# Stochastic features of dissipative large-amplitude dynamics and nuclear
fission

V. M. Kolomietz and S. V. Radionov Electronic address:
<EMAIL_ADDRESS>Institute for Nuclear Research, 03680 Kiev,
Ukraine

###### Abstract

Within a density matrix approach for nuclear many–body system, it is derived
non–Markovian Langevin equations of motion for nuclear collective parameters,
where memory effects are defined by memory time. The developed stochastic
approach is applied to study both the nuclear descent from fission barrier to
a scission point and thermal diffusive overcoming of the barrier. The present
paper is partly a review of our results obtained earlier and contains new
results on the non–Markovian generalization of Kramers’ theory of escape rate
and on time features of the collective dynamics in the presence of periodic
external modulation.

###### pacs:

21.60.Ev, 25.85.Ca

##  I Introduction

Nuclear large-scale dynamics (nuclear fission, heavy ion collisions etc.) is a
good probe for the investigation of the complex time evolution of finite Fermi
systems. The principal question here is how macroscopic (collective) modes of
motion sije87 ; hamy88 ; hofm97  are affected by complex microscopic
(intrinsic) excitations of the many–body Fermi systems. Due to dissipative and
fluctuating character of the collective modes of motion, manifesting in
non–zero widths of the nuclear giant multipole resonances and non–zero
variances of the kinetic energy of the nuclear fission fragments, one uses
transport approaches of the Fokker–Planck and Langevin types bosu93 ; bosu94 ;
abay96 ; ayno82 ; kolo95 ; kora06 ; koab08 ; kora09  to the study of nuclear
large–amplitude dynamics. In general, basic equations for the macroscopic
collective variables are non–Markovian kora01 ; kosh04 , implying complex
energy flow between the slow collective and fast intrinsic degrees of freedom
of the nuclear many–body system.

In the present paper, following the ideology of the random matrix theory
weiden80 ; bulgac01  we discuss how non–Markovian (memory) and stochastic
aspects of the nuclear fission dynamics are defined by quantum–mechanical
diffusion of energy in the space of occupancies of complex many–body states.
The general problem of decay of a metastable state (like a decay of nuclear
compound state) has gotten a lot of attention in the functional integral
approach grab84 ; hanggi85  and in its application to the nuclear case rumm02
. Quantum decay rate of the nuclear compound state has been derived within the
local harmonic approximation hofm83 , where also non–Markovian effects were
considered. In spite of such wide literature on the subject, the problem of
classical activated (escape) rate over the nuclear fission barrier in the
presence of the non–Markovian effects has been left without appropriate
attention. We are going to renew this deficiency by measuring a general
quantitative impact of memory effects on the classical thermal rate and time
characteristics of the nuclear fission dynamics.

The plan of the paper is as follows. In Sect. II, we consider a many-body
dynamics by use the Zwanzig’s projection technique and derive the basic
non–Markovian Langevin equations of motion within the cranking approach to
nuclear many–body dynamics. In Sect. III, we discuss the application of
non–Markovian dynamics to the nuclear descent from the fission barrier, the
memory effect on the Kramers’ diffusion over the fission barrier, the
stochastic penetration over oscillating barrier and the diffusion of
occupation probabilities within Landau-Zener approach. The non–Markovian
dynamics of nuclear Fermi liquid is considered in Sect. IV. Summary and
conclusions are given in Sect. V.

##  II Many-body dynamics in a moving frame

We assume that dynamics of nuclear many–body system can be described as a
coupled motion of several slow macroscopic (collective) modes and intrinsic
nucleonic ones. Slow collective modes are treated in terms of a set of
classical time–dependent variables
$q(t)\equiv\\{q_{1}(t),q_{2}(t),...,q_{N}(t)\\}$. The fast intrinsic modes are
described quantum mechanically through the Liouville equation for the density
matrix operator $\hat{\rho}$,

$\frac{\partial\hat{\rho}(t)}{\partial t}+i\hat{L}(t)\hat{\rho}(t)=0,$ (1)

where $\hat{L}$ is the Liouville operator defined as

$\hat{L}\hat{\rho}=\frac{1}{\hbar}\left[\hat{H},\hat{\rho}\right].$ (2)

Here $\hat{H}(q)$ is the nuclear many–body Hamiltonian. We introduce a moving
(adiabatic) basis of the Hamiltonian $\hat{H}(q)$,

$\hat{H}(q)\Psi_{n}(q)=E_{n}(q)\Psi_{n}(q),$ (3)

determined by a set of eigenfunctions $\Psi_{n}(q)$ and eigenenergies
$E_{n}(q)$ for each fixed value of the macroscopic variables $q(t)$. Within
this basis one can introduce a non–diagonal, $\rho_{nm}$, and diagonal,
$\rho_{nn}$, parts of the density matrix through the relations,

$\rho_{nn}=\langle\Psi_{n}|\hat{\rho}|\Psi_{n}\rangle,~{}~{}~{}~{}~{}\rho_{nm}=\langle\Psi_{n}|\hat{\rho}|\Psi_{m}\rangle,$
(4)

and whose time evolution may be determined within the Zwanzig’s projection
technique zwan60 . The final results read, see Refs. kolo95 ; kora10 ,

$\displaystyle\rho_{nm}(t)=\rho_{nm}(t=0)-i\sum_{j=1}^{N}\int_{0}^{t}dt^{\prime}\dot{q}_{j}(t^{\prime})\frac{\mathrm{exp}[-i\omega_{nm}(t-t^{\prime})]}{\omega_{nm}}$
$\displaystyle\times\left[h_{j,mn}(t^{\prime})\rho_{nn}(t^{\prime})-h_{j,nm}(t^{\prime})\rho_{mm}(t^{\prime})\right],$
(5)

and its diagonal part,

$\displaystyle\frac{\partial\rho_{nn}(t)}{dt}$ $\displaystyle=$
$\displaystyle\frac{2}{\hbar^{2}}\sum_{i,j=1}^{N}\dot{q}_{i}(t)\int_{0}^{t}dt^{\prime}\dot{q}_{j}(t^{\prime})\sum_{m\neq
n}h_{i,nm}(t)h_{j,mn}(t^{\prime})\frac{\mathrm{cos}[\omega_{nm}(t-t^{\prime})]}{\omega_{nm}^{2}}$
(6)
$\displaystyle\times\left[\rho_{mm}(t^{\prime})-\rho_{nn}(t^{\prime})\right].$

Here, $\omega_{nm}=(E_{n}-E_{m})/\hbar$ and matrix elements
$h_{i,nm}=\left(\partial\hat{H}/\partial q_{i}\right)_{nm}$ measure the
coupling between the quantum nucleonic and the macroscopic collective
subsystems. Note that the equations (5) and (6) were derived in the
weak–coupling limit kora10 . This condition can be violated near the avoided
crossings of two nearest levels. One can go beyond the second–order
perturbation result (5)–(6) by considering the Landau–Zener model of two
crossing levels as shall be done below.

A complexity of a quantum system is understood as the absence of any special
symmetries in a system. Such a system is expected to have some universal
statistical properties which can be modeled by the random matrix ensembles.
Within the random matrix approach goel65 , we will average the right–hand side
of master equation (6) over suitably chosen statistics of the randomly
distributed matrix elements $h_{i,nm}$ and the energy spacings $E_{n}-E_{m}$.
First, we perform the ensemble averaging over the matrix elements. They are
treated as complex random numbers with the real and the imaginary parts which
are independently Gaussian distributed, and with wilk89

$\overline{h_{i,nm}(q)h_{j,n^{\prime}m^{\prime}}^{\ast}(q^{\prime})}=\delta_{nn^{\prime}}\delta_{mm^{\prime}}\sigma_{ij}^{2}(E_{n},E_{m},q+q^{\prime})Y_{ij}(|q-q^{\prime}|),$
(7)

where $Y_{ij}(|q-q^{\prime}|)$ is a correlation function, measuring how strong
the ensemble averaged matrix elements correlate at different collective
deformations $q$ and $q^{\prime}$, and
$\sigma_{ij}^{2}(E_{n},E_{m},q+q^{\prime})$ is the energy distribution of the
squared matrix elements. It is rather clear that at high excitation energies
the coupling matrix elements between the complex many–body states should drop
out with increasing energy distance between them. In order to characterize the
energy distribution of the matrix elements, we introduce the strength of the
distribution $\sigma_{0,ij}^{2}$ and its width $\Gamma_{ij}$ assuming

$\sigma_{ij}^{2}(E_{n},E_{m},q+q^{\prime})=\frac{\sigma_{0,ij}^{2}(E_{n}+E_{m},q+q^{\prime})}{\sqrt{\Omega(E_{n})\Omega(E_{m})}\Gamma_{ij}}f_{ij}(|E_{n}-E_{m}|/\Gamma_{ij}),~{}$
(8)

where $\Omega(E)$ is the average level–density at given excitation energy $E$
and $f(E)$ is a shape of the energy distribution. The correlations of the
matrix elements, existing at different values of the macroscopic parameter,
$q$ and $q^{\prime}$, are measured with the help of the correlation function
$Y_{ij}$. Since the energy correlations between two different states $n$ and
$m$ drop out with the rise of a distance between them, it is rather obvious
that $f_{ij}\rightarrow 0$ with $|E_{n}-E_{m}|/\Gamma_{ij}\rightarrow\infty$
and $f_{ij}(E)\sim 1$ at $|E_{n}-E_{m}|/\Gamma\ll 1$. As is discussed in Refs.
brne79 ; zelev95 , for nuclear shell–model Hamiltonians typical values of the
spreading width $\Gamma$ are of several (tens) MeV.

The energy spacings part of the ensemble averaging procedure is defined
through the two–level correlation function, $R(|E_{n}-E_{m}|\Omega(E))$, that
is the probability density to find the state $m$ with energy $E_{m}$ within
the interval $[E_{m},E_{m}+dE_{m}]$ at the average distance $|E_{n}-E_{m}|$
from the given state $n$ with energy $E_{n}$. In the case of quite dense
spectrum of the adiabatic states $n$, it is more convenient to use continuous
energy variables $E$ and $e$, that measure a total excitation and distances
between different states, correspondingly,

$E\equiv E_{n},~{}~{}~{}e\equiv E_{n}-E_{m},$ (9)

In this variables, the basic equation (6) for the occupancies
$\overline{\rho}(E,t)$ of adiabatic states is transformed as

$\Omega(E)\frac{\partial\bar{\rho}(E,t)}{\partial
t}=\sum_{i,j=1}^{N}\dot{q}_{i}(t)\int_{0}^{t}dt^{\prime}K_{ij}(t,t^{\prime})\dot{q}_{j}(t^{\prime})\frac{\partial}{\partial
E}\left[\Omega(E)\frac{\partial\bar{\rho}(E,t^{\prime})}{\partial E}\right].$
(10)

In Eq. (10), the memory kernel, $K_{ij}(t,t^{\prime})$, is defined as

$K_{ij}(t,t^{\prime})=\frac{\sigma_{0,ij}^{2}(E,q+q^{\prime})}{\Gamma_{ij}}Y_{ij}(|q-q^{\prime}|)\int_{\infty}^{+\infty}de\
f_{ij}(|e|/\Gamma_{ij})R[|e|\Omega(E)]\mathrm{cos}(e[t-t^{\prime}]/\hbar).$
(11)

The integration limits over the energy spacing $e$ in Eq. (11) were extended
to infinities since the time changes of the occupancy $\bar{\rho}(E,t)$ of the
given state with the energy $E$ are mainly due to the direct interlevel
transitions from the close–lying states located at the distances $|e|\ll E$.
The explicit form of the two–level correlation function $R(x)$ in Eq. (11)
depends on the statistical ensemble of levels pame83 . It can be established
the following result for three often used ensembles:

(i) Gaussian Orthogonal Ensemble (GOE)

$R_{\mathrm{GOE}}(x)=1-\left(\frac{\mathrm{sin}(\pi x)}{\pi
x}\right)^{2}+\left(\int_{0}^{1}dy\frac{\mathrm{sin}(\pi
xy)}{y}-\frac{\pi}{2}\right)\left(\frac{\mathrm{cos}(\pi x)}{\pi
x}-\frac{\mathrm{sin}(\pi x)}{(\pi x)^{2}}\right),$ (12)

(ii) Gaussian Unitary Ensemble (GUE)

$R_{\mathrm{GUE}}(x)=1-\left(\frac{\mathrm{sin}(\pi x)}{\pi x}\right)^{2},$
(13)

(iii) Gaussian Symplectic Ensemble (GSE)

$R_{\mathrm{GSE}}(x)=1-\left(\frac{\mathrm{sin}(2\pi x)}{2\pi
x}\right)^{2}+\int_{0}^{1}dy\frac{\mathrm{sin}(2\pi
xy)}{y}\left(\frac{\mathrm{cos}(2\pi x)}{2\pi x}-\frac{\mathrm{sin}(2\pi
x}{(2\pi x)^{2}}\right),$ (14)

where $x\equiv|E_{n}-E_{m}|\Omega(E_{n})$.

