Non-Gaussian Fluctuation and Non-Markovian Effect in the Nuclear Fusion Process: Langevin Dynamics Emerging from Quantum Molecular Dynamics Simulations
Kai Wen, Fumihiko Sakata, Zhu-Xia Li, Xi-Zhen Wu, Ying-Xun Zhang, Shan-Gui Zhou
aa r X i v : . [ nu c l - t h ] J un Non-Gaussian Fluctuation and Non-Markovian Effect in the Nuclear Fusion Process:Langevin Dynamics Emerging from Quantum Molecular Dynamics Simulations
Kai Wen, Fumihiko Sakata,
2, 1
Zhu-Xia Li, Xi-Zhen Wu, Ying-Xun Zhang, and Shan-Gui Zhou
1, 4, ∗ State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China Institute of Applied Beam Science, Graduate School of Science and Technology, Ibaraki University, Mito 310-8512, Japan China Institute of Atomic Energy, Beijing 102413, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China (Dated: June 18, 2018)Macroscopic parameters as well as precise information on the random force characterizing theLangevin type description of the nuclear fusion process around the Coulomb barrier are extractedfrom the microscopic dynamics of individual nucleons by exploiting the numerical simulation of theimproved quantum molecular dynamics. It turns out that the dissipation dynamics of the relativemotion between two fusing nuclei is caused by a non-Gaussian distribution of the random force. Wefind that the friction coefficient as well as the time correlation function of the random force takesparticularly large values in a region a little bit inside of the Coulomb barrier. A clear non-Markovianeffect is observed in the time correlation function of the random force. It is further shown that anemergent dynamics of the fusion process can be described by the generalized Langevin equation withmemory effects by appropriately incorporating the microscopic information of individual nucleonsthrough the random force and its time correlation function.
PACS numbers: 24.60.-k, 24.10.Lx, 25.60.Pj, 25.70.Lm
The fusion of two nuclei is one of the major non-equilibrium processes in low energy nuclear reactionswhere the fluctuation and dissipation play importantroles. It is rather difficult to describe the fusion pro-cess without significant simplifications. Under variousassumptions, several macroscopic transport models havebeen introduced to evaluate the formation of a compoundnucleus in heavy-ion fusion reactions [1, 2]. However, themicroscopic mechanism on how two colliding nuclei fuse,especially how the relevant kinetic energy dissipates intothe intrinsic degrees of freedom (DoF), remains a subjectrequiring further research.On the other hand, it is becoming feasible to get var-ious information out of microscopic numerical simula-tions, like time-dependent Hartree-Fock (TDHF) theo-ries [3–7], the many-body correlation transport (MBCT)theory [8], the quantum molecular dynamics (QMD) [9],the antisymmetrized molecular dynamics [10], and thefermion molecular dynamics [11]. The TDHF theory ismainly based on the mean-field concept; in TDHF, fluc-tuations of collective variables are considerably underes-timated. Much effort has been made to give a beyond-mean-field description of fluctuations [12]. The n -bodycorrelations are incorporated in the MBCT theory [8]which has only been used in very light systems [13].The QMD is a microscopic dynamical n -body the-ory which was successfully used in intermediate-energyheavy-ion collisions (HIC) [9]. An improved QMD(ImQMD) has been developed in order to extend the ap-plication of QMD to low-energy HICs near the Coulombbarrier [14]. A series of improvements were made in theImQMD; in particular, by using the phase space occupa-tion constraint method [15], the fermionic properties of nucleons is remedied, which is important for low-energycollisions. Making full use of the microscopic informa-tion provided by ImQMD simulations, in this Letter, wetry to understand how the macroscopic fusion dynamicsemerges out of the microscopic one.We focus on a simplest case of symmetric fusion pro-cess with the impact parameter equal to zero. In thiscase, the system can be divided into the left- and right-half parts instead of a projectile and a target [16]. Therelative motion between two centers of mass (CoM) ofthe left and right parts is chosen as the relevant DoF tobe described by the Langevin equation. Our analysis islimited in a stage where the relative distance R is muchlarger than its width.The one-dimensional generalized Langevin equationwith memory effects reads [17–19] du ( t ) dt = − Z t −∞ γ ( t − t ′ ) u ( t ′ ) dt ′ + 1 µ δF ( t ) − µ dV ( R ) dR , (1)where u ( t ) is the relative velocity between the two parts, δF ( t ) the random force felt by either part, µ the reducedmass of the system, γ ( t − t ′ ) the friction kernel, and V ( R )the potential for the relative motion.In the ImQMD model [14], a trial