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Inertial and dissipative properties of chaotic system
Nuclear Physics A545 (1992) 561c-56~Sc North-Holland, Amsterdam
NUCLEAR PH YSICS A
INERTIAL AND DISSIPATIVE PROPERTIES OF CHAOTIC SYSTEM V . M . Kolomietz and V. N . Kondratyev Institute for Nuclear Research of Ukraine, Kiev
Abstract Stochastic
approximation
approach is
developed
based
on
for
the
semiclassical
the
description
one-body
relaxation
of
finite
Fermi-system dynamics . The considerable exceeding of the smooth term of the cranking-model inertial coefficient under the hydrodynamic value is demonstrated . Connection between the friction
coefficient
and
established
time
is
i . INTRODUCTION The equations
interacting
of
motion
particles .
investigation of these
work we utilize such
However
are
the
not
intégrable
stochastic
methods
systems can be developed ti,2J .
approach
for
t~~e
calculations
inertial mass parameter B and friction coefficient 1~. framework of cran>
gcr - 2 ~ dE2 Here ~d ( g) and
s-.o
~~( g)
response functions :
for
many of
In this
of
In
the
the
dx (~) ~ . ~ -SLdE ~ ~-.o are
the
dissipat ve
and
polarization
0375-9474/92/$05 .00 © 1992 - Elscvicr Scicncc Publishers B .V. All rights rcservcd .
SC~`~~ t~.~i. r~to~~aicatz, V.dV. o®adraryev l A~tertial and dissipa~rive properties of a chaotic .system
P
d~, d ( ~° ) ?L
' g -E
a
and are where n~ are the occupat ion numbers, (p01, A g~. single-particle wave functions and eigenvalues, A is the one-body operator of cranking field . Usually the sum in eq .(2) is localized close to the Ferrai surface . In this case
In a semiclassical approximation the smooth term dissipative response function C4) of chaotic system written as t1,2] (2
of the can be
)
where C(g~,,t) is the classical correlation function C(~~,t)=
dp dr ~(g~, - h(~i,r))A(p,r)A(pt~rt)
(ë~)
Here (~~, r t ) is a paint of the phase space at time t, belongs to the classical trajectory with Hamiltonian and initial point The c .l.assical value (P, r) A. corresponds to the operator A .
which h(p,r) A(p, r)
CfI®
I?~F~2TIA1_
S
Let us consider a spherical poten~,ial ~( r) = tfo A( -r ; with a
o
(7)
small quadrupole deformation of the potential surface
V.M. Kolomietz, V.N. Kondratyev / Inertial and dissipative properties of a chaotic system 563c R(t) = R The
1 + ß(t)Y
a
following
m
(g)
perturbation
motion [4l
A= ~5/ 16~
(~ ) r
2o
P2-
operator
A
corresponds to this
7pa
The correlation function eq .(6) can be now written as C(~,t)
1
=
The index at t=d .
a P dp dr ~(E- 2m~p2`12a(~~)P2(t)`l2~(~P(t))Q(RG-r) af
r
means the initial point
r
of classical
Integrating in eq .(1d) over the space solid
(10)
trajectory
and momentum p we get
angle
s PF Ro C ( g~, t ) = m f drrz f d~~~ PZ ( cos P, Pt 0
where pue=
2mgF and (p, pt ) is the angle between r,m ""-~ mermen tcsm a_~,dl ~a,r . . t Lii .,e ~ .~ P t at i, me t .
the
initial
(i) Specular reflection
The trajectory turns
arcsin ~RG sin~ipl~ at each
through
surface
the
angle
reflection
7G-2ßf ~Qf point for
= the
specular reflection conditions . Therefore we have (P,Pt)
t+e - ti~2 ti ,2 (~-2Q f ) f t+d+~ (7L-2Bf)
L
ti
for +.,<0
(12)
far t>a
where , [ aJ i s the in+.,egAr part of a,
ti =
2R ~
cosQ f/vF
rcosQp/v~, , v~. = P~./m .