The main difference between the statistics is its behavior of $R(x)$ at small
energy spacings $x$. For the GOE statistics one has the linear repulsion
between levels, $R_{\mathrm{GOE}}\sim x$, the GUE statistics implies the
quadratic level repulsion, $R_{\mathrm{GUE}}\sim x^{2}$, while in the GSE case
we have $R_{\mathrm{GSE}}\sim x^{4}$. On the other hand, $R_{\mathrm{GOE}}$,
$R_{\mathrm{GUE}}$ and $R_{\mathrm{GSE}}$ are similar at moderate spacings
$x$, when the spectral correlations between levels consistently disappear, see
also Refs. goel65 ; koab08 .

Note that the dynamic process (10) may be treated as a quantum mechanical
diffusion of energy in space of the occupancies of quantum adiabatic states,
where a function

$W(E,t)\equiv\Omega(E)\bar{\rho}(E,t),~{}~{}~{}~{}~{}\int_{E_{0}}^{+\infty}W(E,t)dE=1,$
(15)

gives a probability density to find the intrinsic quantum system with an
excitation energy lying in the interval $[E,E+dE]$ at the moment of time $t$.
The time features of the non–Markovian quantum diffusive dynamics (10)–(11)
are defined by the relationship between characteristic time scales,
$\tau_{ij}\sim\hbar/\Gamma_{ij}$ (caused by the finite width $\Gamma_{ij}$ of
the coupling matrix elements’ energy distribution $f_{ij}(E)$) and the typical
time interval $\tau_{\mathrm{macr},ij}$ of the macroscopic variables’
variations. For nuclear many–body Hamiltonians, showing statistical
properties, the energy distribution of the Breit–Wigner shape zelev95 ,
$f_{ij}(|e|/\Gamma_{ij})=(1/\pi)/(1+[|e|/\Gamma_{ij}]^{2})$. This gives rise
to an exponentially decaying with time memory kernel (11) (here we restrict
ourselves by the one-dimension case to simplify the notation)

$K(t,t^{\prime})=\sigma_{0}^{2}(E,q+q^{\prime})\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)$
(16)

with $\tau=\hbar/\Gamma$. If one now assumes that $\tau$ is the shortest time
scale in a system, one easily obtains the Markovian limit of the quantum
diffusion process (10) in terms of the probability density function of the
intrinsic quantum system (15). Namely,

$\frac{\partial W(E,t)}{\partial t}=-\frac{\partial}{\partial
E}\left[r(E,t)W(E,t)\right]+\frac{\partial^{2}}{\partial
E^{2}}\left[D(E,t)W(E,t)\right],~{}~{}~{}~{}~{}\frac{\hbar}{\Gamma}\ll\tau_{\mathrm{macr}},$
(17)

where the drift coefficient $r(E,t)$ is equal to

$r(E,t)=\frac{dD(E,t)}{dE}+\frac{D(E,t)}{\Omega(E)}\frac{d\Omega(E)}{dE}$ (18)

and the diffusion coefficient $D(E,t)$ is given by

$D(E,t)=(\hbar/\Gamma)\cdot\sigma_{0}^{2}(E,q)\dot{q}^{2}.$ (19)

The drift term in the diffusion equation (17) leads to parametric excitation
of the intrinsic quantum system that becomes possible either due to the
energy–dependence of the distribution of the slopes $\sigma_{0}^{2}(E)$ in Eq.
(19), or due to the average level–density $\Omega(E)$ that grows with an
intrinsic excitation $E$. The diffusion coefficient $D(E,t)$ (19) depends
quadratically on the velocity $\dot{q}$ of the parametric driving as a result
of the assumed weak–coupling regime kora10 . It should be pointed out that the
quantum–mechanical diffusion in space of many–body states’ occupancies gets
stronger dependence on the velocity of the driving in the case of Landau–Zener
transitions between states (see Subsection III.5).

###  II.1  Energy rate

To obtain equations of motion for the macroscopic collective parameters
$q(t)$, we first find the average energy of the many–body system,
$\mathcal{E}(t)=\mathrm{Tr}[\hat{H}\\{q(t)\\}\hat{\rho}(t)]$. Calculating the
time change of $\mathcal{E}(t)$, we obtain

$\frac{d\mathcal{E}(t)}{dt}=\sum_{i}\dot{q}_{i}\frac{\partial
E_{\mathrm{0}}(q)}{\partial
q_{i}}+\sum_{i}\dot{q}_{i}\sum_{nm}\left(\frac{\partial\hat{H}}{\partial
q_{i}}\right)_{mn}\rho_{nm}+\sum_{n}E_{n}\frac{\partial\rho_{nn}}{\partial
t}+\sum_{i}\dot{q}_{i}\sum_{n}\left(\frac{\partial\hat{H}}{\partial
q_{i}}\right)_{nn}\rho_{nn}.\quad$ (20)

The first term in the right–hand side of Eq. (20) describes a change of the
macroscopic potential energy $E_{\mathrm{0}}(q)=E_{\mathrm{pot}}(q)$. The
second contribution to the energy rate $d\mathcal{E}/dt$ is defined by the
non–diagonal components of the density matrix $\rho_{nm}(t)$. Its time
evolution is caused by the virtual transitions among the adiabatic states.
Such a term is a microscopic source for the appearance of the macroscopic
kinetic energy. To demonstrate that, we write it as

$\displaystyle\left(\frac{d\mathcal{E}}{dt}\right)^{\mathrm{virt}}\equiv\sum_{i}\dot{q}_{i}\sum_{nm}\left(\frac{\partial\hat{H}}{\partial
q_{i}}\right)_{mn}\rho_{nm}$
$\displaystyle=\sum_{i}\dot{q}_{i}(t)\xi_{i}(t)+2\sum_{i}\dot{q}_{i}\sum_{j}\sum_{nm}\int_{0}^{t}dt^{\prime}\mathcal{V}_{ij,nm}(t,t^{\prime})\dot{q}_{j}(t^{\prime})[\rho_{mm}(t^{\prime})-\rho_{nn}(t^{\prime})],$

where

$\xi_{i}(t)=\left(\frac{\partial\hat{H}}{\partial
q_{i}}\right)_{mn}\rho_{nm}(t=0),~{}~{}~{}\mathcal{V}_{ij,nm}(t,t^{\prime})=h_{i,nm}(t)h_{j,mn}(t^{\prime})\frac{\mathrm{cos}(\omega_{nm}[t-t^{\prime}])}{\omega_{nm}}.$
(22)

We formally extend the lower limit of the time integration in Eq. (II.1) to
$-\infty$. In this way, we would like to study stationary dynamics of the
complex quantum system, i. e., when the dynamics of the system does not depend
on the choice of initial time. It should be pointed out that this does not
imply a loss of any possible memory effects in collective motion. Thus,
integrating by parts the time integral in the r.h.s of Eq. (II.1), one can
show that

$\int_{-\infty}^{t}dt^{\prime}\mathcal{V}_{ij,nm}(t,t^{\prime})\dot{q}_{j}(t^{\prime})[\rho_{mm}(t^{\prime})-\rho_{nn}(t^{\prime})]\approx\sum_{l=0}^{+\infty}\omega_{nm}^{-(2l+3)}\times\frac{d^{(2l+1)}\left(\dot{q}_{j}h_{i,nm}h_{j,mn}[\rho_{mm}-\rho_{nn}]\right)}{dt^{(2l+1)}}.$
(23)

In the weak–coupling limit kora10 , we obtain

$\left(\frac{d\mathcal{E}}{dt}\right)^{\mathrm{virt}}\approx\sum_{i}\dot{q}_{i}(t)\xi_{i}(t)+\sum_{i}\dot{q}_{i}\sum_{j}(B_{ij}(q)\ddot{q}_{j}+\sum_{k}\frac{\partial
B_{ij}(q)}{\partial q_{k}}\dot{q}_{j}\dot{q}_{k}),$ (24)

where the term

$B_{ij}(q)=\sum_{n,m}h_{i,nm}h_{j,mn}\omega_{nm}^{-3}[\rho_{mm}-\rho_{nn}]$
(25)

can be associated with a macroscopic inertia tensor.

The third term on the right–hand side of Eq. (20) is determined by the real
transitions between the adiabatic states $\Psi_{n}(q)$ and it defines how the
energy of macroscopic motion is transferred into the energy of the intrinsic
excitations of the quantum system:

$\left(\frac{d\mathcal{E}}{dt}\right)^{\mathrm{real}}=\sum_{i}\dot{q}_{i}(t)\sum_{j}\int_{0}^{t}dt^{\prime}K(t,t^{\prime})\dot{q}_{j}(t^{\prime})\int_{E_{0}}^{+\infty}dE~{}E\frac{\partial}{\partial
E}\left(\Omega(E)\frac{\partial\bar{\rho}(E,t^{\prime})}{\partial E}\right),$
(26)

where Eq. (10) was used.

The fourth term in the r.h.s of Eq. (20) is given by the distribution of
slopes of the adiabatic eigenstates $E_{n}$. Within the random matrix model,
the negative and positive slopes of the adiabatic states are assumed to be
equally distributed. Therefore, under the averaging over all random
realizations of the random matrices, modeling the nuclear many body spectrum,
one can neglect the contribution from the fourth term in the r.h.s. of Eq.
(20).

