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A transport model description of Time-Dependent Generator Coordinate under
Gaussian overlap approximation
Fangyuan Wang,1 Yingxun Zhang,1, ∗ and Zhipan Li2
1Department of Nuclear Physics, China Institute of Atomic Energy, Beijing 102413, People’s Republic of China
2School of Physical Science and Technology, Southwest University, Chongqing 400715, People’s Republic of China
(Dated: January 3, 2023)
In this work, we derived a transport equation based on a generalized equation of time-dependent
generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). The
transport equation is obtained by using quantum-mechanics phase space distributions under a
“quasi-particle” picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hierar-
chy. The theoretical advantage of this transport equation is that time evolution of s-body phase
space density distribution is coupled with s + 1-body phase space density distributions, and thus,
non-adiabatic effects and dynamical fluctuations could be involved by more collective degrees and
entanglement of phase space trajectories. In future, we will perform the numerical calculations for
fission nuclei after obtaining collective inertia and potential energy surface (PES).
I. INTRODUCTION
After the discovery of nuclear fission by Hahn and
Strassmann [1] in 1939, nuclear fission has become one
of the most challenging topics in physics since it is a key
ingredient for modeling nucleosynthesis [2], energy pro-
duction [3], medicine [4], and nuclear safeguard [5]. Even
with recent progress in experimental techniques, mea-
surements of nuclear fission are not possible for all fis-
sion nuclei. Thus, theoretical simulations are mandatory
for fully understanding the fission dynamics and comple-
menting the missing data [6–12].
One kind of models is useful macroscopic model by
taking into account shell effects, collective variables, and
correlations between collective degree and single parti-
cles motions, such as Brownian shape motion [13–15]
and Langevin model [16–18]. With four or five collective
degrees, it could depict fission dynamics process appro-
priately and reproduce fission yields distribution well.
Another kind of models is microscopic model, which
based on nucleonic Hamiltonian and solve fission dy-
namics with Schrödinger or Dirac equation in time
domain. For example, time-dependent density func-
tional theory, such as time-dependent superfluid lo-
cal density approximation (TDSLDA) [19, 20], Con-
strained and time-dependent Hartree-Fock calculations
with dynamical Bardeen–Cooper–Schrieffer pairing cor-
relations (CHF+BCS) [21, 22], time dependent Hartree-
Fock-Bogliubov (TDHFB)/TDHF+BCS [23], the adia-
batic time-dependent HFB (ATDHFB) [24] and time-
dependent covariant density functional theory (TD-
CDFT) [25] describe fission process with full quantum
microscopic approaches. At tremendous costs of compu-
tations, these models successfully describe the fission dy-
namics and predict the most probable fission yields. The
great understanding of fission dynamics from microscopic
model are obtained [19, 22]. However, describing fission
∗ zhyx@ciae.ac.cn
yields distribution with these models is still a theoreti-
cal challenge due to the lacking of fluctuation in initial
state and fission process. The efforts on this direction is
to describe quantum fluctuation by a sampling of initial
conditions followed by TDDFT[26] but without quantum
interference.
An alternative method to include the correlation is
to represent a many-body wave function of the system
with a mixture of states with different shapes. It stim-
ulates the description of fission dynamics with time-
dependent generator coordinate method under Gaussian
overlap approximation (TDGCM+GOA) [27–29]. In the
TDGCM+GOA, fission is assumed as adiabatic process
since the typical time for the motion of individual nucle-
ons inside the fission nucleus (roughly 10−22 s) is roughly
ten times smaller than the time scale of the system’s col-
lective deformation (10−21 s) [29]. Thus, the fission dy-
namics are approximately described in terms of a few
shape coordinates.
Currently, most of the TDGCM+GOA calculations
were performed by using only two degrees of freedom,
usually (q20, q30) or (β2, β3), under adiabatic assump-
tion [30–32]. While, the semiphenomenological and fully
microscopic approaches illustrate that at least four or
five collective variables play a role in the dynamics of
fission [19, 33–37]. Regnier et al. [38] have started some
trials on the rigorous three degrees of freedom calculation
of the PES for 240Pu in the collective space (q20, q30, q40),
and their work is still in progress. In Ref. [39], Zhao et
al. did the calculations with the dynamical pairing de-
gree of freedom as the third degree of freedom besides
(β2, β3), and their results also demonstrate the impor-
tance of including more degree of freedom. Furthermore,
one should note that an ad hoc Gaussian smoothing have
to be used at the end of the TDGCM+GOA calculations,
to account for the fluctuations in particle number of the
fragments due to both pairing effects and the finite num-
ber of particles in the neck region for points along the
fission line. Thus, one would expect to develop a micro-
scopic method that can include more collective degrees
http://arxiv.org/abs/2301.00202v1
mailto:zhyx@ciae.ac.cn
2
and the correlations to account for dynamical fluctua-
tion and non-adiabatic effects, and reasonably describing
fission dynamics and distribution of fission yields.
In this work, we derive a transport equation based on
a generalized N-dimensional TDGCM+GOA equation to
describe fission dynamics, in which non-adiabatic effects
are introduced by more collective degrees, fluctuations
are introduced by initial state fluctuation and entan-
glement of phase space trajectories. The paper is or-
ganized as follows: in Sec.II, the transport equation is
obtained by using quantum-mechanics phase space dis-
tributions under a “quasi-particle” picture and strategy
of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hi-
erarchy. One of the advantages of this hierarchy is that
correlations from high-order degree can be involved in
the evolution of the one-body phase-space density dis-
tribution. In Sec.III, a numerical recipes for solving the
transport equation is provided. Sec.IV is the summary
and outlook.
