend in any
signicant way on h, and apparently reflects a dierent underlying mechanism. It will be
highly desirable to identify the classical limit of the present approach.
In conclusion, we have presented numerical and analytical solutions of the time de-
pendent evolution of a driven complex quantum system, such as a nucleus, under the
assumption that the number of levels is large. The parametric random matrix approach
chosen here incorporates the essential attributes of the intrinsic dynamics, namely: an
6
exponentially increasing level density, GOE spectral fluctuations and loss of correlations
during large amplitude collective motion. We have shown that the resulting energy dif-
fusion process is highly non{Gaussian in character, that the energy distribution has very
long tails and also that the energy diusion constant is proportional to the square of the
collective velocity.
We thank H.A. Weidenmu¨ller for stimulating discussions and correspondence, the
Centre National de la Recherche Scientique (C.N.R.S.) and Universite de Paris-Sud for
nancial support, and the computer facilities at NERSC. This work was also partially
supported by NSF and DOE. The Laboratoire de Physique Theorique et Hautes Energies
is a Laboratoire associe au C.N.R.S., URA 0063.
References
[1] D.L. Hill and J.A. Wheeler, Phys. Rev. 89, 1102 (1953).
[2] Representative works in the nuclear context include: V.M. Strutinsky, Z. Phys.
A280, 99 (1977); V.M. Strutinsky, V.I. Abrosimov, and F.A. Ivanyuk, Z. Phys.
A306, 273 (1982); V.M. Strutinsky, F.A. Ivanyuk and S.M. Vydrug-Vlasenko, Z.
Phys. A341, 267 (1992); L. Wilets, Theories of Nuclear Fission, Clarendon, (1964);
G. Schu¨tte and L. Wilets, Z.Phys. A286, 313 (1978); R.W. Hasse, Rep. Prog. Phys.
41, 1027 (1978); S.E. Koonin and J.R. Nix, Phys. Rev. C13, 209 (1976); T. Lederg-
erber, Z. Paltiel, Z. Fraenkel and H.C. Pauli, Nucl. Phys. A275, 280 (1977); M.C.
Nemes and H.A. Weidenmu¨ller, Phys. Rev. C24, 944 (1981); M. Wilkinson, J. Phys.
A21, 4021 (1988).
[3] P.J. Siemens, A.S. Jensen and H. Hofmann, Nucl. Phys. A441, 410 (1985).
[4] B.W. Bush and G.F. Bertsch, Phys. Rev. C45, 1709 (1992).
[5] D.M. Brink, J. Neto and H.A. Weidenmu¨ller, Phys. Lett. 80B, 170 (1979) and ref-
erences therein.
[6] J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A. J. Sierk and W.J. Swiatecki,
Ann.Phys. 113, 330 (1978); J. Randrup and W.J. Swiatecki, Ann.Phys. 125, 193
(1980).
[7] H.A. Weidenmu¨ller, in Progress in Particle and Nuclear Physics, Vol. 3, p. 49 Perg-
amon, Oxford (1980) and references therein.
[8] A. Lukasiak, W. Cassing and W. No¨renberg, Nucl. Phys. A426, 181 (1984).
[9] M.V. Berry and J.M Robbins, Proc. R. Soc. Lond. A442, 659 (1993).
[10] A. Bulgac, G. Do Dang and D. Kusnezov, Ann. Phys. (in press) (1995).
7
[11] A. Bulgac, G. Do Dang and D. Kusnezov, unpublished.
8
Figure 1: Instantaneous eigenvalue spectrum En(X) as a function of the \shape" (X),
for a Hamiltonian of the ensemble dened by Eqs. (1{3), with [H0]jk = kjk, using the
correlator F (x) = exp(−x2=2) (top) and F (x) = exp(−jxj) (bottom).
9
Figure 2: The time dependence of the occupation probabilities nk(t), for k = 1; : : : ; 51
(in this case nk(t) = n102−k(t)), where k counts from bottom to top in the gure, for the
case of fast, V0 = 4 (top), and slow, V0 = 1=16 (bottom), driving velocities.
10
Figure 3: The time dependence of the energy variance E(t) for a range of velocities
V0 = 4; 2; 1; 0:5; 0:25; 0:125 and 0:0625. The highest curve corresponds to the larger
velocity. The insert shows the diusion constant D(V 20 ) as a function of V
2
0 , indicating
D / V 20 .
11


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