Rewriting the time change of the average energy of the nuclear many–body
system (20) as

$\displaystyle\frac{d\mathcal{E}(t)}{dt}=\sum_{i}\dot{q}_{i}(t)F_{i}(q,t)$
$\displaystyle\equiv\sum_{i}\dot{q}_{i}(t)\big{(}\frac{\partial
E_{\mathrm{pot}}}{\partial
q_{i}}+\xi_{i}(t)+\sum_{j}\left[B_{ij}\ddot{q}_{j}+\sum_{k}\frac{\partial
B_{ij}}{\partial q_{k}}\dot{q}_{j}\dot{q}_{k}\right]$
$\displaystyle+\sum_{j}\int_{0}^{t}dt^{\prime}K_{ij}(t,t^{\prime})\dot{q}_{j}(t^{\prime})\int_{E_{0}}^{+\infty}dE\Omega(E)E\frac{\partial}{\partial
E}\left[\Omega(E)\frac{\partial\bar{\rho}(E,t^{\prime})}{\partial
E}\right]\big{)}$ (27)

and assuming that all the partial contributions $F_{i}(q,t)$ to the energy
rate (having a meaning of the forces acting on the macroscopic collective
subsystem) are equal zero, we obtain transport description of the macroscopic
collective dynamics

$\displaystyle\sum_{j}\left[B_{ij}\ddot{q}_{j}+\sum_{k}\frac{\partial
B_{ij}}{\partial q_{k}}\dot{q}_{j}\dot{q}_{k}\right]=-\frac{\partial
E_{\mathrm{pot}}}{\partial q_{i}}$
$\displaystyle-\sum_{j}\int_{0}^{t}dt^{\prime}K_{ij}(t,t^{\prime})\dot{q}_{j}(t^{\prime})\int_{E_{0}}^{+\infty}dEE\frac{\partial}{\partial
E}\left[\Omega(E)\frac{\partial\bar{\rho}(E,t^{\prime})}{\partial
E}\right]-\xi_{i}(t).$

###  II.2 Fluctuation–dissipation theorem

The transport equations (II.1) for the classical collective parameters $q(t)$
should be considered selfconsistently with the equation of motion for the
occupancies $\bar{\rho}(E,t)$ of the quantum many–body states (10). As was
stated above (see Eq. (17) and comment to it), the intrinsic quantum system is
excited by time variations of the collective parameters $q(t)$ provided that
the averaged density of the many–body states $\Omega(E)$ grows with the
excitation $E$. Importantly that such a parametric excitation is
time–irreversible in the sense that the intrinsic quantum system (10) does not
go back to the initial state with time–reversing of the macroscopic collective
parameters but continues to be excited. This is so because under the
parametric time variations (defined by the absolute value of the parameters’
velocity $|\dot{q}(t)|$, see Eqs. (18) and (19)), quantum–mechanical
transitions to higher–lying states occur more often than to lower–lying
states. In turn, due to the energy conservation condition (20), the increase
of the intrinsic excitation

$E^{*}(t)\equiv\int_{E_{0}}^{+\infty}dE\Omega(E)E\bar{\rho}(E,t)$ (29)

of the nuclear many–body system may be intrepreted as a corresponding decrease
(dissipation) of an energy associated with the time variations of the
macroscopic collective parameters $q(t)$:

$E_{\rm
coll}\equiv\sum_{i,j}\frac{1}{2}B_{i,j}(q)\dot{q}_{i}\dot{q}_{j}+E_{\rm
pot}(q).$ (30)

In this way, we justified microscopically dissipative character of the nuclear
collective motion. For quite high initial excitations of the nucleus
$E^{*}(t=0)$, when the averaged density of many–body states is given by

$\Omega(E^{*}(t=0))\sim e^{E^{*}(t=0)/T},$ (31)

the collective dissipation is determined by memory kernels

$\mathcal{K}_{ij}(t,t^{\prime})=\frac{\sigma_{0,ij}^{2}(E^{*}(t=0),q[t]+q[t^{\prime}])}{T}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau_{ij}}\right),$
(32)

of the retarded friction forces in the transport equations of motion

$\sum_{j}\left[B_{ij}\ddot{q}_{j}+\sum_{k}\frac{\partial B_{ij}}{\partial
q_{k}}\dot{q}_{j}\dot{q}_{k}\right]=-\frac{\partial E_{\mathrm{pot}}}{\partial
q_{i}}-\sum_{j,k}\frac{\partial B_{ij}}{\partial
q_{k}}\dot{q}_{j}\dot{q}_{k}-\sum_{j}\int_{0}^{t}dt^{\prime}\mathcal{K}_{ij}(t,t^{\prime})\dot{q}_{j}(t^{\prime})-\xi_{i}(t),$
(33)

In Eqs. (31)–(33), a quantity $T$ is considered as thermodynamic temperature
of the nucleus.

To complete the transport description of the macroscopic collective dynamics
(33) one has to define statistical properties of the terms $\xi_{i}(t)$ (22),
which are proportional to the non–diagonal components $\rho_{nm}$ of the
density matrix operator $\hat{\rho}(t)$ at initial moment of time $t=0$. If
the initial distribution of the diagonal components $\rho_{nn}$ is uniquely
set in by the initial excitation energy of the nuclear many–body system
$E^{*}(t=0)$ (29), the initial values of the non–diagonal components
$\rho_{nm}$ are left without specifying. For each fixed value of the initial
excitation $E^{*}(t=0)$ , one has a buch of the values $\rho_{nm}(t=0)$ which
may be treated as random numbers. In this case, we can treat the terms
$\xi_{i}(t)$ in the transport equations (33) as random forces acting on the
macroscopic collective parameters $q_{i}(t),~{}i=1,2,...$. By using Eqs.
(22),(7) and (32), we obtain for correlation functions of the random forces
$\xi_{i}(t)$,

$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=T\mathcal{K}_{ij}(t,t^{\prime}),$
(34)

where it was assumed for simplicity that $|\rho_{nm}(t=0)|^{2}=1$. The result
(34) represents the classical fluctuation–dissipation relation between
dissipative and fluctuating properties of the macroscopic collective dynamics
both caused by the quantum–mechanical diffusion of energy in the space of the
occupancies of many–body states (10).

##  III Application to large–amplitude collective motion

###  III.1 Descent from the fission barrier

We are going to apply the developed transport approach (33)–(34) to the
nuclear fission in two dimension space of shape variables $\\{q_{1},q_{2}\\}$.
We consider the symmetric fission of heavy nuclei whose shape is obtained by
rotation of a profile function $\mathcal{L}(z)$ around $z$–axis. The Lorentz
parameterization for $\mathcal{L}(z)$ is used in the following form hamy88 ,

$\mathcal{L}^{2}(z)=(z^{2}-\zeta_{0}^{2})(z^{2}+\zeta_{2}^{2})/Q\,,$ (35)

where the multiplier $Q$ guarantees the volume conservation,

$Q=-[\zeta_{0}^{3}({\frac{1}{5}}\zeta_{0}^{2}+\zeta_{2}^{2})]/R_{0}^{3}\,.$
(36)

Here, all quantities of the length dimension are expressed in the $R_{0}$
units, where $R_{0}$ is the radius of equal-sized sphere. The shape variables
$q_{1}=\zeta_{0}$ and $q_{2}=\zeta_{2}$ are related to the nuclear elongation,
$\zeta_{0}$, and the neck radius
$\rho_{\mathrm{neck}}=\zeta_{2}/\sqrt{\zeta_{0}(\zeta_{0}^{2}/5+\zeta_{2}^{2})}.$
The basic equations of motion (33) in two dimensions is written as

$\sum_{j=1}^{2}\left[B_{ij}(q)\ddot{q}_{j}+\sum_{k=1}^{2}\frac{\partial
B_{ij}(q)}{\partial q_{k}}\
\overset{\cdot}{q_{j}}\overset{\cdot}{q_{k}}\right]=-\frac{\partial
E_{\mathrm{pot}}(q)}{\partial
q_{i}}-\sum_{j=1}^{2}\int_{0}^{t}dt^{\prime}\mathcal{K}_{ij}(t,t^{\prime})\dot{q}_{j}(t^{\prime})-\xi_{i}(t).$
(37)

The numerical calculations are performed for the symmetric fission of the
nucleus ${}^{236}\mathrm{U}$ at temperature $T=2\ \mathrm{MeV}$. Considering
the nuclear descent from the saddle point to the scission, we use the
potential energy of deformation $E_{\mathrm{pot}}(\zeta_{0})$ from Ref. hamy88
assuming that $\partial
E_{\mathrm{pot}}(q_{1},q_{2})/\partial\rho_{\mathrm{neck}}=0$. The scission
line is derived from the condition of the instability of the nuclear shape
with respect to any variations of the neck radius:

$\left.\frac{\partial^{2}E_{\mathrm{pot}}(q_{1},q_{2})}{\partial\rho_{\mathrm{neck}}^{2}}\right|_{\mathrm{scis}}=0$
(38)

Here, we wish to study the random force effect on the non–Markovian dynamics
(37). To define properly a scission condition, we first introduce a
deterministic (i. e., without the random force) path of the system (37) and
evaluate the nuclear neck radius at the scission point
$\left(\rho_{\mathrm{neck}}^{\mathrm{det}}\right)_{\mathrm{scis}}$. Then, a
bunch of $2\times 10^{4}$ stochastic trajectories $\zeta_{0}(t),\zeta_{2}(t)$
is stopped as far as

$\rho_{\mathrm{neck}}(\zeta_{0}(t),\zeta_{2}(t))=(\rho_{\mathrm{neck}}^{\mathrm{det}})_{\mathrm{scis}}.$
(39)

As a result of that, we get a distribution of moments of time,
$t_{\mathrm{sc}}$, when the system reaches the scission point (38). The
corresponding probability density $p$ of the scission events is shown in
figure 1 for the case of quite weak memory effects $\tau_{1}=2\times
10^{-23}~{}s$ and fairly strong memory effects $\tau_{2}=8\times
10^{-23}~{}s$. For comparison, it is also shown (by vertical lines) the
corresponding scission times in the absence of the random force.

Figure 1: Histogram, showing a probability density $p$ of moments of time
$t_{\mathrm{sc}}$, when the stochastic trajectories
$\zeta_{0}(t),\zeta_{2}(t)$ (37) hit the scission line
$\rho_{\mathrm{neck}}(\zeta_{0},\zeta_{2})=\left(\rho_{\mathrm{neck}}^{\mathrm{det}}\right)_{\mathrm{scis}}$,
is given at two values of the memory time $\tau$. The dashed histogram is
found for $\tau_{1}=2\times 10^{-23}~{}\mathrm{s}$ (when the memory effects in
the system are quite weak) and the solid histogram corresponds to the memory
time $\tau_{2}=8\times 10^{-23}~{}s$ (when the memory effects are fairly
strong). The corresponding times of descent in the absence of the random force
are given by small vertical arrows. Taken from Ref. kora09 .

We see that with the growth of the memory effects in (37) the distribution of
the scission times $t_{\mathrm{sc}}$ becomes wider and the centroid of the
distribution shifts to the left compared to the deterministic values of the
scission time. One can interpret this as a ”stochastic” acceleration of the
nuclear descent from fission barrier, caused by the presence of the random
force term in equations of motion (37), see also Ref. kora09 .

Note that within the two–dimensional non–Markovian Langevin approach (37)–(38)
one can also calculate other characteristics of nuclear fission process. Thus,
in Ref. kora09  we have calculated two experimentally observable quantities
like the mean value and variance of the total kinetic energy of fission
fragments at infinity. By that, we have estimated the value of memory time and
found that $\tau\approx 8\times 10^{-23}~{}s$.

###  III.2 Escape rate problem within the Langevin approach

To measure the role of memory effects in collective dynamics of the nuclear
system on the way from ground state to saddle point, we restrict ourselves by
considering a one–dimensional collective motion $q(t)$ over a schematic
parabolic barrier shown in figure 2. The potential energy $E_{\mathrm{pot}}$
presents a single–well barrier formed by a smoothing joining at $q=q^{\ast}$
of the potential minimum oscillator with the inverted oscillator hofm97

$E_{\mathrm{pot}}=\begin{cases}\frac{1}{2}B\omega_{A}^{2}(q-q_{A})^{2},~{}~{}~{}~{}q\leq
q^{\ast},\\\
=E_{\mathrm{pot,B}}-\frac{1}{2}B\omega_{B}^{2}(q-q_{B})^{2},~{}~{}~{}~{}q>q^{\ast}.\end{cases}$
(40)

Figure 2: Dependence of the potential energy $E_{\mathrm{pot}}$ (Kramers’
potential) on the shape parameter $q$.

