Using a recently developed method for the inclusion of fluctuation in the BUU
dynamics, we study the self-consistent propagation of inherent thermal noise of
unstable nuclear matter. The large time behaviour of the evolving system
exhibits synergism between fluctuation and non-linearities in a universal
manner which manifest in the appearance of macroscopic structure in the average
description.Comment: 12 pages Revtex. Two figures, uuencoded, are enclosed in a separate
  fil

C.J. Pethick

D. Kiderlen

E. Suraud

G. F. Burgio

G.F. Burgio

H. Risken

J. Randrup

L.G. Moretto

M. Baldo

M. Colonna

M. Gong

M. Suzuki

R. Donangelo

S. Ayik

S. Chattopadhyay

W. Bauer

W.U. Schröder

English

arXiv

Crossref

Physical Review C

Spinodal instabilities within the Boltzmann-Langevin approach

Using a recently developed method for the inclusion of fluctuation in the BUU dynamics, we study the self-consistent propagation of inherent thermal noise of unstable nuclear matter. The large time behaviour of the evolving system exhibits synergism between fluctuation and non-linearities in a universal manner which manifest in the appearance of macroscopic structure in the average description

Chattopadhyay, S

CERN Document Server

Spinodal instabilities within BUU approach
S. Chattopadhyay
S. N. Bose Centre for Basic Sciences, DB-17, Bidhannagore, Calcutta-700064, India
Abstract
Using a recently developed method for the inclusion of fluctuation in the
BUU dynamics, we study the self-consistent propagation of inherent thermal
noise of unstable nuclear matter. The large time behaviour of the evolving
system exhibits synergism between fluctuation and non-linearities in a uni-
versal manner which manifest in the appearance of macroscopic structure in
the average description.
PACS NO. 21.65. +f, 24.60.Ky
Typeset using REVTEX
1
In recent years a considerable interest has grown in the understanding of the role of fluc-
tuation in the dynamics of multifragmentation. In the decompression phase of the collision,
the density of the composite nuclear system eventually attains such a low value so that it
crosses the boundary of the so called spinodal zone where a small perturbation can grow
rapidly until the fragments form. The fluctuation certainly plays a signicant role in the
process of fragmentation. To unravel some basic features of such delicate process simple
model study seems encouraging. For this purpose, dierent theoretical approaches based on
Landau [1,2] or Boltzmann-Uhling-Ulhenbeck (BUU) [3] equation have been utilized.
Let us concentrate on the BUU approach to the problem. In this model, the description
of the process is governed by the evolution of reduced one body density distribution. In the
mean eld approximation, the fluctuations arising from the stochastic part of the collision
term are ignored and evolution of a single average eective density is considered only. In this
description the bundle of trajectories are conned in a certain region of the associated phase
space. The presence of collision term in the equation corresponds to the dissipative eect
in the evolution. When the system reaches the point of instability where any fluctuation,
however, small it is, amplies indenitely so that trajectories may bifurcate and drive the
system to some new xed point. In other words, dierent manifestation of the system are
now become accessible dynamically. Therefore, the mean trajectory description has been
lost. To encounter such a situation, the general strategy is to study the evolution of a
collection of initially prepared samples. Due to the very presence of the stochasticity in
the process their fates are dierent. To make this picture consistent, a usual method is to
introduce a fluctuation term in the equation of motion, so that the nature of the solution
become probabilistic. Within the framework of BUU equation this prescription is properly
incorporated and the resultant evolution equation for phase space density is commonly
known as the Boltzmann-Langevin (BL) equation [4].
Guided by the same physical principle, several computational schemes have been pro-
posed to incorporate fluctuation in the BUU equation [5,6]. In this context, we want to
mention a very recent development (see Ref. [7] where a novel method of simulation is pro-
2
posed to incorporate fluctuations in the BUU equation. The occupancy factor n(r;p) at
a particular elementary cell s of size rp=hD (D the dimension of the physical space)
around a point (r;p) in the associated phase space of the one body description of the system
is considered to be discrete in nature so that it takes the value 1 (say success) or 0 (failure)
otherwise. The basic transition probability (say, 12! 1020) within a small interval of time is
given by T (12 ! 1020) = !12!1020ts4n1n2n10n20 where !12!1020s4 is the basic transition
rate. For a particular transition, every sample is tried with a probability of success !ts4.
However due to the presence of four occupancy factors in the above relation a successful
transition occur when an appropriate conguration of particle and hole states appeared in a
sample. By this way enough fluctuation in sample space can be generated and at the same
time, the relevant statistical criteria i.e. 2i = hniihnii can be fullled exactly for every cell
i.
Another essential aspect of the simulation is to generate the solution of the Vlasov part
of the BUU equation in the presence of the mean eld potential. Due to the discrete nature
of the phase space occupancy n(r;p), the usual procedure of matrix method [3,5] cannot
be applied here properly. To consider Vlasov propagation of phase space distribution we
follow the Nordheim’s approach, closely related to the well known test particle method of
solving Vlasov equation. A lled unit phase cell may be considered as a particle. The time
evolution of these particles can be described by the standard leap-frog algorithm in the
following manner,
Xik(t + t) = X
i
k(t) +
1
m
P ikt ; (1)
P ik(t + t) = P
i
k(t) +rkU((r; t))t (2)
where (Xik(t); P
i
k(t)) represent the kth component of position and momentum of the ith
particle at time t. It is important to note that unlike to the test particle method the position
and also momentum of the particle are represented by the grid points here, therefore, its