Using egs .(5), (11) ând (12) we obtain s ~0 7G P F 3~ (~~cos8f ) cos ( ~~cosPP) ~d ( g) = f dr rz ~d®~sinQp s in m$~ ( 27C`~) ` 0 0 x ~ n
Ô(?~cosQf+2Q f +n7G) +Ô(2~cos(3 f-2Qf-n7L)
r
( 13)
St~4~ V
olomietz, V.N. ondratyev l inertial and dissipative properties ofa chaotic system
From
.(
the polarization response function is
) and (1~)
s ~ 7G p ~ 127L r 2 d8psin$~,sin(2$f) dr ,~t E) _(2 ) m o (-i) n ( 2 8t+n 7L) a
x ~dere
a =
~' = gsg~, E
rcos$
cos
(14 )
ß ( 2 yf+n7G) ~
i1~./R Q . In the adiabat ic 1 imit
( v«i )
we
have ~,~( E)
(O) +27Cv~
_
2 5°i pF` ï0 (2
R3 o
~'a + . . .
)
(15 )
Using this expression we find the smooth part
of the cranking
mass eq . ( i ) as BC ~,
_
(25?d70)
2
(87C
(16)
AmRô) .a 5, 5H h
where A is the number of particles and Hh is ~..he usual inertial coefficient for irrota~ional liquid drop [5) . The much
value
of
the
cranking
hydrodinamical one
can
be
mass
in
explained
comparison as
the
greater
with
vortex
the
motion
contribution to HcP . (ii) Isotropic reflection Let us assume that
the
reflections
toward
the
sphere
interior angle-independent . This reflection property should be taken into account in eq .(11) as the additional averaging aver
a scattering angle d~~ at the arbitrary reflection point k : C(E,t)
s Ro p~ . di~~ dr r 2 d~ jl ~ ~~ = m
47C 5
0
where the prime near
an
integral
Y2~(~1p) Ya (~p (t) ) %1 over
~k
means
(17)
that
the
integration is fulfiled over the internal region of the sphere,
The corresponding response functions are given by ~(E)
E~ 2
sinzv_
sin(2~)
V.M. Kvlomietz, V.N. Kondratyev l Inertial and dissipative properties ofa chaotic system
x,it( B)
kFR° 3 SF
=27G ~
27L
~v
a
sin2?~ - cos ( 2v) 2v
565c
(sg)
Now the smooth cranking mass is B~r = (3/1a7L)
(AmR~) _ (4/5)B~,
(2d)
Thus, the vortex contribution vanishes due to the isotropic reflection conditions, and the average cranking mass is close to the irrotational fluid mass B h.
3 .ONB-SODY RELAXATION The behaviour of the time-dependent response function xd( t) = plays
2~ the
Im dW e l ~t
xd (i~)
_m important
collective motion .
role
_
(2 in
~
) a
d
4
dt
G(E F , t)
theory
of
(21) the
adiabatic
It determines the relaxation time tir and the memory effects by the collective motion . Using egs .(11), (12) and (Z7) we obtain ?G
k F.R a 3 E~BR f ( T) ~ 2 2~
sign { T) ,
where the function f(T) in the considered (i) Specular re±~lection f(T) S 8
( 22 ) above cases is :
_ (1-3{T/2) ~+2(T/2) s ) Q(2-~T~ ) cos(2(2k+3)~) 2{2k+3)
cos(2(2k-s)d~) 2 ( 2k-1)
(23) cos(2C2k+1)~) ~=~k+s , 2k+1 ~~k
where = 8(2k-!T~ ) arc cas (T/2k) g T=~~t/ . (ii)
Isotropic reflection
olo tet`® f.1V. vndratyev i loae~rt~al aaad dissipatsve pro~erties of a ci:antic systerr~ '(T) = ( -t
(24)
Th se
e tassions give the estimate of the relaxation time ®Av e for both cases . Finally, we can evaluate . tom eqs . C i) and (i~) the Z'riction coefficient k
(25)
This expression coincides with the well-known adiabatic s'riction coe~'ficient .for the one-body dissipation [6] . However, the relat ;nn between ~~ and ti t differs from the one in the kinetic theory .