We also adopt constant value $B$ for the collective inertia tensor (25) and
for the energy distribution $\sigma^{2}_{0}$ of the coupling matrix elements
in (32), leading us to the equation of motion for the collective variable
$q(t)$,

$B\ddot{q}=-\frac{\partial E_{\mathrm{pot}}}{\partial
q}-\mathcal{K}_{0}\int_{0}^{t}dt^{\prime}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)\dot{q}(t^{\prime})-\xi(t),$
(41)

where $\mathcal{K}_{0}=\sigma^{2}_{0}/T$ and

$\langle\xi(t)\xi(t^{\prime})\rangle=T\mathcal{K}_{0}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right),$
(42)

see Eq. (34).

We studied the non–Markovian Langevin dynamics (41), (42) by calculating a
distribution of times $t_{\rm pre-sad}$ of the first crossing of the barrier
top (pre–saddle times). For that, the equation of motion (41) is solved
numerically by generating a bunch of the trajectories, all starting at the
potential well (point $A$ in figure 2) and having the initial velocities
distributed according to the Maxwell–Boltzman distribution. On left panel of
figure 3, it is shown the mean pre–saddle time $\langle t_{\rm pre-
sad}\rangle$ as a function of the memory time $\tau$ and right panel of figure
3 gives the ration between the standard deviation $\sigma(t_{\rm pre-sad})$
and the mean value $\langle t_{\rm pre-sad}\rangle$ of the pre–saddle time
distribution.

Figure 3: The mean pre–saddle time $\langle t_{\mathrm{pre-sad}}\rangle$ (left
panel) and the standard deviation $\sigma(t_{\rm pre-sad})$ over $\langle
t_{\rm pre-sad}\rangle$ (right panel) of the non–Markovian diffusion dynamics
(41), (42) are shown as a function of the memory time $\tau$.

An increase of the mean time $\langle t_{\mathrm{pre-sad}}\rangle$ of motion
from the potential minimum $q_{A}$ to the saddle point $q_{B}$ with the memory
time $\tau$ means that the memory effects in the Langevin dynamics (41),(42)
hinders the diffusion over barrier. With growth of the memory time $\tau$ the
adiabatic conservative force $-(\partial E_{\mathrm{pot}}/\partial q)$ in
equation of motion (41) gets an additional contribution from the time–retarded
force

$-\mathcal{K}_{0}\int_{0}^{t}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)\dot{q}(t^{\prime})dt^{\prime}\rightarrow-\mathcal{K}_{0}\left[q(t)-q_{A}\right],~{}~{}~{}~{}~{}\omega_{B}\tau\rightarrow\infty,$
(43)

leading to the subsequent slowing down of motion to the saddle point
$q_{\mathrm{b}}$. In the opposite limit of quite small values of the memory
time $\tau$, the slowing down of motion is exclusively due to effect from an
usual friction force since

$-\mathcal{K}_{0}\int_{0}^{t}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)\dot{q}(t^{\prime})dt^{\prime}\rightarrow-\mathcal{K}_{0}\tau\dot{q}(t),~{}~{}~{}~{}~{}\omega_{B}\tau\rightarrow
0.$ (44)

Also note that with the growth of the memory time $\tau$, the pre–saddle time
distribution becomes wider as it is followed from right panel of figure 3.

###  III.3 Non–Markovian extension of the Kramers model

One can also study the non–Markovian thermal diffusion over the parabolic
barrier (41), (42) with the help of an escape rate characteristics, $R_{0}$,
defining an exponential time decay of the probability
$\mathrm{Prob}(q[t]<q_{B})$,

$\mathrm{Prob}(q[t]<q_{B})=e^{-R_{0}t},~{}~{}~{}~{}~{}t>1/\gamma_{0},$ (45)

that the system $q[t]$ does not still reach the barrier top at $q_{B}$. Note
that the exponential decay with time of the survival probability (45) is
reached at sufficiently large times $t$, when $t$ is larger than the inverse
characteristic friction coefficient $\gamma_{0}$, that will be derived below.

To get an analytical estimate for the escape rate $R_{0}$, one can naturally
try to use the formalism of the Kramers’ theory kram40  and that must be
extended to the non–Markovian system (41),(42). With this purpose, we write
down the equation of motion (41) for the collective variable $q(t)$ in the
vicinity of the barrier top $q_{B}$,

$B\ddot{q}=B\omega_{B}^{2}(q-q_{B})-\mathcal{K}_{0}\int_{0}^{t}dt^{\prime}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)\dot{q}(t^{\prime})-\xi(t).$
(46)

General solution to the integro–differential equation (46) is written as

$q(t)=q_{B}+\mathcal{A}(t)q_{0}+\mathcal{B}(t)v_{0}-\int_{0}^{t}\mathcal{B}(t-t^{\prime})\xi(t^{\prime})$
(47)

or, in terms of velocity

$v(t)=\dot{\mathcal{A}}(t)q_{0}+\dot{\mathcal{B}}(t)v_{0}-\int_{0}^{t}\dot{\mathcal{B}}(t-t^{\prime})\xi(t^{\prime})dt^{\prime},$
(48)

where $q_{0}$ is an initial coordinate and $v_{0}$ is an initial velocity of
the system. Two functions $\mathcal{A}(t)$ and $\mathcal{B}(t)$ are given by

$\mathcal{A}(t)=1+\omega_{B}^{2}\int_{0}^{t}\mathcal{B}(t^{\prime})dt^{\prime},~{}~{}~{}\mathcal{B}(t)=D_{1}e^{s_{1}t}+D_{2}e^{s_{2}t}+D_{3}e^{s_{3}t},$
(49)

where $s_{1},s_{2}$ and $s_{3}$ are roots of the secular equation

$s^{3}+\frac{1}{\tau}s^{2}+\frac{\mathcal{K}_{0}-\omega_{B}^{2}}{B}s-\frac{\omega_{B}^{2}}{B\tau}=0$
(50)

and the coefficients $D_{1},D_{2}$ and $D_{3}$ equal to

$D_{1}=\frac{(s_{1}+1/\tau)}{(s_{1}-s_{2})(s_{1}-s_{3})},~{}~{}D_{2}=\frac{-(s_{2}+1/\tau)}{(s_{1}-s_{2})(s_{2}-s_{3})},~{}~{}D_{3}=\frac{(s_{3}+1/\tau)}{(s_{1}-s_{3})(s_{2}-s_{3})}.$
(51)

The solutions to the secular equation (50) have a threshold behavior. For
small enough memory times $\tau<\tau_{\mathrm{thresh}}$ (weak memory effects
in the equation of motion (46), the first root $s_{1}$ of the secular equation
(50) is a positive number, while the other two roots $s_{2}$ and $s_{3}$ are
negative ones. In this case, the mean collective path $\langle q(t)\rangle$
becomes to grow exponentially with time as

$\langle
q(t)\rangle=a_{1}\mathrm{exp}(s_{1}t)+a_{2}\mathrm{exp}(-|s_{2}|t)+a_{3}\mathrm{exp}(-|s_{3}|t).$
(52)

In the case of fairly large memory times $\tau>\tau_{\mathrm{thresh}}$ (strong
enough memory effects), $s_{1}$ is still a positive number, whereas $s_{2}$
and $s_{3}$ become complex conjugated numbers, that results in appearing of
characteristic time oscillations of $\langle q(t)\rangle$:

$\langle
q(t)\rangle=a_{1}\mathrm{exp}(s_{1}t)+a_{4}\mathrm{exp}(-\mathrm{|Im}[s_{2}]|t)\mathrm{sin}(\mathrm{Re}[s_{2}]t)+a_{5}\mathrm{exp}(-|\mathrm{Im}[s_{2}]|t)\mathrm{cos}(\mathrm{Re}[s_{2}]t).$
(53)

Since the random force $\xi(t)$ is Gaussian (but non–Markovian because of Eq.
(42)), the two–dimensional process $\\{q(t),v(t)\\}$ (47)–(48) is also
Gaussian and is defined by the probability distribution function risk89 ,

$\mathcal{W}(q\equiv
q-q_{B},q_{0};v,v_{0};t)=\frac{1}{2\pi\sigma_{q}(t)\sigma_{v}(t)\sqrt{1-r_{qv}(t)}}$
(54)
$\times\mathrm{exp}\left(-\frac{1}{2(1-r_{qv}^{2}(t))}\left\\{\frac{[q-\langle
q(t)\rangle]^{2}}{\sigma_{q}^{2}(t)}+\frac{[v-\langle
v(t)\rangle]^{2}}{\sigma_{v}^{2}(t)}-\frac{2r_{qv}(t)[q-\langle
q(t)\rangle][v-\langle
v(t)\rangle]}{\sigma_{q}(t)\sigma_{v}(t)}\right\\}\right),$ (55)

where

$\sigma_{q}(t)=\sqrt{\langle q^{2}(t)\rangle-\langle
q(t)\rangle^{2}},~{}~{}~{}\sigma_{v}(t)=\sqrt{\langle v^{2}(t)\rangle-\langle
v(t)\rangle^{2}},~{}~{}~{}r_{qv}(t)=\langle q(t)v(t)\rangle.$ (56)

All the time–dependent functions in (55) can be expressed in terms of the
functions $\mathcal{A}(t)$ and $\mathcal{B}(t)$ (49) through the relations
(47)–(49) and (42). Knowing the probability distribution function
$\mathcal{W}(q,q_{0};v,v_{0};t)$, one can obtain the corresponding
Fokker–Planck equation. We omit several intermediate steps in deriving of the
Fokker–Planck equation (for details, see e.g. Ref. adel76 ) and give the final
result

$\displaystyle\left[\frac{\partial}{\partial t}+v\frac{\partial}{\partial
q}+\tilde{\omega}_{B}^{2}(t)q\frac{\partial}{\partial
v}\right]\mathcal{W}(q\equiv
q-q_{B},q_{0};v,v_{0};t)=\frac{\gamma(t)}{B}\frac{\partial}{\partial
v}[v\mathcal{W}]$
$\displaystyle+\frac{T\gamma(t)}{B^{2}}\frac{\partial^{2}\mathcal{W}}{\partial
v^{2}}+\frac{T}{B\omega_{B}^{2}}[\tilde{\omega}_{B}^{2}(t)-\omega_{B}^{2}]\frac{\partial^{2}\mathcal{W}}{\partial
v\partial q}.$ (57)

Here, the time–dependent friction coefficient $\gamma(t)$ and the renormalized
frequency parameter $\tilde{\omega}_{B}(t)$ of the parabolic potential barrier
are given by

$\gamma(t)=B\frac{\ddot{\mathcal{A}}(t)\mathcal{B}(t)-\mathcal{A}(t)\ddot{\mathcal{B}}(t)}{\mathcal{A}(t)\dot{\mathcal{B}}(t)-\dot{\mathcal{A}}(t)\mathcal{B}(t)},~{}~{}~{}~{}~{}\tilde{\omega}_{B}^{2}(t)=\frac{\dot{\mathcal{A}}(t)\ddot{\mathcal{B}}(t)-\ddot{\mathcal{A}}(t)\dot{\mathcal{B}}(t)}{\mathcal{A}(t)\dot{\mathcal{B}}(t)-\dot{\mathcal{A}}(t)\mathcal{B}(t)}.$
(58)

The non–Markovian character of the system shows up in the time–dependence of
the friction coefficient $\gamma$ and frequency parameter
$\tilde{\omega}_{B}^{2}$, and in the presence of a cross–term
$\sim\partial^{2}\mathcal{W}/\partial v\partial q$. The Markovian limit of the
Fokker–Planck equation (57) is reached at $\omega_{B}\tau\rightarrow 0$, when
the retarded force in Eq. (46) simply turns to an ordinary friction force
(44). In general case, of course, the retarded force contains both the
time–dependent friction and conservative contributions,