evolution with time take places in discrete steps. However to provide a reasonably good
energy conserving evolution, we incorporate a modication in the leap-frog routine. The
3
error comes mainly due to the rounding o the quantitiesXik=xd or P
i
k=pd where xd and pd are
the co-ordinate and momentum grid spacing. The accumulated error that arise in momentum
due to the incoming particles at a particular spatial cell (say l) is given as ~plk =
P
i p
i
k. Here
pik is the residual part of P
i
k so that P
i
k = pd  nearest integer of(P
i
k=pd) + ~p
i
k. Therefore,
to reduce this error substantially, one may assume that the momentum of all the particles
in a cell l are modied as pd  nearest integer of(
P i
k
pd
) +
~pl
k
Nl
. Here Nl is the total number of
incoming particles in cell l. To check the reliability of this method we consider the evolution
of the breathing mode of a spherical nucleus in 2-dimension having small skin of about 2
fm. We observe two interesting points: 1) the initial spherical symmetry in momentum
and co-ordinate space are maintained exactly and 2) no two particles appear simultaneously
in the same phase cell throughout the evolution. However, it is to be noted that when
fluctuation in phase cell is taken into account by allowing the the collisions among the
particles the above mentioned features are gradually lost. In this situation, the smearing of
the distribution becomes necessary. This is incorporated by simply moving a particle in the
neighbouring sites in momentum space if it tries to access a cell in the process of evolution
which is already lled. In this simulation, the values of xd and pd are taken as 1 fm. and
4060 MeV/c respectively. The total energy per particle is found to be conserved up to
95% even at time t = 100 fm/c.
Using the above mentioned simulation procedure the present study revisits the problem of
nuclear matter in spinodal zone. This work mainly concerned about the large time behaviour
of the system and the non-linearities that appear in the process of evolution. To make this
study consistent with earlier investigation we adopt the scenario that was studied rigorously
by several authors [3,5] namely, a gas of fermions situated inside a two dimensional torus.
For the eective one body eld at two dimensional grid point (x,y) we employ a simplied
Skyrme interaction U(x; y) = A(x;y)
0
+ B((x;y)
0
)2 with A = −100:3 MeV, B = 48 MeV,
the saturation density 0 = 0:55 fm−2. To evaluate eective potential on the lattice an
averaging over 2-dimensional Gaussian function having width 0.87 is done. The in-medium
cross section is taken as 2.4 fm and Fermi-momentum Pf = 260 MeV/c. For simulation
4
we consider the square lattice of size (2121) in co-ordinate space having width xd = 1
fm., (3131) in momentum space with pd = 40 MeV/c and the number of samples Ns for
parallel runs are taken to be 100. To ensure that our system is initially situated inside the
spinodal zone we prepare the samples having uniform (in space) average density hi (or the
number density hNi)=0:50 (N0) and momentum distribution same as that of a fermi gas
at temperature T = 3 MeV.
At a given temperature T 6= 0 there exists thermal fluctuation in the initial state. To
incorporate this important feature within our simulation procedure we prepare the sam-
ples with the assumption that the distribution of N is given by a Gaussian function with
mean hNi where as the sample average value of occupancy hnii is given by the fermi dis-
tribution of appropriate temperature T . The variance 2N satises the standard relation
2N =
P
N
P
2i =
P
N
P
hniihnii. To maintain spatial uniformity in initial state, the value
of N or the momentum distribution is considered to be the same at every spatial cell of
a particular sample. Inspite of the fact that at T = 0 all collisions are Pauli blocked, the
symmetry of the system remains intact even by BL treatment used here or the same used
in Ref. [3]. It is to be noted that in this case 2N = 0 i.e. all samples are identical. The
presence of the thermal fluctuation in the initial state is in fact, the source of disturbance
which generates irregularities subsequently in co-ordinate space and may amplify due to
the action of the mean eld. Hence, to make the picture consistent, the preparation of the
samples in the aforesaid manner is therefore an essential part of our simulation procedure.
Let us now turn to the investigation of the Boltzmann-Langevin dynamics on lattice. The
normal mode analysis of the evolving density will allow us to study the interplay between
instabilities and fluctuations. Following the Ref. [3] we introduce two point correlation
function to the associated density fluctuation as
k(t) = hjk(t)j
2i =
1
L4
Z L2
dr0
Z L2
dre−k:jr
0−rjh(r0)(r)i (3)
For theoretical analysis of the time evolution of the quantity k the starting point is to
linearise the Vlasov equation and derive the equation for normal modes (for example see Ref.
5
[8]). In the presence of fluctuation in the transport process the evolution of the amplitude of
the Fourier components (or in other words, that of the collective mode ) can be represented
by the following Langevin equation
dA(t)
dt
= i!A(t) + ~B(t) (4)
where A is the two component amplitude matrix (with elements A
+
 and A
−
 ) of the Fourier
components of the density fluctuation f so that f = A+ e
i!t + A− e
−i!t. ~B(t) is the
kicking term of the Langevin equation. The associated correlation or diusion matrix for
Gaussian Markov process can be given by a Hermition matrix with components
hBi(t)B
j
(t
0)i = Dij (t− t
0) (5)
where i (j) runs for states ’+’ or ’-’.
Let us now concentrate on the situation of instability. For innite system one can nd a
range of  , for which all the modes are unstable, the associated the frequencies !(= −iΩ)
are purely an imaginary numbers and beyond that range all modes are stable. Therefore, in
course of time the fate of the system is dictated by the evolution of few such most unstable
modes. The probabilistic evolution of A (see eq.(4)) can be described as well in terms of
Fokker-Planck equation. The evolution equations for rst and second moment related to it
can be written as [9]
dA
dt
=
0@Ω 0
0 −Ω
1A hAi(t) (6)
dij (t)i
dt
= Ωil
lj
 (t) + Ω
jm
 