II. THEORY FRAMWORK
A. Overview of TDGCM for fission
For convenience, we briefly review the TDGCM +GOA
theory which describes induced fission as a slow adiabatic
process determined by a small number of collective de-
grees of freedom. Under the Griffin-Hill-Wheeler ansatz,
the many-body state of fissioning system at any time
reads
|Ψ(t)〉 =
∫
q∈E
dq|φ(q)〉f(q, t). (1)
The set {|φ(q)〉} is a family of the generator states
which are the solutions of a constrained Hartree-Fock-
Bogoliubov equation. f(q, t) is the complex-valued
weights of the quantum mixture of states. The gener-
ator coordinate q = {q1, ..., qN}, and each of these qi is a
collective variable chosen based on the physics of fission.
The time-dependent Schrödinger equation for the
many-body state of fission system |Ψ(t)〉,
(Ĥ − i~
d
dt
)|Ψ(t)〉 = 0, (2)
can yield an equation of the unknown weight func-
tion f(q, t), i.e., the Hill-Wheeler equation with time-
dependent form,
∫
dq′〈φq|
(
Ĥ − i~
d
dt
)
|φq′〉f(q
′, t) = 0. (3)
Here, Ĥ is the Hamiltonian acting on the full many-body
system. Principally, Eq. (3) can be solved numerically,
but it needs a tremendous amount of computations. To
overcome these difficulties, a popular approach named
as Gaussian overlap approximation (GOA) is used. The
simplest formulation of GOA assumes that the overlap
between two generator states 〈φq|φq′〉 has a Gaussian
shape,
N (q,q′) = 〈φq|φq′〉 ≡ exp
[
−
1
2
(q− q′)tG(q̄)(q − q′)
]
.
(4)
N (q,q′) = 〈φq|φq′〉 is peaked functions for q = q′, and
q̄ = (q+ q′)/2. By changinga new collective coordinate
α by the relation
α(q) =
∫
a∈Cq
0
G1/2(a)da, (5)
in terms of which the overlap matrix becomes
N (α,α′) = exp
[
−
1
2
(α−α′)2
]
, (6)
G (q) is the metric of new coordinates of α(q), and G (q)
is the determinant of G.
Within this approximation, the time-dependent Hill-
Wheeler equation is reduced to a local, time-dependent
Schrödinger-like equation as
i~
∂g(q, t)
∂t
= Ĥcoll(q)g(q, t). (7)
g(q, t) is related to the weight function f(q, t) as g =
N 1/2f , and contains all the information about the fis-
sion dynamics of system [29]. The collective Hamiltonian
Ĥcoll(q) is a local operator acting on g(q, t),
Ĥcoll(q) = (8)
[
−
~
2
2
∑
kl
1
√
G (q)
∂
∂qk
√
G (q)Bkl (q)
∂
∂ql
+ V (q)
]
.
The potential V (q),
V (q) = 〈q|Ĥ |q〉 − ǫ0(q), (9)
with the zero-point energy-correction
ǫ0 =
1
2
Gij(q)
∂2h
∂qi∂q′j
∣
∣
∣
∣
q=q′
. (10)
The symmetric collective inertial tensor B(q) ≡ Bij(q),
Bkl(q) =
1
2~2
Gkm(q)
(
∂2h(q,q′)
∂qm∂q′n
−
∂2h(q,q′)
∂qm∂qn
+
{
i
mn
}
∂h(q,q′)
∂qi
)∣
∣
∣
∣
q=q′
Gnl(q), (11)
the expression in braces is the Christoffel symbol of the
second kind. h(q,q′) is
h(q,q′) =
〈
φq|Ĥ |φq′
〉
〈φq|φq′〉
. (12)
3
They are usually calculated from the nuclear Hamilto-
nian Ĥ and the generator states |φq〉 with HFB [30] or
RMF+BCS [40].
The number of collective degree of freedom are usually
selected as N = 2, and shape coordinates q are the mul-
tipole moments Q20 and Q30 in Ref. [30–32], or β2 and
β3 as in Refs. [39, 40], and G (q) = 1 Ref. [31]. In this
case, the equation of TDGCM+GOA is
i~
∂
∂t
g (q1, q2; t) = (13)
[
−
~
2
2
∑
kl
∂
∂qk
Bkl (q1, q2)
∂
∂ql
+ V (q1, q2)
]
g (q1, q2; t) .
This equation has been solved by the software package
FELIX-1.0 [31] or FELIX-2.0 [32] with finite element
method.
B. A transport equation for N-dimensional
TDGCM+GOA
The TDGCM has achieved great progress on describ-
ing the fission dynamics [30–32, 39–41], but previous cal-
culations in Refs. [33–36, 38] also showed it is necessary
to include more degrees to describe nonadiabatic effects
which may arise from the coupling between collective and
intrinsic degrees of freedom, and invovle dynamical fluc-
tuations to describe fission products distributions. Now,
the question is that can we effectively involve more collec-
tive degrees of freedom into the equation with two degrees
that we are currently using?