$-\mathcal{K}_{0}\int_{0}^{t}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)\dot{q}(t^{\prime})\
dt^{\prime}=-\gamma(t)\dot{q}(t)+B(\tilde{\omega}_{B}^{2}(t)-\omega_{B}^{2})(q(t)-q_{0}).$
(59)

We are looking for the stationary solution to Eq. (57) in the form

$\mathcal{W}_{\mathrm{stat}}(q;v)=\mathrm{const}\cdot\mathfrak{F}(q,v)\cdot\mathrm{exp}\left(-\frac{Bv^{2}/2}{T}-\frac{[E_{\mathrm{pot,B}}-B\tilde{\omega}_{B,\mathrm{sat}}^{2}q^{2}/2]}{T(1+\epsilon)}\right),$
(60)

where
$\epsilon=[\tilde{\omega}_{B,\mathrm{sat}}^{2}-\omega_{B}^{2}]/\omega_{B}^{2}$,
Substituting the solution (60) into Eq. (57), we obtain an equation for the
function $\mathfrak{F}(q,v)$

$(1+\epsilon)v\frac{\partial\mathfrak{F}}{\partial
q}+\left(\frac{1}{1+\epsilon}\tilde{\omega}_{B,\mathrm{sat}}^{2}q+\frac{\gamma_{0}v}{B}\right)\frac{\partial\mathfrak{F}}{\partial
v}=\frac{T\gamma_{0}}{B^{2}}\frac{\partial^{2}\mathfrak{F}}{\partial
v^{2}}+\frac{T\epsilon}{B}\frac{\partial^{2}\mathfrak{F}}{\partial v\partial
q}.$ (61)

In Eqs. (60) and (61), $\gamma_{0}$ and $\tilde{\omega}_{B,\mathrm{sat}}^{2}$
are long time values of the corresponding quantities (58) taken at
$\omega_{B}t\gg 1$. We found that

$\gamma_{0}=-2B(s_{1}+\mathrm{min}(s_{2},s_{3})),~{}~{}~{}~{}~{}\tilde{\omega}_{B,\mathrm{sat}}^{2}=-(s_{1}-\mathrm{min}(s_{2},s_{3})),~{}~{}~{}~{}~{}~{}~{}\tau<\tau_{\mathrm{thresh}}.$
(62)

To be precise, at $\tau>\tau_{\mathrm{thresh}}$ the quantities $\gamma_{0}$
and $\tilde{\omega}_{B,\mathrm{sat}}^{2}$ do not exist as far as the friction
coefficient $\gamma(t)$ and frequency parameter $\tilde{\omega}_{B}^{2}(t)$
are strongly oscillating functions of time, taking positive as well as
negative values. The oscillations of $\gamma(t)$ and
$\tilde{\omega}_{B}^{2}(t)$ are caused by the structure of functions
$\mathcal{A}(t)$ and $~{}\mathcal{B}(t)$, see Eqs. (58) and (49). Formally, we
will use the quantities $\gamma_{0}$ and $\tilde{\omega}_{B,\mathrm{sat}}^{2}$
even in the case of $\tau>\tau_{\mathrm{thresh}}$ in view of the fact that one
actually needs not the values $\gamma_{0}$ and
$\tilde{\omega}_{B,\mathrm{sat}}^{2}$ itself but some combination of them,
that can be properly defined at the long time $\omega_{B}t\gg 1$ limit.

Looking now for the solution of Eq. (61) in the form

$\mathfrak{F}\equiv\mathfrak{F}(v-aq)\equiv\mathfrak{F}(\zeta)$ (63)

and repeating all derivation’s steps of the Kramers’ theory, we get

$\mathfrak{F}(\zeta)=\mathfrak{F}_{0}\int_{-\infty}^{\zeta}\mathrm{exp}\left[-\frac{\\{(1+\epsilon)a-\gamma_{0}/B\\}}{[2T/B]\\{\gamma_{0}/B-\epsilon
a\\}}\zeta^{\prime 2}\right]d\zeta^{\prime},$ (64)

with

$\mathfrak{F}_{0}=\sqrt{\frac{\\{(1+\epsilon)a-\gamma_{0}/B\\}}{[2\pi
T/B]\\{\gamma_{0}/B-\epsilon a\\}}}$ (65)

and

$a=\frac{1}{1+\epsilon}\left[\sqrt{\frac{\gamma_{0}^{2}}{4B^{2}}+\tilde{\omega}_{B,\mathrm{sat}}^{2}}+\frac{\gamma_{0}}{2B}\right].$
(66)

Therefore, the escape rate $R_{0}$ over parabolic barrier is found as

$R_{0}=\frac{\omega_{A}}{2\pi\omega_{B}}\left[\sqrt{\frac{\gamma_{0}^{2}}{4B^{2}}+\tilde{\omega}_{B,\mathrm{sat}}^{2}}-\frac{\gamma_{0}}{2B}\right]e^{-E_{\mathrm{pot,B}}/T}=\frac{\omega_{A}}{2\pi\omega_{B}}s_{1}e^{-E_{\mathrm{pot,B}}/T}$
(67)

that differs from the standard Kramers’ result,

$R_{\mathrm{Kr}}=\frac{\omega_{A}}{2\pi\omega_{B}}\left[\sqrt{\frac{\gamma_{0}^{2}}{4B^{2}}+\omega_{B}^{2}}-\frac{\gamma_{0}}{2B}\right]e^{-E_{\mathrm{pot,B}}/T},$
(68)

by memory effects’ modification $\tilde{\omega}_{B,\mathrm{sat}}$ of the
frequency parameter $\omega_{B}$ at the top of barrier. We would like to
stress that the transition in Eq. (67) from a nonlinear combination of the
friction coefficient $\gamma_{0}$ and frequency parameter
$\tilde{\omega}_{B,\mathrm{sat}}$ to the largest positive root $s_{1}$ of the
secular equation (50) is mathematically precise at any size $\tau$ of the
memory effects, see discussion after Eq. (62).

In figure 4, we show by solid spline–fitting curve the escape rate $R_{0}$
derived numerically from the Langevin collective dynamics (41), (42).
Analytical estimate of $R_{0}$, given by the non–Markovian extension of the
Kramers’ rate formula (67) is shown in figure 8 by dashed line. For
comparison, we also plotted in figure by dotted line the standard Kramers’
expression for the escape rate (68) with the $\tau$–dependent friction
coefficient

$\gamma_{0}(\tau)=\frac{\mathcal{K}_{0}\tau}{1+[\mathcal{K}_{0}/B]\tau^{2}},$
(69)

that correctly covers two limiting situations (44) and (43) of the retarded
friction force in Eq. (41).

Figure 4: Escape rate $R_{0}$ vs memory time $\tau$. Solid spline–fitting
curve represents direct numerical determination of $R_{0}$ (45) from the
Langevin dynamics Eqs. (41) and (42). Dashed line shows the result of the
non–Markovian extension of the Kramers’ formula (67). The improved formulation
(68)– (69) of the standard Kramers’ formula is given by dotted line.

One can see that the result (67) correctly reproduces the decrease of the rate
$R$ of non–Markovian diffusion over barrier with the memory time $\tau$
(dashed line). On the other hand, the improved standard Kramers’ formula,
given by Eqs. (68) and (69), overestimates the value of $R_{0}$ at fairly
large memory times $\tau$, see dotted line. From that, one can conclude an
importance of the taking into account the memory–renormalization of the
stiffness of the system near the top of a potential barrier.

In the discussion above, we did not consider for a quantum tunneling mechanism
of the fission barrier penetration. This is justified by the fact that the
considered nucleus temperature of $2~{}{\rm MeV}$ is above the characteristic
crossover temperature $T_{0}$ grab84 ; hanggi85 ,

$T_{0}=\frac{\hbar s_{1}}{2\pi},$ (70)

which defines region where tunneling transitions dominate over thermally
activated transitions across the barrier. We found that $T_{0}\leq 0.13~{}{\rm
MeV}$ at any value of the memory time $\tau$.

###  III.4 Stochastic penetration over oscillating barrier

One of the applications of the stochastic approach to the large amplitude
motion is the study of the features of the response of complex nonlinear
systems on periodic external field. The very famous example of such features
is the stochastic resonance phenomenon bsv81 ; mwr88 , when the response of
the nonlinear system on the harmonic perturbation is resonantly activated
under some optimal level of a noise. The resonant activation of the system
occurs when the frequency of the modulation is near the Kramers’ escape rate
of the transitions from one potential well to another. The prototype of the
stochastic resonance studies is a model of overdamped motion between potential
wells of the bistable system. The frequency of the transitions between wells
is given by the Kramers’ rate and the stochastic resonance is achieved when a
frequency of an external periodic modulation is of the order of the Kramers’
rate.

We start from the general Langevin formulation (41), (42) of the problem of
diffusive overcoming of the potential barrier (40) in the presence of a
harmonic time perturbation:

$B\ddot{q}(t)=-\frac{\partial E_{\mathrm{pot}}}{\partial
q}-\mathcal{K}_{0}\int_{0}^{t}\mathrm{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)\dot{q}(t^{\prime})dt^{\prime}-\xi(t)+\alpha\cdot\mathrm{sin}(\omega
t),$ (71)

where $\alpha$ and $\omega$ are an amplitude and frequency of the perturbation
and the correlation function of the random force $\xi(t)$ is given by Eq.
(42). Considering the diffusion over the barrier (71),(42) in the presence of
the external harmonic force, we assume that the amplitude $\alpha$ of the
periodic in time force $\alpha\cdot\mathrm{sin}(\omega t)$ is so small that
still the reaching the top of the barrier is caused exclusively by diffusive
nature of the process. In figure 5, we show the typical dependencies of the
mean pre–saddle time $t_{\mathrm{pre-sad}}$ on the frequency $\omega$ of the
external harmonic force. The calculations were performed for the weak,
$\tau=2\times 10^{-23}~{}\mathrm{s}$, (lower curve in figure 5) and strong,
$\tau=10\times 10^{-23}~{}\mathrm{s}$, (upper curve in figure 5) memory
effects in the non–Markovian diffusive motion over barrier.

Figure 5: The mean pre–saddle time $\langle t_{\mathrm{pre-sad}}\rangle$ of
the non–Markovian diffusion process (71), (42) is given as a function of the
frequency $\omega$ of the harmonic time perturbation at two values of the
correlation time $\tau=2\times 10^{-23}~{}\mathrm{s}$ (lower curve) and
$\tau=10\times 10^{-23}~{}\mathrm{s}$ (upper curve).

In both cases the mean pre–saddle time $\langle t_{\mathrm{pre-sad}}\rangle$
non–monotonically depends on the frequency of the perturbation that is
character for the stochastic resonance phenomenon observed in a number of
different physical systems. From figure 5 one can conclude that the diffusion
over potential barrier in the presence of a harmonic time perturbation is
maximally accelerated at some definite so to say resonant frequency
$\omega_{\mathrm{res}}$ of the perturbation,

$\omega_{\mathrm{res}}\approx\frac{1.5}{\langle t_{\mathrm{pre-
sad}}\rangle(\omega=0)}$ (72)

see also figure 2. In fact, the quantity $\langle t_{\mathrm{pre-
sad}}\rangle(\omega=0)$ presents the characteristic time scale for the
diffusion dynamics (71). In the case of adiabatically slow time variations of
the harmonic force, $\omega\langle t_{\mathrm{pre-sad}}\rangle(\omega=0)\ll 1$
and $t<\langle t_{\mathrm{pre-sad}}\rangle$, one can approximately use
$\alpha\cdot\mathrm{sin}(\omega t)\approx\alpha\cdot\omega t$ and the
diffusion over the barrier is slightly accelerated. As a result of that, the
mean pre–saddle time $\langle t_{\mathrm{pre-sad}}\rangle(\omega)$ is smaller
than the corresponding unperturbed value $\langle t_{\mathrm{pre-
sad}}\rangle(\omega=0)$. The same feature is also observed at the fairly large
modulation’s frequencies. Thus, in the case of $\omega\langle t_{\mathrm{pre-
sad}}\rangle(\omega=0)\gg 1$, the harmonic perturbation
$\alpha\cdot\mathrm{sin}(\omega t)$ may be treated as a random noise term with
the zero mean value and variance $\alpha^{2}$. Such a new stochastic term will
lead to additional acceleration of the diffusion over the barrier.