Z
4 5 6
Feingold and A .Peres, Phys .Rev . A34 (i9~6) 59?2 ilkinson, J . Plays . A20 (i9~?) 2415 V. olo iet? and P .Siemens, Nucl .Phys .A314 (19i7) i :i2 A. igdal and V .P .1(rainov, The approximation methods of quant machete i cs : auka, Moscow, 1966 ohr and B . ottelson . Nuclear structure : Plenum, Mew York, 1965 S . . ~oonin, R . L . Hatch and J . Randrup, Nucl . Plays . A2B~ (19?7) 7
NUCLEAR PH YSICS A
INERTIAL AND DISSIPATIVE PROPERTIES OF CHAOTIC SYSTEM V . M . Kolomietz and V. N . Kondratyev Institute for Nuclear Research of Ukraine, Kiev
Abstract Stochastic
approximation
approach is
developed
based
on
for
the
semiclassical
the
description
one-body
relaxation
of
finite
Fermi-system dynamics . The considerable exceeding of the smooth term of the cranking-model inertial coefficient under the hydrodynamic value is demonstrated . Connection between the friction
coefficient
and
established
time
is
i . INTRODUCTION The equations
interacting
of
motion
particles .
investigation of these
work we utilize such
However
are
the
not
intégrable
stochastic
methods
systems can be developed ti,2J .
approach
for
t~~e
calculations
inertial mass parameter B and friction coefficient 1~. framework of cran>
gcr - 2 ~ dE2 Here ~d ( g) and
s-.o
~~( g)
response functions :
for
many of
In this
of
In
the
the
dx (~) ~ . ~ -SLdE ~ ~-.o are
the
dissipat ve
and
polarization
0375-9474/92/$05 .00 © 1992 - Elscvicr Scicncc Publishers B .V. All rights rcservcd .
SC~`~~ t~.~i. r~to~~aicatz, V.dV. o®adraryev l A~tertial and dissipa~rive properties of a chaotic .system
P
d~, d ( ~° ) ?L
' g -E
a
and are where n~ are the occupat ion numbers, (p01, A g~. single-particle wave functions and eigenvalues, A is the one-body operator of cranking field . Usually the sum in eq .(2) is localized close to the Ferrai surface . In this case
In a semiclassical approximation the smooth term dissipative response function C4) of chaotic system written as t1,2] (2
of the can be
)
where C(g~,,t) is the classical correlation function C(~~,t)=
dp dr ~(g~, - h(~i,r))A(p,r)A(pt~rt)
(ë~)
Here (~~, r t ) is a paint of the phase space at time t, belongs to the classical trajectory with Hamiltonian and initial point The c .l.assical value (P, r) A. corresponds to the operator A .
which h(p,r) A(p, r)
CfI®
I?~F~2TIA1_
S
Let us consider a spherical poten~,ial ~( r) = tfo A( -r ; with a
o
(7)
small quadrupole deformation of the potential surface
V.M. Kolomietz, V.N. Kondratyev / Inertial and dissipative properties of a chaotic system 563c R(t) = R The
1 + ß(t)Y
a
following
m
(g)
perturbation
motion [4l
A= ~5/ 16~
(~ ) r
2o
P2-
operator
A
corresponds to this
7pa
The correlation function eq .(6) can be now written as C(~,t)
1
=
The index at t=d .
a P dp dr ~(E- 2m~p2`12a(~~)P2(t)`l2~(~P(t))Q(RG-r) af
r
means the initial point
r
of classical
Integrating in eq .(1d) over the space solid
(10)
trajectory
and momentum p we get
angle
s PF Ro C ( g~, t ) = m f drrz f d~~~ PZ ( cos P, Pt 0
where pue=
2mgF and (p, pt ) is the angle between r,m ""-~ mermen tcsm a_~,dl ~a,r . . t Lii .,e ~ .~ P t at i, me t .
the
initial
(i) Specular reflection
The trajectory turns
arcsin ~RG sin~ipl~ at each
through
surface
the
angle
reflection
7G-2ßf ~Qf point for
= the
specular reflection conditions . Therefore we have (P,Pt)
t+e - ti~2 ti ,2 (~-2Q f ) f t+d+~ (7L-2Bf)
L
ti
for +.,<0
(12)
far t>a
where , [ aJ i s the in+.,egAr part of a,
ti =
2R ~
cosQ f/vF
rcosQp/v~, , v~. = P~./m .