Principally, nuclear shape can be described by an ex-
pansion in spherical harmonics, i.e.,
R(θ, φ, t) = R0

1 +
N
∑
λ=0
λ
∑
µ=−λ
α∗λµ(t)Yλµ(θ, φ)

 . (14)
The number of shape coordinates (or collective degree
of freedom) N depends on the choice of collective co-
ordinates or generator coordinates and stage of fission.
In the stage of fissionning system from the quasista-
tionary initial state to the outer fission barrier, evolu-
tion is slow and fission process can be described by a
small number of collective degree, i.e., a small N , with
adiabatic approximation[42]. In the stage of saddle-to-
scission, the nucleus quickly elongates toward scission
and non-adiabatic effects have to be considered and N
may vary with the stage of fission process. Thus, the col-
lective wave function is presented with N degrees, i.e.,
g(q1, q2, ..., qN ), for fissioning system, and Eq.(7) will be-
come a generalized N-dimensional TDGCM+GOA equa-
tion since the degrees are not limited to a few.
In this work, we interpret the wave function of
g(q1, q2, ..., qN ) for fissioning system as a wave function
for ‘N-body quasiparticles in 1-Dimension space’ sys-
tem (NB1D), and a convention gN(q1, q2, ..., qN ) is used
in following to represent the wave function with parti-
cle from 1 to N. Then, we derive a transport equation
which can effectively couple one more collective degree
of freedom to the degrees currently used. Firstly, we per-
form Wigner transformation [43] on NB1D wave function
gN (q1, · · · , qN ) to get their quantum mechanically phase
space density fN as,
fN (q1, · · · , qN ; p1, · · · , pN ) (15)
=
1
(π~)N
∫
· · ·
∫
dy1 · · · dyNg
∗
N(q1 − y1, · · · , qN − yN )
gN (q1 + y1, · · · , qN + yN)
× exp[−2i(p1 · y1 + · · ·+ pN · yN )/~].
Here pi is the conjugated momentum of qi for quasi-
particle i. After a trivial deviation, the corresponding
transport equation reads,
∂fN
∂t
= −
∑
kl
B̄
(b)
kl (q)pk
∂fN
∂ql
(16)
+
∑
λ=1,3,···
(
~
2i
)λ1+···+λN−1
1
λ1! · · ·λN !
×
∂λ1+···+λNVN (q)
∂qλ11 · · · ∂q
λN
N
∂λ1+···+λN fN
∂pλ11 · · ·∂p
λN
N
= −
∑
kl
B̄
(b)
kl (q)pk
∂fN
∂ql
+
∑
l
∂VN
∂ql
∂fN
∂pl
+
∑
λ=3,···
(
~
2i
)λ1+···+λN−1
1
λ1! · · ·λN !
×
∂λ1+···+λNVN (q)
∂qλ11 · · · ∂q
λN
N
∂λ1+···+λN fN
∂pλ11 · · ·∂p
λN
N
.
Here, λ =
∑N
i=1 λi and B̄
(b)
kl (q) is an effective collective
inertia, which is defined as
B̄
(b)
kl (q) =
(
Bkl(q− y
∗
(1b)) +Bkl(q− y
∗
(2b))
)
/2. (17)
y∗(1b) and y
∗
(2b) are corrections on q, and their origins can
be found in appendix A. VN (q) is N-body potential.
Principally, VN (q) = V (q1, q2, · · · , qN ). If the N-body
potential is calculated from the two-body interaction, i.e.,
V (q1, q2, · · · , qN ) =
∑
i≤j
V (qi, qj), (18)
the transport equation is simplified as,
∂fN
∂t
= −
∑
kl
B̄
(b)
kl (q)pk
∂fN
∂ql
+
1
2
∑
k,m 6=k
∂Vkm
∂qk
∂fN
∂pk
.(19)
Vkm is two-body potential between quasi-particle k and
m, which can be obtained by HFB/RMF+BCS as in
Refs.[30, 40]. When the N-body potential is obtained
from multi-dimensional PES by HFB/RMF+BCS, trans-
port equation of Eq.(16) should be used.
4
A standard procedure to solve the N -body transport
equation is to use BBGKY hierarchy, in which one-body
degrees of freedom (DOF) is coupled to two-body DOF
that are themselves coupled to three-body DOFs and
so forth. As an example, we present the time evolu-
tion of fs under the condition of V (q1, q2, · · · , qN ) =
∑
i≤j V (qi, qj). The s-body phase space density distri-
bution fs is defined as,
fs(q1, · · · , qs, p1, · · · , ps) (20)
=
1
ΩN−s
∫
fN (q1, · · · , qN , p1, · · · , pN )dΓs+1 · · · dΓN ,
dΓi = dqidpi,
here, Ω is volume in phase space. Thus,
∂fs(q1, · · · , qs, p1, · · · , ps)
∂t
(21)
=
1
ΩN−s
∫
∂fN (q1, · · · , qN , p1, · · · , pN )
∂t
dΓs+1 · · · dΓN
=
1
ΩN−s
∫ [
−
∑
kl
B̄
(b)
kl (q)pk
∂fN
∂ql
+
∑
1≤k<m≤N
∂Vkm
∂qk
∂fN
∂pk
]
dΓs+1 · · · dΓN .