The existence of the resonant regime (72) in the periodically modulated
diffusion process (71) is even more clear visible through the time evolution
of the survival probability $\mathrm{Prob}(q[t]<q_{B})$. For convience, in
figure 6 we plotted $\mathrm{lnProb}(q[t]<q_{B})$ for the resonant frequency
(72) $\omega=\omega_{\mathrm{res}}$ (curve 1), for fairly large frequency
$\omega=10\omega_{\mathrm{res}}$ (curve 2) and for quite small frequency
$\omega=\omega_{\mathrm{res}}/10$ (curve 3) of the periodic time modulation.

Figure 6: Natural logarithm of the survival probability
$\mathrm{lnProb}(q[t]<q_{B})$ is given as a function of time $t$ for the
periodically modulated escape dynamics (71), (42). Curve 1 corresponds to the
modulation on the resonance frequency (72) $\omega=\omega_{\mathrm{res}}$,
curve 2 – on the frequency $\omega=10\omega_{\mathrm{res}}$ and curve 3 – on
the frequency $\omega=\omega_{\mathrm{res}}/10$. The time dependencies are
calculated for the memory time $\tau=2\times 10^{-23}~{}\mathrm{s}$.

At very rare ($\omega=\omega_{\mathrm{res}}/10$) or frequent
($\omega=10\omega_{\mathrm{res}}$) periodic perturbations the probability
$\mathrm{Prob}(q[t]<q_{B})$ of finding the system $q(t)$ (71) on the left from
the saddle point $q_{B}$ exponentially decay with time and characterized by
almost constant escape (decay) rate $R_{0}$, see Eq. (45). On the other hand,
periodic modulation of the non–Markovian diffusive dynamics (71), (42) at the
resonance frequency $\omega=\omega_{\mathrm{res}}$ provides much stronger
decay with time for the survival probability,
$\mathrm{Prob}(q[t]<q_{B})\sim{\rm exp}(-at^{2})$. In that case, the
probability flow over the saddle point is resonantly amplified.

###  III.5 Diffusion of occupation probabilities within Landau-Zener approach

We restrict our analysis of the Landau-Zener approach to the one-dimensional
case and reduce the basic master equation (6). To eliminate the strong time
oscillations of the observed quantitates $\rho_{nn}(t)$ etc. we introduce the
time averaging as

$\bar{A}(t)={\frac{1}{\pi}}\int\limits_{-\infty}^{\infty}dt^{\prime}\,{\frac{\Delta}{{(t-t^{\prime
2}+\Delta^{2}}}}\,A(t^{\prime}).$ (73)

The master equation (6) can be then presented in the following form kolo95

$\frac{\partial\overline{\rho}_{nn}(t)}{dt}=\sum_{m}\overline{P}_{nm}(t)\left[\overline{\rho}_{mm}(t)-\overline{\rho}_{nn}(t)\right],$
(74)

where

$\overline{P}_{nm}(t)=2\,\mathrm{Re}\int\limits_{o}^{\infty}dt^{\prime}\,\overline{D_{nm}(t)D_{mn}(t-t^{\prime})}\exp[i(\omega_{nm}+i\Gamma_{nm})t^{\prime}]\quad$
(75)

and

$D_{nm}(t)=\left\langle\Psi_{n}\right|i\hbar{\frac{\partial}{{\partial
t}}}\left|\Psi_{m}\right\rangle,\quad\Gamma_{nm}=\Delta\,\dot{\omega}_{nm}.$
(76)

The transition rate (75) is a result of second order perturbation theory in
$\dot{q}$. However, the dimensionless cranking model parameter
$\dot{q}|\left\langle\Psi_{n}\right|\partial\hat{H}/\partial
q\left|\Psi_{m}\right\rangle|/\hbar\omega_{mn}^{2}(q)$ is increasing for two
neighboring terms and therefore, the perturbation theory criterion can not be
fulfilled. We may improve the result (75) by considering the Landau-Zener
model zene32  of two crossing levels. In the case of the Landau-Zener
transitions the probability of a non-adiabatic transition is

$R^{\mathrm{LZ}}(e,\eta).=\exp\left(-{\frac{{\pi
e^{2}}}{{2\eta\mid\dot{q}\mid}}}\right),$ (77)

where $e$ is the gap size and $\eta$ is the slope of the avoided crossings of
two nearest levels. The energy difference $\Delta E$ of these levels as a
function of $q$ can be represented as

$\Delta E=(e^{2}+\eta^{2}q^{2})^{1/2}.$ (78)

The highly excited energy levels $E_{n}$ in Eqs. (74) and (75) exhibit many
avoided crossings when the parameter $q$ is varied. We introduce the average
transition rate $\left\langle P^{\mathrm{LZ}}\right\rangle$ as the probability
of transitions per unit time along of the collective path $q(t)$ where the gap
sizes $e$ and slopes $\eta$ are randomly distributed

$\left\langle
P^{\mathrm{LZ}}\right\rangle=\dot{q}\int\limits_{0}^{\infty}d\eta\int\limits_{0}^{\infty}de\
\>N(\eta,e)\,R^{\mathrm{LZ}}(e,\eta).$ (79)

Here $N(\eta,e)\,d\eta\,de$ is the number of avoided crossings encountered per
unit length and slopes in the interval $[\eta,\eta+d\eta]$ and gap sizes in
the interval $[e,e+de]$. The distribution $N(\eta,e)$ has been calculated
wilk89  in the limit where the gap $e$ is small (the asymptotic slope $\eta$
is only well defined in this limit). Assuming that the matrix elements
$\left\langle\Psi_{n}\right|\partial\hat{H}/\partial
q\left|\Psi_{m}\right\rangle$ are independently Gaussian distributed, it was
shown that

$N(\eta,e)=\Omega(E)\,\eta\,P(e)\,P^{\prime}(\eta).$ (80)

The functions $P(e)$ and $P^{\prime}(\eta)$ are distribution functions of gap
sizes and slopes, respectively. In the case of chaotic systems the
distributions $P(e)$ and $P^{\prime}(\eta)$ depend on the spectral statistics
wilk89 .

Using the transition rate $\left\langle P^{\mathrm{LZ}}\right\rangle$ of (79)
we can reduce the master equation (74) into a diffusion equation. Summation of
both sides of Eq. (74) over the states $n$ with $E_{n}<E$ gives,

$\sum_{n\ (E_{n}<E)}{\frac{\partial f_{n}}{{\partial
t}}}=-\sum_{n(E_{n}<E)}\left\langle
P_{nm}^{\mathrm{LZ}}\right\rangle(f_{n}-f_{m}),$ (81)

where $f_{n}=\bar{\rho}_{nn}$ is the occupation probability. Let us introduce
$l(E)$ as the flux of probability from a level with energy less than $E$ to
levels with energy greater than $E$,

$l(E)=\sum_{n(E_{n}<E),\ m(E_{m}>E)}\left\langle
P_{nm}^{\mathrm{LZ}}\right\rangle(f_{n}-f_{m})$ (82)
$=\int\limits_{0}^{E}de_{1}\,\Omega(e_{1})\int\limits_{E}^{\infty}de_{2}\,\Omega(e_{2})\left\langle
P^{\mathrm{L.Z.}}(e_{1},e_{2})\right\rangle\ [f(e_{1})-f(e_{2})].$ (83)

Using the cut-off properties of the transition rate (77) we transform the flux
$l(E)$ as

$l(E)=-\Omega(E)\,{\frac{\partial f}{{\partial E}}}\,D(E)$ (84)

where $D(E)$ is the diffusion coefficient

$D(E)=\dot{q}\,\Omega(E)\int\limits_{0}^{\infty}de\
\>e^{2}\,P(e)\int\limits_{0}^{\infty}d\eta\>\
\eta\,P^{\prime}(\eta)\,\exp\left(-{\frac{\pi
e^{2}}{{2\eta\mid\dot{q}\mid}}}\right).$ (85)

Finally, we obtain the basic diffusion equation

$\Omega(E)\,{\frac{\partial f}{{\partial t}}}={\frac{\partial}{{\partial
E}}}\,\Omega(E)\,D(E)\,{\frac{\partial f}{{\partial E}}}.$ (86)

This equation gives the Landau-Zener’s evolution of the occupation
probabilities which is caused by slowly varying set of macroscopic variables
$q(t)$.

The distribution functions $P(e)$ and $P^{\prime}(\eta)$ can be evaluated
dependently on the spectral statistics:

(i) For systems with time-reversal invariance (Gaussian orthogonal ensemble)

$P(e)\approx{\frac{\pi^{2}}{{6}}}\Omega(E)\qquad(\mathrm{for}\>\
\mathrm{small}\text{ \ }\>e),$ (87)
$P^{\prime}(\eta)={\frac{\eta}{{4\sigma^{2}}}}\exp\\{-\eta^{2}/8\sigma^{2}\\}.$
(88)

(ii) For systems without time-reversal invariance (Gaussian unitary ensemble)

$P(e)\approx{\frac{\pi^{2}}{6}}\,e\,\Omega^{2}(E)\qquad(\mathrm{for}\>\
\mathrm{small}\text{ \ }\>e),$ (89)
$P^{\prime}(\eta)={\frac{\eta}{{2\sqrt{\pi}}\sigma^{3}}}\,\exp\\{-\eta^{2}/4\sigma^{2}\\}.$
(90)

We will calculate the rate of dissipation of energy due to Landau-Zener
transitions. Let us introduce the jump probability per unit time into an
energy interval $[e,e+de]$ as

$\mathcal{P}(e)\,de=\dot{q}\int\limits_{0}^{\infty}d\eta\,N(\eta,e;q)\,\exp\left[-{\frac{{\pi
e^{2}}}{{2\mid\dot{q}\mid\eta}}}\right]\,de,$ (91)

where $N(\eta,e;q)$ is defined in Eq. (80). The dissipation energy,
$E_{\mathrm{diss}}$, is connected with the internal transitions which
accompany the macroscopic collective motion. The definition of
$E_{\mathrm{diss}}$ in terms of the jump probability $P(e)$ depends on the
total energy $E$ of the nucleus. We will consider two limiting cases.

(i) $E\approx E_{\mathrm{eq}}$ (motion close to the ground state)

The dissipation rate averaged over the spectral statistics is defined in this
case as

$\left\langle\dot{E}_{\mathrm{diss}}\right\rangle_{E}\,=\int\limits_{0}^{\infty}de\>\
e\>\mathcal{P}(e)$ (92)

Using the expressions (91), (80) and (87)-(90) we obtain

$\mathrm{GOE}:\,\,\,\,\left\langle\dot{E}_{\mathrm{diss}}\right\rangle_{E}\,\,=\mathrm{const}\cdot\Omega^{2}(E)\,\sigma^{2}(E)\,\dot{q}^{2},$
(93)
$\mathrm{GUE}:\,\,\,\,\left\langle\dot{E}_{\mathrm{diss}}\right\rangle_{E}\,\,=\mathrm{const}\cdot\Omega^{3}(E)\,\sigma^{5/2}(E)\,\mid\dot{q}\mid^{5/2}.$
(94)

Thus, the friction force is proportional to $\dot{q}$ for the case of
$\mathrm{GOE}$ and is proportional to $\mid\dot{q}\mid^{3/2}$ for
$\mathrm{GUE}$ statistics. This result is greatly reduced in the case of high
excitation energy.