Using egs .(5), (11) ând (12) we obtain s ~0 7G P F 3~ (~~cos8f ) cos ( ~~cosPP) ~d ( g) = f dr rz ~d®~sinQp s in m$~ ( 27C`~) ` 0 0 x ~ n
Ô(?~cosQf+2Q f +n7G) +Ô(2~cos(3 f-2Qf-n7L)
r
( 13)
St~4~ V
olomietz, V.N. ondratyev l inertial and dissipative properties ofa chaotic system
From
.(
the polarization response function is
) and (1~)
s ~ 7G p ~ 127L r 2 d8psin$~,sin(2$f) dr ,~t E) _(2 ) m o (-i) n ( 2 8t+n 7L) a
x ~dere
a =
~' = gsg~, E
rcos$
cos
(14 )
ß ( 2 yf+n7G) ~
i1~./R Q . In the adiabat ic 1 imit
( v«i )
we
have ~,~( E)
(O) +27Cv~
_
2 5°i pF` ï0 (2
R3 o
~'a + . . .
)
(15 )
Using this expression we find the smooth part
of the cranking
mass eq . ( i ) as BC ~,
_
(25?d70)
2
(87C
(16)
AmRô) .a 5, 5H h
where A is the number of particles and Hh is ~..he usual inertial coefficient for irrota~ional liquid drop [5) . The much
value
of
the
cranking
hydrodinamical one
can
be
mass
in
explained
comparison as
the
greater
with
vortex
the
motion
contribution to HcP . (ii) Isotropic reflection Let us assume that
the
reflections
toward
the
sphere
interior angle-independent . This reflection property should be taken into account in eq .(11) as the additional averaging aver
a scattering angle d~~ at the arbitrary reflection point k : C(E,t)
s Ro p~ . di~~ dr r 2 d~ jl ~ ~~ = m
47C 5
0
where the prime near
an
integral
Y2~(~1p) Ya (~p (t) ) %1 over
~k
means
(17)
that
the
integration is fulfiled over the internal region of the sphere,
The corresponding response functions are given by ~(E)
E~ 2
sinzv_
sin(2~)
V.M. Kvlomietz, V.N. Kondratyev l Inertial and dissipative properties ofa chaotic system
x,it( B)
kFR° 3 SF
=27G ~
27L
~v
a
sin2?~ - cos ( 2v) 2v
565c
(sg)
Now the smooth cranking mass is B~r = (3/1a7L)
(AmR~) _ (4/5)B~,
(2d)
Thus, the vortex contribution vanishes due to the isotropic reflection conditions, and the average cranking mass is close to the irrotational fluid mass B h.
3 .ONB-SODY RELAXATION The behaviour of the time-dependent response function xd( t) = plays
2~ the
Im dW e l ~t
xd (i~)
_m important
collective motion .
role
_
(2 in
~
) a
d
4
dt
G(E F , t)
theory
of
(21) the
adiabatic
It determines the relaxation time tir and the memory effects by the collective motion . Using egs .(11), (12) and (Z7) we obtain ?G
k F.R a 3 E~BR f ( T) ~ 2 2~
sign { T) ,
where the function f(T) in the considered (i) Specular re±~lection f(T) S 8
( 22 ) above cases is :
_ (1-3{T/2) ~+2(T/2) s ) Q(2-~T~ ) cos(2(2k+3)~) 2{2k+3)
cos(2(2k-s)d~) 2 ( 2k-1)
(23) cos(2C2k+1)~) ~=~k+s , 2k+1 ~~k
where = 8(2k-!T~ ) arc cas (T/2k) g T=~~t/ . (ii)
Isotropic reflection
olo tet`® f.1V. vndratyev i loae~rt~al aaad dissipatsve pro~erties of a ci:antic systerr~ '(T) = ( -t
(24)
Th se
e tassions give the estimate of the relaxation time ®Av e for both cases . Finally, we can evaluate . tom eqs . C i) and (i~) the Z'riction coefficient k
(25)
This expression coincides with the well-known adiabatic s'riction coe~'ficient .for the one-body dissipation [6] . However, the relat ;nn between ~~ and ti t differs from the one in the kinetic theory .
Z
4 5 6
Feingold and A .Peres, Phys .Rev . A34 (i9~6) 59?2 ilkinson, J . Plays . A20 (i9~?) 2415 V. olo iet? and P .Siemens, Nucl .Phys .A314 (19i7) i :i2 A. igdal and V .P .1(rainov, The approximation methods of quant machete i cs : auka, Moscow, 1966 ohr and B . ottelson . Nuclear structure : Plenum, Mew York, 1965 S . . ~oonin, R . L . Hatch and J . Randrup, Nucl . Plays . A2B~ (19?7) 7