One should note that the derivation in this case is dif-
ferent than a system with fixed-mass many particles, be-
cause the inertial B̄
(b)
kl (q) depends on collective coordi-
nates. To overcome this difficulty, we move the B̄
(b)
kl (q)
out from the integration by assuming the following rela-
tionship, i.e.,
∫
Bkl(q)O(q,p)dΓs+1 · · · dΓN = (22)
Bkl(qs, q
∗
s+1, · · · , q
∗
N )
∫
O(q,p)dΓs+1 · · · dΓN ,
The q∗s+1, · · · , q
∗
N depend on O(q,p), and its values will
be fixed once the O(q,p) is determined.
By using above relationship, we get
∂fs
∂t
= −
s
∑
k=1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )pk
∂fs
∂ql
(23)
−
N
∑
k=s+1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )p
∗
k
∂fs
∂ql
+
∑
1≤k<m≤s
∂Vkm
∂qk
∂fs
∂pk
+
N − s
Ω
∫ s
∑
k=1
(
∂Vk,s+1
∂qk
)
×
(
∂fs+1
∂pk
)
dΓs+1.
The collective inertial B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N ) only varies
with qs, since the values of q
∗
s+1, . . . , q
∗
N will be fixed
once the integrand was selected. But the values of
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N ) could be different from Bkl(qs).
The details of the derivation are in appendix B.
By expressing the fourth terms on the right side of
Eq. (23) as δIcoll, the transport equation can be rewritten
as,
∂fs
∂t
+
s
∑
k=1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )pk
∂fs
∂ql
(24)
+
N
∑
k=s+1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )p
∗
k
∂fs
∂ql
−
∑
1≤k<m≤s
∂Vkm
∂qk
∂fs
∂pk
= δIcoll.
with
δIcoll =
N − s
Ω
∫ s
∑
k=1
(
∂Vk,s+1
∂qk
)(
∂fs+1
∂pk
)
dΓs+1.(25)
As one can see that the δIcoll is related to the phase space
density fs+1, and the potential betweenk and s+1, i.e.,
Vk,s+1.
III. NUMERICAL RECIPE FOR SOLVING
TRANSPORT EQUATION
In this section, we discuss practical numerical recipe of
the time evolution of f2. In the following discussions, we
take q1 = β2 and q2 = β3.
Given s = 2, the time evolution of f2 becomes,
∂f2(q1, q2; p1, p2)
∂t
(26)
+
2
∑
k=1
2
∑
l=1
pkB̄
(b)
kl (q1, q2, q
∗
3 , · · · , q
∗
N )
∂f2
∂ql
+
N
∑
k=3
2
∑
l=1
p∗kB̄
(b)
kl (q1, q2, q
∗
3 , · · · , q
∗
N )
∂f2
∂ql
−
2
∑
k=1
2
∑
l 6=k
1
2
∂Vkl
∂qk
∂f2
∂pk
= δIcoll.
The time evolution of f2 is not only related to the col-
lective inertia Bkl and potential Vkl, but also related to
f3 and potential between qi=1,2 and q3 which actually
reflect the high order correlation between different shape
coordinates.
A. Initialization
For the TDGCM+GOA equation, the starting point is
a collective wave packet at initial time, which represents
the compound nucleus after excitation by absorption of
a low-energy neutron or photon. One choice is to use
the quasibound state [30], i.e., collective ground state
g0(q, t = 0), and q = {q1, q2}. Its modulus is roughly
a Gaussian centered on the minimum of the potential
5
Vmin(q), which is achieved by extrapolating inner poten-
tial barrier with a quadratic form. The width of this
Gaussian is characterized by a width close to the dimen-
sion of the first potential well. To describe fission, g0
has to boost in q1(β2) direction for simulating the fission
events, i.e.,
g(q, t = 0) = g0(q) exp(ikq1), (27)
since its average energy is below the fission barrier. The
amplitude k of the boost is determined so that the av-
erage energy of the initial state lies few MeV above the
inner fission barrier.
For the transport equation described in this work, we
need to do the initialization in {q1, q2; p1, p2} space ac-
cording the phase space density f2. The initial f2, which
represents the compound nucleus after the absorption of
a low-energy neutron, can be realized by doing Wigner
transformation on g0(q), i.e.,
f2(q1, q2; p1, p2, t = 0) = (28)
1
(π~)2
∫
dy1dy2g
∗
0(q+ y)g0(q− y)e
2ip·y/~.
Numerically, a test particle method is used to describe
f2(q1, q2; p1, p2, t = 0), which means each quasiparticle
is replaced by a large number of test particles and the
method was first proposed by Wong in nuclear Vlasov
model [44].
f2(q1, q2; p1, p2, t = 0) = (29)
1
Ntest
2
∑
k=1
Ntest
∑
i=1
δ(qki − q̄ki(t))δ(pki − p̄ki(t)).
qki and pki are the time-dependent coordinates and mo-
menta of the test-particle i for particle k=1 or 2. q̄ki(t =
0) and p̄ki(t = 0) are sampled according to the f2. Ntest
is the number of test particles. Once p̄ki(t = 0)’s are
obtained, p̄1i(t = 0) is boosted as p̄1i(t = 0) + k. k can
be obtained as same as in TDGCM initialization.
B. Time evolution
To solve the transport equations with test particle
method, we separately treat the left and right hand of
Eq.(26) as same as in the transport model used for sim-
ulating the heavy ion collisions[45].