(ii) $E\gg E_{\mathrm{eq}}$ (high excitation energy regime)

The definition of $\left\langle E_{\mathrm{diss}}\right\rangle_{E}$ is
different in this case from that of Eq. (92) since transitions with $e<0$
appear. We have,

$\left\langle\dot{E}_{\mathrm{diss}}\right\rangle_{E}\,=\int\limits_{-\infty}^{\infty}de\
e\>\mathcal{P}(e)$ (95)

This expression and (91), (80) and (87)-(90) lead to

$\mathrm{GOE}:\,\,\,\,\left\langle\dot{E}_{\mathrm{diss}}\right\rangle_{E}\,=\mathrm{const}\cdot\Omega(E)\,\frac{d\Omega(E)}{dE}\,\sigma^{5/2}(E)\,\mid\dot{q}\mid^{5/2},$
(96)
$\mathrm{GUE}:\,\,\,\,\left\langle\dot{E}_{\mathrm{diss}}\right\rangle_{E}\,=\mathrm{const}\cdot\Omega^{3}(E)\,\frac{d\Omega(E)}{dE}\,\sigma^{3}(E)\,\mid\dot{q}\mid^{3}.$
(97)

The main feature of this result is the dependence of the dissipation energy on
the derivative $d\Omega(E)/dE$ of the level density. This means that time
irreversible exchange between the collective and internal degrees of freedom
is possible if the phase space volume is increasing with the excitation energy
of the system.

##  IV Non-Markovian Langevin–like dynamics of Fermi–liquid drop

The large amplitude motion of a nucleus can be studied in terms of fluid
dynamic approaches which allow us to reduce the problems of the avoided
crossings of adiabatic energies $E_{n}(q)$, see e.g. Eq. (25) for the inertia
tensor $B(q)$. In general, the nuclear fluid dynamics is influenced strongly
by the Fermi motion of nucleons and is accompanied by the dynamic distortion
of the Fermi-surface in momentum space kosh04 ; kolo83 . The presence of the
dynamic Fermi-surface distortion gives rise to some important consequences in
the nuclear dynamics which are absent in classical liquids. The dynamics of a
nuclear Fermi liquid is determined by the pressure tensor instead of the
scalar pressure as in a classical liquid. This fact changes the conditions for
the propagation of the isoscalar and isovector sound excitations and creates a
strong transverse component in the velocity field of the particle flow.
Furthermore, because of the Fermi-surface distortion, the scattering of
particles on the distorted Fermi-surface becomes possible and the relaxation
of collective motion occurs abkh59 . The equations of motion of nuclear Fermi-
liquid take then a non-Markovian form. The memory effects depend here on the
relaxation time and provide a connection between both limiting cases of the
classical liquid (short relaxation time limit) and the quantum Fermi-liquid
(long relaxation time limit). The Markovian dynamics only exist in these two
limiting cases.

Under the description of the properties of a drop of quantum Fermi liquid, one
can start from the collisional kinetic equation for the distribution function
in phase space $f(\mathbf{r},\mathbf{p};t)$. Using the standard
$\mathbf{p}$-moment procedure, one can derive the continuity equation for the
particle density $\rho$ and Euler–like equation for the velocity field
$u_{\nu}$(for details, see Refs. kora01 ; kosh04 ; kota81 ),

${\frac{\partial}{\partial t}\rho}=-{\nabla}_{\nu}(\rho u_{\nu}),$ (98)
$m\rho{\frac{\partial}{\partial t}}u_{\nu}+m\rho\ u_{\mu}\nabla_{\mu}\
u_{\nu}+\nabla_{\nu}\mathcal{P}+\rho\nabla_{\nu}{\frac{\delta\epsilon_{\mathrm{pot}}}{\delta\rho}}=-\nabla_{\mu}P_{\nu\mu}^{\prime},$
(99)

where $\mathcal{P}$ is the isotropic part of the pressure,
$P_{\nu\mu}^{\prime}$ is its anisotropic part caused by the dynamic distortion
of Fermi surface and $\epsilon_{\mathrm{pot}}$ is the potential energy density
of inter-particle interaction. The pressure tensor $P_{\nu\mu}^{\prime}$ is
derived by the distribution function in phase space
$f(\mathbf{r},\mathbf{p};t)$. Considering the dynamic Fermi–surface
distortions up to multipolarity $l=2$ one can establish kosh04 ; kolo83  the
following equation the pressure tensor $P_{\nu\mu}^{\prime}$

${\frac{\partial}{\partial t}}P_{\nu\mu}^{\prime}+\mathcal{P\
}\left(\nabla_{\nu}u_{\mu}+\nabla_{\mu}u_{\nu}-{\frac{2}{3}}\delta_{\nu\mu}\nabla_{\alpha}u_{\alpha}\right)=I_{\nu\mu}+Y_{\nu\mu},$
(100)

where $I_{\nu\mu}$ is the second moment of the collision integral $I[\delta
f]=-\delta f/\tau$:

$I_{\nu\mu}={\frac{1}{m}}\int{\frac{d\mathbf{p}}{(2\pi\hbar)^{3}}}p_{\nu}p_{\mu}I[\delta
f]$ (101)

and $Y_{\nu\mu}$ gives the contribution from the random force

$Y_{\nu\mu}={\frac{1}{m}}\int{\frac{d\mathbf{p}}{(2\pi\hbar)^{3}}}p_{\nu}p_{\mu}Y.$
(102)

The equation of motion (98)-(100) are closed with respect to the velocity
field $u_{\nu}$.

For an incompressible and irrotational flow of the nuclear fluid with a sharp
surface, the local equation of motion (98)-(100) can be reduced to the
equations for the variables $q_{i}(t),~{}i=\overline{1,N}$ that specify the
shape of the nucleus kora01 . The continuity equation (98) has to be
complemented by the boundary condition on the moving nuclear surface $S$.
Below we will assume that the axially symmetric shape of the nucleus is
defined by rotation of the profile function
$\rho=\mathcal{L}(z,\\{q_{i}(t)\\})$ around the $z$-axis in the cylindrical
coordinates $\rho,z,\varphi$. The velocity of the nuclear surface is then
given by ivko95

$u_{S}=\sum_{i=1}^{N}\bar{u}_{i}\dot{q}_{i},$ (103)

where

$\bar{u}_{i}=({{\partial}\mathcal{L}/{\partial q_{i}}})/\Lambda,\ \ \
\Lambda=\sqrt{1+({{\partial}\mathcal{L}/{\partial z}})^{2}}.$ (104)

The potential of the velocity field takes the form

$\phi=\sum_{i=1}^{N}{}_{i}\overline{\phi}\ \dot{q}_{i},$ (105)

where the potential field
$\overline{\phi}_{i}\equiv\overline{\phi}_{i}(\mathbf{r,}q)$ is determined by
the equations of the following Neumann problem

$\nabla^{2}\overset{\\_}{\phi}_{i}=0\,,\,\,\ \ \
(\mathbf{n\cdot}\nabla\overset{\\_}{\phi}_{i})_{S}={\frac{1}{\Lambda}}{\frac{{\partial}\mathcal{L}}{{\partial
q_{i}}}},$ (106)

where $\mathbf{n}$ is the unit vector which is normal to the nuclear surface.
Using Eqs. (98) and (99) with multiplying Eq. (99) by
$\nabla_{\mu}\overline{\phi}_{i}$ and integrating over $\mathbf{r}$, one
obtains

$\sum_{j=1}^{N}[B_{ij}(q)\ddot{q}_{j}+\sum_{k=1}^{N}\frac{\partial
B_{ij}}{\partial q_{k}}\
\overset{\cdot}{q_{j}}\overset{\cdot}{q_{k}}+\int_{t_{0}}^{t}dt^{\prime}\exp(\frac{t^{\prime}-t}{\tau})\kappa_{ij}(t,t^{\prime})\
\overset{\cdot}{q_{j}}(t^{\prime})]=-\frac{\partial
E_{\mathrm{pot}}(q)}{\partial q_{i}}.$ (107)

Here $B_{ij}(q)$ is the inertia tensor

$B_{ij}(q)=m\rho_{0}\oint ds\bar{u}_{i}\ \overline{\phi}_{j}$ (108)

and we ignore the random force. The memory kernel $\kappa_{i,j}(t,t^{\prime})$
in Eq. (107) is given by

$\kappa_{ij}(t,t^{\prime})=2\ \int d\mathbf{r\
}\mathcal{P}(\mathbf{r,}q(t^{\prime}))\
\left[\nabla_{\nu}\nabla_{\mu}\overset{\\_}{\phi}_{i}(\mathbf{r,}q(t))\right]\left[\nabla_{\nu}\nabla_{\mu}\overset{\\_}{\phi}_{j}(\mathbf{r,}q(t^{\prime}))\right]\
.$ (109)

The potential field
$\overline{\phi}_{i}\equiv\overline{\phi}_{i}(\mathbf{r,}q)$ are determined by
a solution to the Neumann problem (106).

To solve Eq. (107) we will rewrite it as a set of two equations. Namely,

$\sum_{j=1}^{2}[B_{ij}(q)\ddot{q}_{j}+\sum_{k=1}^{2}\frac{\partial
B_{ij}}{\partial q_{k}}\
\overset{\cdot}{q_{j}}\overset{\cdot}{q_{k}}]=-\frac{\partial
E_{\mathrm{pot}}(q)}{\partial q_{i}}+R_{i}(t,q)$ (110)

and

$\frac{\partial R_{i}(t,q)}{\partial
t}=-\frac{R_{i}(t,q)}{\tau}+\sum_{j=1}^{2}\kappa_{ij}(q,q)\dot{q}_{j}\ \
\qquad\mathrm{at\qquad}R_{i}(t=0,q)=0,$ (111)

where $q=\\{q_{1},q_{2}\\}=\\{\zeta_{0},\zeta_{2}\\}$ and the terms
$\sim\dot{q}_{i}\dot{q}_{j}$ were omitted in Eq. (111), as the next order
corrections. The memory kernel $\kappa_{ij}(q,q)$ is given by

$\kappa_{ij}(q,q)=\frac{2}{5}m\rho_{0}v_{F}^{2}\ \int d\mathbf{r\
}(\nabla_{\nu}\nabla_{\mu}\overset{\\_}{\phi}_{i}(\mathbf{r,}q))\
(\nabla_{\nu}\nabla_{\mu}\overset{\\_}{\phi}_{j}(\mathbf{r,}q)).$ (112)

We will apply this approach to the case of symmetric nuclear fission
described, assuming the Lorentz parameterization for the profile function
$\mathcal{L}(z)$ in the form of Eq. (35). We have performed numerical
calculation for symmetric fission of the nucleus ${}^{236}\mathrm{U}$. We
solved Eqs. (110) and (111) numerically using the deformation energy
$E_{\mathrm{pot}}(q)$ from Refs. hamy88 . The scission line was derived from
the condition of the instability of the nuclear shape with respect to the
variations of the neck radius, see Eq. (38). The equations of motion (110) and
(111) were solved with the initial conditions corresponding to the saddle
point deformation and the initial kinetic energy
$E_{\mathrm{kin,0}}=1~{}\mathrm{MeV}$ (initial neck velocity
$\dot{\zeta_{2}}=0$).

Figure 7: Trajectories of descent from the saddle point of the nucleus 236U in
the $\zeta_{0},\zeta_{2}$ plane. Dashed line represents the result of the
calculation in presence of the memory effects and dotted line is for the case
of Markovian (no memory) motion with the friction forces. We have used the
relaxation time $\tau=4\times 10^{-22}~{}\mathrm{s}$ and the initial kinetic
energy $E_{\mathrm{kin}}=1\,\mathrm{MeV}$. Dot-dashed line is the scission
line derived from the condition (38).