mi
 (t) + 2D
ij
 (7)
where ij is a symmetric matrix. The initial condition i.e at t= 0, d=dt = 0 ( =
P
i>j 
ij
 )
satises if D++ = D
−−
 = −D
+−
 . These equations are the same as those derived in Ref [8],
the solutions of which provide a good t to the simulation result [10].
Within the framework of Langevin or FP equation, further study can be directed to
the non-linear regime of the dynamics by introducing non-linearity in the drift term. Since
−dU=dA correspond the drift coecient of FP equation this can be done by adding a
6
quartic term to the associated quadratic potential. This stochastic model of non-linear
Langevin equation, is rich in physics capable of explaining the general behaviour of the
system in the presence of instability far from the equilibrium and onset of macroscopic
structure [11]. The presence of the quartic term prevents an indenite fall of a system and
drive it towards any one of the two newly appeared stable xed points which correspond
to the minima of the dynamically generated cusps. The co-operative behaviour or the
synergism of nonlinearity and random force simulate this typical feature i.e. slowing down
the fluctuation near dynamical phase transition. The fluctuation deviate gradually from
its initial Gaussian prole as the system approaches to the second or scaling regime. The
modied equation for second moment with positive value of Ω is given by
d
dt
++ (t) = 2fΩ − g
++
 (t)g
++
 + 2D
++
 (8)
which provides a scaling solution
++ (t) =
Ω
g