The equation of motion of test particles under the
mean field can be obtained by comparing,
df2
dt
=
∂f2(q1, q2; p1, p2)
∂t
+
2
∑
k=1
[
∂f2
∂qk
∂qk
∂t
+
∂f2
∂pk
∂pk
∂t
]
= 0,
(30)
to Eq.(26) without considering the collision term, i.e.,
∂f2
∂t
= −
2
∑
k=1
2
∑
l=1
B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N )pk
∂f2
∂ql
(31)
−
N
∑
k=3
2
∑
l=1
B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N )p
∗
k
∂f2
∂ql
+
∑
1≤k<m≤2
∂Vkm
∂qk
∂f2
∂pk
.
The equation of motion of test particle ki becomes,
q̇ki =
2
∑
l=1
B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N )pli (32)
+
N
∑
l=3
B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N )p
∗
li
ṗki = −
1
2
∑
1≤l≤2
∂Vkl
∂qki
, (33)
The abbreviation of ki in the lower index means par-
ticle k and its ith test-particle, i.e., k = 1, 2 and i =
1, · · · , Ntest.
As shown in Eq.(32), the evolution of position of
test particle ki not only depend on the collective
inertia B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N ) and momentum of pli
with l = 1 and 2, but also on collective inertia
B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N ) and effective momentum of parti-
cle of p∗ki with l ≥ 3. In the calculations, one can approx-
imate B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N) = η(q
∗
3 , . . . , q
∗
N )Bkl(q1, q2)
for l ≤ 2, in which the parameter η depend on
the selection of qk≥3. Alternatively, one can use η
as a phenomenological parameter to fit the data of
fission yield. The value of B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N ) =
η(q∗3 , . . . , q
∗
N )Bkl(q1, q2) for l ≥ 3 can be learned if
the three-dimensional collective inertia can be provided.
Within the framework of this equation, the correlations
beyond q1 and q2 are involved via B̄
(b)
kl (q1, q2, q
∗
3 , . . . , q
∗
N ).
In the second term of Eq.(32), the contribution of p∗ki can
also be thought as friction effects.
The momentum of test particle update according to
Eq.(33), and it will depend on the potential Vkm. Inside
the scission line (hypersurface), potential Vkm can be ob-
tained by RMF+BCS/HFB model. Out of the scission
hypersurface, the fissioning trajectories will not go back
and one can set the potential ∂Vkm/∂qk = 0.
For high order correlation term in Eq. (25), i.e., the
collision term in our approach, it comes from,
δIcoll =
N−2
Ω
∫ ∑2
k=1
(
∂Vk,3
∂qk
)(
∂f3
∂pk
)
dΓ3. (34)
Suppose f3 can be expressed as,
f3(q1, q2, q3; p1, p2, p3) = f2(q1, q2; p1, p2)f1(q3; p3),
(35)
6
Thus,
δIcoll =
2
∑
k=1
(
∂Φ̄k
∂qk
)(
∂f2
∂pk
)
, (36)
and potential Φ̄k means,
Φ̄k =
N − 2
Ω
∫
Vk,3(qk, q3)f1(q3, p3)dΓ3, (37)
which reflect the potential of k particle felt by surround-
ing particles. However, exact calculations of Eq.(36)
and Eq.(37) are impossible since they always beyond one
more degree we have.
One effective way to handle the collision term is to use
a random collision among test particles. For example,
one first select three particles among 2Ntest test parti-
cles according to “collision section”, and then, perform a
random collision as follows,
pk1 + pk2 + pk3 = p
′
k1 + p
′
k2 + p
′
k3. (38)
p′k1, p
′
k2 and p
′
k3 will be determined by using the Monte-
Carlo sampling under the momentum conservation,
p′k1 = (pk1 + pk2 + pk3) ∗ ξ1, (39)
(p′k2 + p
′
k3) = (pk1 + pk2 + pk3) ∗ (1− ξ1), (40)
p′k2 = (p
′
k2 + p
′
k3) ∗ ξ2, (41)
p′k3 = (p
′
k2 + p
′
k3) ∗ (1− ξ2). (42)
ξ1 and ξ2 are the random number satisfy a certain dis-
tribution. The collision among test particles during the
evolution describe entanglement among different trajec-
tories of test particles. In practical calculations, the col-
lision probability can be adjusted by introducing a ‘cross
section’ of three-body collision. After random collision,
the values of momentum of test particle will be randomly
modified. As a result, the fluctuation on q will be au-
tomatically involved and the mass distribution of fission
fragment can be expected.
C. Fission fragments distributions
In this work, we only focused on fission fragment
mass/charge distribution. First, one need to search scis-
sion line (or hypersurface) on potential energy surface,
which is composed from many scission points qsci. In-
side the scission line(hypersurface), the nucleus is whole.
Out of scission line (hypersurface), the system becomes
two well-separated fragments which are connected by a
thin neck. Thus, each scission points qsci is associated
with a given fragmentation (AL, AR). AL and AR means
the mass of fission fragment in the left and right of neck,
respectively. The fission fragment mass can be obtained
from the integration of single-body density over the do-
main of left or right of neck,
AL =
∫
r∈L
drρ(r), (43)
AR =
∫
r∈R
drρ(r).
Here, R and Lmeans the region of right and left of neck of
fissioning system. ρ(r) is constructed from the collective
coordinates.