In Figure 7 we show the dependence of the fission trajectory, i.e., the
dependence of the neck parameter $\zeta_{2}$ on the elongation $\zeta_{0},$
for the fissioning nucleus ${}^{236}\mathrm{U}$ for two different values of
the relaxation time $\tau$: $\tau=4\times 10^{-22}~{}\mathrm{s}$ (dashed line)
and $\tau=0\times 10^{-23}~{}{\rm s}$ (dotted line). The scission line (dot-
dashed line in Figure 7) was obtained as a solution to Eq. (38). We define the
scission point as the intersection point of the fission trajectory with the
scission line. As can be seen from figure 11 the memory effect hinders
slightly the neck formation and leads to a more elongated scission
configuration. To illustrate the memory effect on the observable values we
have evaluated the translation kinetic energy of the fission fragments at
infinity, $E_{\mathrm{kin}}$, and the prescission Coulomb interaction energy,
$E_{\mathrm{Coul}}$. The value of $E_{\mathrm{kin}}$ is the sum of the Coulomb
interaction energy at scission point, $E_{\mathrm{Coul}}$, and the prescission
kinetic energy $E_{\mathrm{kin,ps}}$. Namely,

$E_{\mathrm{kin}}\mathrm{=}E_{\mathrm{Coul}}+E_{\mathrm{kin,ps}}\mathrm{.}$
(113)

Figure 8: Fission-fragment kinetic energy, $E_{\mathrm{kin}}$, (curves 1) and
the Coulomb repulsive energy at the scission point, $E_{\mathrm{Coul}}$,
(curves 2) versus the relaxation time $\tau$ for the nucleus 236U. Solid lines
represent the result of the calculation in presence of the memory effects and
dashed lines are for the case of Markovian (no memory) motion with the
friction forces. The initial kinetic energy is
$E_{\mathrm{kin,0}}=1\,\mathrm{MeV}$.

The influence of the memory effects on the fission-fragment kinetic energy,
$E_{\mathrm{kin}}$, and the prescission Coulomb interaction energy,
$E_{\mathrm{Coul}}$, is shown in figure 8. As seen from figure 8, the memory
effects are neglected at the short relaxation time regime where the memory
integral is transformed into the usual friction force. In the case of the
Markovian motion with friction (dashed line), the yield of the potential
energy, $\Delta E_{\mathrm{pot}}$, at the scission point is transformed into
both the prescission kinetic energy, $E\mathrm{{}_{kin,ps},}$ and the time
irreversible dissipation energy, $E_{\mathrm{dis}}$, providing $\Delta
E_{\mathrm{pot}}=E\mathrm{{}_{kin,ps}}$ $+\ E_{\mathrm{dis}}\mathrm{.}$ In
contrast to this case, the non-Markovian motion with the memory effects (solid
line) produces an additional time reversible prescission energy,
$E_{\mathrm{F,ps}},$ caused by the distortion of the Fermi surface. In this
case, the energy balance reads $\Delta E_{\mathrm{pot}}=E\mathrm{{}_{kin,ps}}$
$+\ E_{\mathrm{dis}}+E_{\mathrm{F,ps}}.$ Note that the used parametrization of
the fissioning nucleus at the scission point, leads to the prescission Coulomb
energy $E_{\mathrm{Coul}}$ which is about 5 $\mathrm{MeV}$ lower (for
${}^{236}\mathrm{U}$) than the Coulomb interaction energy of the scission
point shape dama77 . Taking into account this fact and using the experimental
value of the fission-fragment kinetic energy
$E_{\mathrm{kin}}^{\mathrm{\exp}}=168$ $\mathrm{MeV}$ dama77 , one can see
from figure 12 that the Markovian motion with friction (dashed line) leads to
the overestimate of the fission-fragment kinetic energy $E_{\mathrm{kin}}.$ In
the case of the non-Markovian motion with the memory effects (solid line), a
good agreement with the experimental data is obtained at the relaxation time
of about $\tau=8\times 10^{-23}~{}\mathrm{s}$. A small deviation of the
prescission Coulomb energy $E_{\mathrm{Coul}}$ obtained at the non-Markovian
motion (solid line in figure 12) from the one at the Markovian motion (dashed
line in figure 12) is caused by the corresponding deviation of both fission
trajectories in Figure 8.

In figure 9 we illustrate the memory effect on the saddle-to-scission time
$t_{\mathrm{sc}}.$ In the case of the non-Markovian motion (solid line), the
delay in the descent of the nucleus from the barrier grows with the relaxation
time $\tau$ (at $\tau\geq 4\cdot 10^{-23}~{}\mathrm{s}$). This is mainly due
to the hindering action of the elastic force caused by the memory integral.
The saddle-to-scission time increases by a factor of about 2 due to the memory
effect at the experimental value of the relaxation time $\tau=8\times
10^{-23}~{}\mathrm{s}$ which was derived from the fit of the fission-fragment
kinetic energy $E_{\mathrm{kin}}$ to the experimental value of
$E_{\mathrm{kin}}^{\mathrm{\exp}}$ (see above).

Figure 9:  Dependence upon relaxation time $\tau$ of the saddle-to-fission
time, $t_{\mathrm{sc}}$, for the descent from the barrier in the case of two-
dimension ($\zeta_{0},\zeta_{2}$) parametrization for the nucleus 236U. Solid
line represents the result of the calculation in presence of the memory
effects and dashed line is for the case of Markovian (no memory) motion with
the friction forces. The initial kinetic energy is
$E_{\mathrm{kin}}=1\,\mathrm{MeV}$.

##  V Summary

In attempt to understand microscopically the origin for dissipative and
fluctuating properties of collective modes of motion in nuclear many–body
systems, we have developed an approach, based on the Zwanzig’s projection
technique (5)–(6).

We have averaged the intrinsic nucleonic dynamics over suitably chosen
statistics of the randomly distributed matrix elements (7), measuring the
coupling between the intrinsic nucleonic and macroscopic collective $q(t)$
subsystems, and the energy spacings. In assumption of the weak coupling kora10
, we have derived the diffusion–like equation of motion (10) for the ensemble
averaged occupancies of the adiabatic many–body states, describing a process
of non–Markovian energy diffusion and that occurs due to quantum–mechanical
transitions between the many–body levels. Going beyond the weak–coupling limit
kora10 , we have also studied the energy diffusion process in space of the
occupancies of the adiabatic many–body states for Landau–Zener transitions
between levels (77), where the diffusive properties essentially depend on
quantum statistics of many–body states, see Eqs. (96), (97). It has been
demonstrated that under the growing of the averaged density of levels with
energy $E$, the intrinsic nucleonic subsystem parametrically excites, that
can, in turn, be interpreted as corresponding dissipation of collective energy
due to constancy of the total energy of the nuclear many–body system.

We have shown that memory effects in the diffusion equation (10) is caused by
finite spread $\Gamma$ of the randomly distributed coupling matrix elements
(7) and disappears for quite spread distributions, when $\hbar/\Gamma$ is the
shortest time scale in the system. We have demonstrated that a (memory or
correlation) time $\tau=\hbar/\Gamma$ defines non–Markovian character of
macroscopic collective dynamics (II.1) and also determines statistical
properties of of the random force term, incorporated into equations of motion
for the collective variables (II.1), through the fluctuation–dissipation
theorem (34).

We have further applied our non–Markovian Langevin approach (II.1), (34) to
the description of descent of the nucleus from the fission barrier. We have
showed that the random force accelerate significantly the process of descent
from the barrier for both the Markovian and non–Markovian Langevin dynamics,
see figure 1. This fact may be explained by the correlation properties (34) of
the random force in the non–Markovian Langevin equations.

We have generalized the Kramers’ theory of escape rate over parabolic
potential barrier (of figure 2) to systems with non–Markovian dynamics
(41),(42). The found expression for the escape rate (67) differs from the
standard Kramers result (68) by memory–renormalization of the frequency
parameter $\tilde{\omega}_{B,\mathrm{sat}}$ (58) at the top of barrier. It has
been shown that $\tilde{\omega}_{B,\mathrm{sat}}$ is effectively smaller that
the frequency parameter $\omega_{B}$ of the adiabatic barrier, giving rise to
slowing down of the thermal diffusion over barrier with the increase of the
size $\tau$ of memory effects, see figure 3.

We have also studied the non–Markovian diffusion (71), (42) over parabolic
barrier in the presence of weak periodic time perturbation. The mean
pre–saddle time was calculated as a function of the frequency $\omega$ of the
perturbation (see figure 5) and found that the diffusive motion over the
barrier may be significantly accelerated at some definite (resonance)
frequency $\omega_{\mathrm{res}}$, that is defined by the mean time of motion
in the absence of perturbation.

By use of $p$-moments techniques, we have reduced the collisional kinetic
equation to the equations of motion for the local values of particle density,
velocity field and pressure tensor, see Eqs. (98)–(100). To apply our approach
to the nuclear large amplitude motion, we have assumed that the nuclear liquid
is incompressible and irrotational. We have derived the velocity field
(103)–(106), which depends then on the nuclear shape parameters $q(t)$ due to
the boundary condition on the moving nuclear surface. Finally, we have reduced
the problem to a macroscopic equation of motion for the shape parameters
$q(t)$ (107)–(109). Thus, we consider a change (not necessary small) of the
nuclear shape which is accompanied by a small quadrupole distortion of the
Fermi surface. The obtained equations of motion for the collective variables
$q(t)$ contains the memory integral which is caused by the Fermi-surface
distortion and depends on the relaxation time $\tau$.

We have shown that the development of instability near the fission barrier is
strongly influenced by the memory effects if the relaxation time $\tau$ is
large enough. In this case, a drift of the nucleus from the barrier to the
scission point is accompanied by characteristic shape oscillations which
depend on the relaxation time $\tau$. The shape oscillations appear due to the
elastic force induced by the memory integral. The elastic force acts against
the adiabatic force $-\ \partial E_{\mathrm{pot}}(q)/\partial q$ and hinders
the motion to the scission point. In contrast to the case of the Markovian
motion, the delay in the fission is caused here by the conservative elastic
force and not only by the friction force. Due to this fact, the nucleus loses
a part of the prescission kinetic energy converting it into the potential
energy of the Fermi surface distortion instead of the time-irreversible
heating of the nucleus.

The memory effects lead to the decrease of the fission-fragment kinetic
energy, $E_{\mathrm{kin}},$ with respect to the one obtained from the
Markovian motion with friction, see Figure 7. This is because a significant
part of the potential energy at the scission point is collected as the energy
of the Fermi surface deformation. Note that the decrease of the fission-
fragment kinetic energy due to the memory effects is enhanced in the rare
collision regime (at larger relaxation time) while the effect due to friction
decreases. An additional source for the decrease of the fission-fragment
kinetic energy is caused by the shift of the scission configuration to that
with a larger elongation parameter $\zeta_{0},$ in the case of the non-
Markovian motion. Due to this fact, the repulsive Coulomb energy of the
fission fragments at the scission point in figure 12 decreases with respect to
the case of the Markovian motion.

##  VI Acknowledgments

This study was supported in part by the program Support for the development of
priority areas of scientific research of the National Academy of Sciences of
Ukraine, Grant No. 0120U100434.

##  VII References

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suggestions that improved the content of this paper. We are also grateful to
Professor Jingbo Xia (SUNY-Buffalo) and Professor Kunyu Guo (Fudan University)
for providing useful suggestions. This work was partially supported by NSFC
(grant number: 11701052). The second author was partially supported by the
Fundamental Research Funds for the Central Universities (grant numbers:
2020CDJQY-A039, 2020CDJ-LHSS-003).

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