1 + 
;  =
g
Ω
(++ (0) +
D++
Ω
)e2Ωt (9)
with g > 0. However, in the derivation of Eq. (8) a physical approximation i.e. A++3 (t) =
++ (t)A
++
 (t) is made [11]. The time needed to reach the saturation for a unstable mode 
is given by
t0 =
1
2Ω
log[
g
Ω
f++ (0) +
D++
Ω
g]−1 (10)
In Fig. 1. the open diamonds represents the time evolution of k for unstable modes
with values of kx ranging from 0.3 to 0.6 with ky = 0. For direct comparison to earlier
results [3] we plot the function L2k (see Fig. 1.) so that the unit becomes fm−2. The
solution of Eqs. (7) (valid for linear regime of the dynamics) are shown by dotted curves.
The solid curves represent the solution of non-linear generalization so that the evolution of
++ component is given by Eq. (9). For these solutions we use the initial condition i.e. at
t = 0, hAi =  = d=dt = 0. The general agreement of our t to the simulation result
is quite good. The saturation time t0k for dierent modes are shown in the gure. The
7
most unstable mode i.e. the mode having minimum growth constant tk (= Ω
−1
k ) saturate
earliest and the time t0k gradualy increses as the tk increases. For some cases, the linear
regime of the dynamics persists even at time t = 120 fm/c. Let us consider the evolution
of density (x; y; t) in co-ordinate space. In earlier time up to  75fm/c when spatial
average of
q
h()2i=hi reaches the value 1 we observe a considerable fluctuation in  in
the individual samples. However, the sample average density i.e. h(x; y; t)i is found to
be almost uniform with  3% fluctuation from its initial value of 0.5 which is essentially
the mean (or single) trajectory result valid in the linear regime of the dynamics. As the
time increases the non-linear eect makes the situation dierent. We see structure even in
the sample average distribution of density which develops slowly in a coherent manner. In
Fig. 2 we plot h(x; y; t)i at a time t = 150fm/c. This observed structure almost remains
steady with increasing time, which corresponds to a equilibrium or at least a metastable
state. The fluctuation is seen to be  20% with respect to the initial uniform state. The
appearance of such spatial inhomogeneity in the average density is in conformation with
that of a mean trajectory calculation reported in Ref. [12]. Though, the fluctuation in latter
case is larger than what we observe here. This may be attributed to the dierent initial and
also boundary conditions used in these two cases. However, large scale fluctuation exists in
individual samples, as evident from the attainment of a non-zero steady value of k in the
present formalism.
In conclusion, using a recently proposed simulation scheme for inclusion of fluctuation
in BUU dynamics we study the problem of spinodal decomposition in nuclear matter. Our
work is mainly devoted to the study of the large time behaviour of system. To analyize
the simulation result we apply the well known Suzuki’s model of dynamical phase transition
successfully in the context of spinodal decomposition. Accordingly, the evolution of the
system is guided by the co-operative eect of non-linearity and fluctuation in a universal
manner. To make our stand clear, we would like to mention a recent work done by Baldo et
al. [13] where the evolution of fluctuation is studied in the framework of Vlasov equation. In
8
the simulation work they studied the growth of an initial disturbance imposed on the uniform