In transport model approach, the probability of mea-
sured the fission fragment AL and AR, i.e., Y (AL) and
Y (AR) can also be obtained from the time integrated flux
through the hypersurfaceelement S. By using the test
particle method, it will be obtained by counting the num-
ber of test particles across the hypersurface S at scission
point, i.e., with t → +∞,
Y (A, S) =
∫ ∫
qs>qsci
fs(qs,ps, t)dpsdqs (44)
=
1
2Ntest
2
∑
k=1
Ntest
∑
i=1
Θ(qki − q̄ki,sci(t)).
Thus, the yield of mass of fragment
Y (A) =
∑
S
Y (A, S). (45)
The sum on S runs over the whole scission hypersurface.
IV. SUMMARY AND OUTLOOK
In this work, we derive a transport equation based
on a generalized N-dimensional TDGCM+GOA equation
to describe fission dynamics. The transport equation is
obtained by using quantum-mechanics phase space dis-
tributions under a “quasi-particle” picture and strategy
of BBGKY hierarchy. The advantages of this transport
equation is that time evolution of s-body phase space
density distribution is coupled with s + 1-body phase
space density distributions. Thus, one can expect that
non-adiabatic effects and dynamical fluctuations could
be introduced by involving more collective degrees and
entanglement of phase space trajectories.
Different than directly solving the TDGCM+GOA
equation with finite elements method, our approach is
realized by using the test particle method. The coordi-
nates and momentum of test particles in initialization of
fissioning system are sampled according to initial phase
space density distribution of system. The time evolu-
tion of test particles are governed by a Hamiltonian-like
equation coupled with a random scattering between test
particles, which will naturally provide a fluctuation on
the mass of fission fragments. Finally, the numerical re-
sults on this transport equations are still on the way,
since we need to select reasonable collective coordinates
to obtain the results of PES and collective inertia, and
7
the high order effects of degree of freedom on collective
inertia should also be investigated in this approach before
presenting numerical results.
ACKNOWLEDGEMENTS
The authors thank Zhuxia Li, Zaochun Gao and
Siyu Zhuo for reading the manuscript and providing
useful feedback. This work was partly supported by
the National Natural Science Foundation of China Nos.
11875323, 12275359, 11875225, 11705163, 11790320,
11790323, and 11961141003, the National Key R&D Pro-
gram of China under Grant No. 2018 YFA0404404,
the Continuous Basic Scientific Research Project (No.
WDJC-2019-13, BJ20002501), Key Laboratory of Nu-
clear Data foundation (No.JCKY2022201C158) and
the funding of China Institute of Atomic Energy
YZ222407001301. The Leading Innovation Project of
the CNNC under Grant No. LC192209000701, No.
LC202309000201.
Appendix A: Derivation of transport equation
The equation of motion of gN is determined by the
Schrödinger-like equation, i.e.,
i~
∂
∂t
gN (q, t) = (A1)
[
−
~
2
2
∑
kl
∂
∂qk
Bkl (q)
∂
∂ql
+ VN (q)
]
gN (q, t) .
By using the Wigner transformation [43], fN (q,p) is ob-
tained. The equation of motion of fN (q,p) reads as,
∂fN(q,p, t)
∂t
=
1
(π~)N
∫
dy1 · · · dyNe
−2ip·y/~ (A2)
{
∂g∗N(q− y)
∂t
gN (q+ y) + g
∗
N (q− y)
∂gN(q + y)
∂t
}
.
where p is the conjugate momentum of q.
In the derivations of Eq. (A2), the ∂g∗N/∂t and ∂gN/∂t
are replaced with the right hand side of Eq. (A1) and we
have,
∂fN(q,p, t)
∂t
=
1
(π~)
N
∫
dy1 · · · dyNe
−2ip·y/~ (A3)
{
∑
kl
[
−
i~
2
∂
∂(qk − yk)
(
Bkl (q− y)
∂g∗N (q− y)
∂(ql − yl)
)
gN (q+ y)
+
i~
2
g∗N (q− y)
∂
∂(qk + yk)
(
Bkl (q+ y)
∂gN (q+ y)
∂(ql + yl)
)]
+
i
~
(V (q− y)− V (q+ y)) g∗N (q− y)gN(q + y)
}
.
One should note that the kinetic energy term contains the
collective coordinate q dependence of inertia Bkl, which
is different than the N-body system with fixed particle
mass.
In coordinate space, the kinetic energy terms in
Eq. (A3) are as follows,
Mkl = −
i~
2
1
(π~)N
∫
dy1 · · ·dyNe
−2ip·y/~ (A4)
[
∂
∂yk
(
Bkl(q− y)
∂g∗N (q− y)
∂yl
)
gN(q+ y)
−g∗N(q− y)
∂
∂yk
(
Bkl(q+ y)
∂gN (q+ y)
∂yl
)]
= Mkl,1 +Mkl,2.
Since the gN and g
∗
N depend on the integration vari-
able yk, one can replace the differentiations with respect
to qk+yk by differentiations with respect to yk in deriva-
tions. By doing one partial integration with respect to
yk, one has
Mkl,1 = (A5)
−
i~
2
1
(π~)N
Bkl(q− y)
∂g∗N (q− y)
∂yl
gN (q+ y)e
−2ip·y/~
∣
∣
∣
∣
+∞
−∞
+
i~
2
1
(π~)N
∫
dy1 · · · dyNBkl(q− y)
∂g∗N (q− y)
∂yl
×
∂
∂yk
(
gN (q+ y)e
−2ip·y/~
)
The first term in Eq. (A5) vanishes due to the boundary
condition at infinity, and second term becomes
Mkl,1 =
i~
2
1
(π~)N
∫
dy1 · · ·dyN (A6)
×Bkl(q− y)
∂g∗N (q− y)
∂yl
∂
∂yk
(
gN (q+ y)e
−2ip·y/~
)
=
i~
2
1
(π~)N
∫
dy1 · · ·dyNs
×Bkl(q− y)
∂g∗N (q− y)
∂yl
[
∂gN(q + y)
∂yk
e−2ip·y/~
+gN(q+ y)e
−2ip·y/~(−
2ipk
~
)
]
=
i~
2
Bkl(q− y
∗
(1a))
1
(π~)N
∫
dy1 · · · dyN (A7)
∂g∗N(q − y)
∂yl
∂gN (q+ y)
∂yk
e−2ip·y/~
+
i~
2
Bkl(q− y
∗
(1b))
1
(π~)N
∫
dy1 · · ·dyN
∂g∗N(q − y)
∂yl
gN(q+ y)e
−2ip·y/~(−
2ipk
~
).
In above derivations, the Bkl(q− y) is moved out by
assuming the following relationship, i.e.,
∫
Bkl(q − y)O(q, ∂/∂q)dy1 · · · dyN = (A8)
Bkl(q− y
∗
(I))
∫
O(q, ∂/∂q)dy1 · · · dyN .
8
Similarly, the Mkl,2 becomes
Mkl,2 = −
i~
2
Bkl(q+ y
∗
(2a))
1
(π~)N
∫
dy1 · · · dyN(A9)
∂g∗N(q− y)
∂yk
∂gN(q + y)
∂yl
e−2ip·y/~
−
i~
2
Bkl(q+ y
∗
(2b))
1
(π~)N
∫
dy1 · · · dyN
g∗N(q− y)
∂gN (q+ y)
∂yl
e−2ip·y/~(−
2ipk
~
)
By defining B̄
(a)
kl , B̄
(b)
kl , δB
(a)
kl and δB
(b)
kl as,
B̄
(a)
kl =
(
Bkl(q− y
∗
(1a)) +Bkl(q+ y
∗
(2a))
)
/2,
B̄
(b)
kl =
(
Bkl(q− y
∗
(1b)) +Bkl(q+ y
∗
(2b))
)
/2,
δB
(a)
kl =
(
Bkl(q− y
∗
(1a))−Bkl(q+ y
∗
(2a))
)
,
δB
(b)
kl =
(
Bkl(q− y
∗
(2a))−Bkl(q+ y
∗
(2b))
)
,
and assuming δB
(a)
kl ≈ 0 and δB
(b)
kl ≈ 0, Mkl becomes
Mkl = Mkl,1 +Mkl,2 (A10)
= −
i~
2
B̄
(a)
kl
1
(π~)N
∫
dy1 · · · dyNe
−2ip·y/~
[
∂g∗N (q− y)
∂ql
∂gN(q+ y)
∂qk
−
∂g∗N (q− y)
∂qk
∂gN(q+ y)
∂ql
]
−
i~
2
B̄
(b)
kl
1
(π~)N
∫
dy1 · · · dyNe
−2ip·y/~
[
∂g∗N (q− y)
∂ql
g(q+ y) − g∗N(q − y)
∂gN (q+ y)
∂ql
]
(−
2ipk
~
).
In Eq.(A10), an identical relationship between differen-
tial with respect to yk(yl) and with respect to qk(ql) is
used.
Due to the symmetric property of B̄
(a)
kl and summation
of
∑
kl in Eq.(A3), the contributions from first term in
Eq.(A10) vanishes. Thus, the kinetic part can be written
as,
Mkl = −pkB̄
(b)
kl
∂fN
∂ql
(A11)
For the potential part in Eq. (A3), a Taylor series with
respect to q is performed with potential fields VN (q+ y)
and VN (q− y).
N =
1
(π~)
N
∫
dy1 · · · dyNe
−2ip·y/~ (A12)
×
i
~
[V (q− y)− V (q+ y)] g∗N (q− y)gN(q + y)
=
1
(π~)
N
∫
dy1 · · · dyNe
−2ip·y
×
2i
~
∑
λ
1
λ1! · · ·λN !
∂λ1+···λNVN (q)
∂qλ11 · · · ∂q
λN
N
yλ11 · · · y
λN
N g
∗
N (q− y) gN (q+ y)
=
2i
~
∑
λ
(
~
2i
)λ1+···+λN
1
λ1! · · ·λN !
×
∂λ1+···+λNVN (q)
∂qλ11 · · · ∂q
λN
N
∂λ1+···+λN fN
∂pλ11 · · ·∂p
λN
N
Finally, Eq. (A3) could be written as
∂fN(q,p, t)
∂t
= −
∑
kl
pkB̄
(b)
kl (q)
∂fN
∂ql
(A13)
+
∑
λ
(
~
2i
)λ1+···+λN−1
1
λ1! · · ·λN !