nuclear matter. Without being saturated, the k overshoots very fast from its linear trends
(in log scale). It will be relevant here to note that the nite collision rate, no matter how
small it be, changes the nature of the solutions describing a diusive process as modeled
by the Langevin equation. Therefore, the non-diusive behaviour of fluctuation, although
exhibits some non-standard evolution pattern, but does not indicate conclusively that the
dynamics of fragmentation is dominantly non-linear. However, similar such studies facilitate
us to extract the growth constant tk, (the RPA frequency) of the initial state [10] and also
allow a further scope to check the consistency of our work. The values of tk indicated in
Fig. 1. provide a good t to the dispersion relation shown in Fig 3. of Ref. [10].
9
REFERENCES
[1] C.J. Pethick and D.G. Ravenhall, Nucl. Phys. A471, 19c (1987); R. Donangelo, C.O.
Dorso and H.D. Marta, Phys. Lett. B 263, 19 (1990); R. Donangelo, A. Romanelli and
A.C. Sicardi Schino, ibid., 342 (1990).
[2] D. Kiderlen and H. Hofmann, Phys. Lett. B 332, 8 (1994); S. Ayik, M. Colonna and
Ph. Chomaz, ibid. 353, 417 (1995).
[3] G.F. Burgio, Ph. Chomaz and J. Randrup, Phys. Rev. Lett. 69, 885 (1992); M. Colonna,
Ph. Chomaz, A. Guarnera and B. Jacqot, Phys. Rev. C 51, 2671 (1995).
[4] S. Ayik and C. Gregoire, Nucl. Phys. A513, 187 (1990); J. Randrup and B. Remaud,
ibid. A514, 339 (1990).
[5] G.F. Burgio, Ph. Chomaz and J. Randrup, Nucl. Phys. A529, 157 (1991); M. Colonna,
G.F. Burgio, Ph. Chomaz, M. Di Toro and J. Randrup, Phys. Rev. C 47, 1395 (1993);
G. F. Burgio, Ph. Chomaz and J. Randrup, Phys. Rev. Lett. 73, 3512 (1994).
[6] E. Suraud, S. Ayik, M. Belkacem and J. Stryjawski, Nucl. Phys. A542, 141 (1992).
[7] S. Chattopadhyay, Phys. Rev. C 52, R480 (1995).
[8] M. Colonna, Ph. Chomaz and J. Randrup, Nucl. Phys. 567, 637 (1994); M. Colonna
and Ph. Chomaz, Phys. Rev. C 49, 1908 (1994).
[9] H. Risken, The Fokker-Planck equation, springer series in synergetics, edited by H.
Haken 18, (1984).
[10] G.F. Burgio, Ph. Chomaz, M. Colonna and J. Randrup, Nucl. Phys. A581, 356 (1995)
[11] M. Suzuki, in Order and Fluctuations in Equilibrium and Nonequilibrium Statistical
Mechanics: XVIIthe Solvay conference on physics, edited by G. Nicolis, G. Dewel and
J.W. Ternur, (Wiely; 1981); Prog. Theo. Phys. suppl. 64, 402 (1978).
10
[12] L.G. Moretto, Kin Tso, N. Colonna and G.J. Wozniak, Phys. Rev. Lett. 69, 1884 (1992);
W. Bauer, G.F. Bertsch and H. Schulz, ibid., 1888 (1992).
[13] M. Baldo, G.F. Burgio and A. Rapisarda, Nucl. Phys. A583, 343c (1995).
11
Figure Captions
Fig.1 The fluctuation coecient L2k versus time t for dierent values of unstable mode k
(kx; ky) are shown. The open diamonds correspond to simulation results in steps of
10 fm/c. The dashed curves represent the solution of Eq.(7) while solid curves show
the nonlinear generalization of the same. The open circles represent the simulation
result of a typical stable mode with (kx = 0:8 fm
−1, ky = 0) which exhibits an early
saturation.
Fig.2 The sample average density prole < (x; y; t) > at time t= 150 fm/c versus the
position (x, y) is shown.
12


Spinodal instabilities within BUU approach

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.262.231

Spinodal instabilities within BUU approach

Abstract

Similar works

Full text

Available Versions

Crossref

CERN Document Server