×
∂λ1+···+λNVN (q)
∂qλ11 · · ·∂q
λN
N
∂λ1+···+λN fN
∂pλ11 · · · ∂p
λN
N
= −
∑
kl
pkB̄
(b)
kl (q)
∂fN
∂ql
+
N
∑
l
∂V
∂ql
·
∂fN
∂pl
+
∑
λ
(
~
2i
)λ1+···+λN−1
1
λ1! · · ·λN !
×
∂λ1+···+λNVN (q)
∂qλ11 · · ·∂q
λN
N
∂λ1+···+λN fN
∂pλ11 · · · ∂p
λN
N
..
where all λ1, · · · , λN are non-negative integer values and
λ1 + · · · + λN is an odd number. This is a general form
of transport equation for TDGCM+GOA.
When the potential is calculated from the two-body
interaction, i.e.,
V (q1, q2, · · · , qN ) =
∑
i≤j
V (qi, qj), (A14)
the equation is further simplified as,
∂fN
∂t
= −
∑
kl
B̄
(b)
kl (q)pk
∂fN
∂ql
+
∑
k≤m
∂Vkm
∂qk
∂fN
∂pk
. (A15)
9
Appendix B: Time evolution of s-body phase space
density distribution
The s-body phase space density is defined as,
fs(q1, · · · , qs, p1, · · · , ps) (B1)
=
1
ΩN−s
∫
fN (q1, · · · , qN , p1, · · · , pN )dΓs+1 · · · dΓN ,
dΓi = dqidpi.
When the potential is calculated from the two-bodyin-
teraction, the time-dependent probability distribution
fs is obtained by the similar strategy of the derivation
of the Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY)
hierarchy, i.e.,
∂fs(q1, · · · , qs, p1, · · · , ps)
∂t
=
1
ΩN−s
∫
dΓs+1 · · · dΓN
∂fN(q1, · · · , qN , p1, · · · , pN )
∂t
=
1
ΩN−s
∫
dΓs+1 · · · dΓN
[
−
∑
kl
B̄
(b)
kl (q)pk
∂fN
∂ql
+
∑
k≤m
∂Vkm
∂qk
∂fN
∂pk
]
. (B2)
Different than the transport equation for fixed-mass
many-particle system, the inertia Bkl(q) depends on the
coordinate and it causes the equation much more com-
plexity.
To avoid the difficulty caused by B̄kl(q), we move the
B̄kl out from the integration by using the equivalent of
the integration, i.e.,
∫
B̄kl(q)O(q,p)dΓs+1 · · · dΓN = (B3)
B̄kl(qs, q
∗
s+1, · · · , q
∗
N )
∫
O(q,p)dΓs+1 · · · dΓN .
Consequently, the derivation will be simplified.
For the first term in r.h.s of Eq. (B2), we perform one
partial integration with respect to ql and the result is
I1 =−
1
ΩN−s
∫ [
∑
kl
B̄
(b)
kl (q)pk
∂fN
∂ql
]
dΓs+1 · · · dΓN
=−
∑
kl
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )
×
1
ΩN−s
∫ [
pk
∂fN
∂ql
]
dΓs+1 · · · dΓN
=−
s
∑
k=1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )pk
∂fs
∂ql
−
N
∑
k=s+1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )
×
1
ΩN−s
∫
pk
∂fN
∂ql
dΓs+1 · · · dΓN
=−
s
∑
k=1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )pk
∂fs
∂ql
− δI1 (B4)
In above derivation, we use the term of
∑N
l=s+1 B̄
(b)
kl (q)pkfN |
ql→∞
ql→−∞
= 0. This is because
the finite values of fN , which means fN (q) = 0 at
q → ±∞. The δI1 is defined as,
δI1 =
N
∑
k=s+1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N ) (B5)
·
1
ΩN−s
∫
pk
∂fN
∂ql
dΓs+1 · · · dΓN
=
N
∑
k=s+1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )p
∗
k
∂fs
∂ql
which is connected to the N -body density distribution
fN , and reflects how the momentum field evolve with the
coordinates.
The second term in Eq. (B2) reads,
I3 =
1
ΩN−s
∫
∑
k≤m
∂Vkm
∂qk
∂fN
∂pk
dΓs+1 · · · dΓN (B6)
=
1
ΩN−s
∫
∑
1≤k≤m≤s
∂Vkm
∂qk
∂fN
∂pk
dΓs+1 · · · dΓN
+ (N − s)
1
ΩN−s
∫ l
∑
k=1
(
∂Vk,s+1
∂qk
)(
∂fN
∂pk
)
dΓs+1 · · · dΓN
=
∑
1≤k≤m≤s
∂Vkm
∂qk
∂fs
∂pk
+
N − s
Ω
∫ s
∑
k=1
(
∂Vk,s+1
∂qk
)
×
(
∂fs+1
∂pk
)
dΓs+1
10
Finally, the time evolution of fs is,
∂fs
∂t
= −
s
∑
k=1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )pk
∂fs
∂ql
(B7)
−
N
∑
k=s+1
s
∑
l=1
B̄
(b)
kl (qs, q
∗
s+1, . . . , q
∗
N )p
∗
k
∂fs
∂ql
−
∑
1≤k≤m≤s
∂Vkm
∂qk
∂fs
∂pk
+
N − s
Ω
∫ s
∑
k=1
(
∂Vk,s+1
∂qk
)
×
(
∂fs+1
∂pk
)
dΓs+1
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