wave function is re-stricted within a parameter space { r j , p j } , where r j and p j are mean values of position and momentum opera-tors of the j th nucleon which is expressed by a Gaussianwave packet. The time evolution of the trial wave func-tion under an effective potential is governed by the time-dependent variational principle [9–11]. An expectationvalue of the Hamiltonian is given by using an improvedSkyrme potential energy density functional. In this Let-ter, we concentrate on head-on collisions of Zr+ Zr.Ten thousand collision events were simulated. Each sim-ulation is started at R = R = 30 fm and with an incidentenergy E = 195 MeV. Numerical details can be found inRefs. [20].The potential for the relative motion is defined as, V ( R ) = E tot ( R ) − E left ( R ) − E right ( R ) , (2)where E tot , E left , and E right represent the energy of thesystem and those of the left and right parts, respectively;each of which consists of the kinetic energy, the nuclearand the Coulomb potential energies. The potential V ( R )is shown in Figs. 1 and 3. The TDHF has also been usedto extract microscopic interaction potentials between twonuclei [5, 21] which show similar features as those fromthe ImQMD simulations presented here and in Refs. [22].The random force or the fluctuation of force in the i thevent is defined as δF ( x ) i ≡ F i ( x ) − h F ( x ) i , x = t or R, (3)where F i ( x ) ≡ P Aj =1 f ji ( x ) denotes the total force actingon the left (right) part of the system in the i th event, h F ( x ) i ≡ n P ni =1 F i ( x ) the mean value, and f ji ( x ) theforce on the j th nucleon in the left (right) part. Here A means the number of nucleons contained in the left(right) part and n denotes the total number of events.In Eq. (3) and hereafter, h Q i denotes an average of Q over all events. For low-energy collisions, the fluctua-tion mainly stems from the initialization of each event inwhich the position and the momentum of each particleare chosen randomly under certain conditions. With timethis initial fluctuation propagates and is not smoothedout because in QMD a many-body rather than a meanfield problem is solved [9].Distributions of δF ( R ) at several distances are shownin Fig. 1. The random force at R ( t = 0) = R showsa Gaussian distribution with the full width at half max-imum (FWHM) Γ ≈ R ≈
18 fm, δF has a Gaussian distribution withΓ ≈ . R ≈ . δF ( R ), one may divide the whole process into threeregions. Region 1 represents an approaching phase upto the touching point: The distribution has a Gaussianform with a rather narrow width. Region 2 is from thetouching point to the barrier top: A non-Gaussian shapeappears. Region 3 is from just inside the barrier top tothe fusing phase: The distribution of δF ( R ) has againa Gaussian shape with Γ ≈
15 MeV/fm which is almosttwo orders of magnitude larger than that in Region 1.To make clear what happens in Region 2, we divide thedistribution of δF ( R ) into a symmetric Gaussian and anasymmetric tail parts as is shown in Fig 2. The width of -3 -2 -1 0 1 2 30.00.51.01.52.0 P r ob a b ilit y d i s t r i bu ti on R = 18 fm -3 -2 -1 0 1 2 30.00.51.01.5
R = 15 fm -12 -8 -4 0 4 8 120.00.10.20.30.40.50.6
R = 13.5 fm -30 -20 -10 0 10 200.000.040.080.12
R = 12.5 fm -40 -20 0 20 400.000.010.020.030.040.050.06
R = 11 fm
Random force (MeV/fm) -40 -20 0 20 400.000.010.020.030.040.050.06
R = 8.5 fm
FIG. 1. (color online) Distributions of the random force δF ( R ). Each inset shows the potential V ( R ) with the bluedot representing the position where the system locates. Thecontour plots display the nucleon density distribution of thesystem. -10 0 10-505 F o r ce ( M e V / f m ) z (fm) -505 -10 0 10 F o r ce ( M e V / f m ) z (fm) P r ob a b ilit y d i s t r i bu ti on Random force (MeV/fm)
R = 13.5 fm (a) (b)
FIG. 2. (color online) Distribution of the random force δF ( R )at R = 13.5 fm which is divided into the symmetric Gaus-sian (dark blue) and asymmetric tail (light blue) parts. Twotypical events are shown in the inset: The abscissa and theordinate express relative position z of each nucleon and theforce it feels in the z direction. the Gaussian part is of the same order of magnitude asthat in Region 1. The detailed structure of the randomforce can be studied by examining the strength and direc-tion of the force felt by each nucleon. One typical eventin the symmetric part is shown in Fig. 2(a): All nucleonsare well divided into two separated groups expressing theprojectile and the target, respectively. Moreover, eachnucleon locating in the left side of each nucleus feels aforce toward the right (positive value), and that in theright side feels a force toward the left (negative value), soas to keep a stable mean-field. The resultant force madeby all nucleons in each nucleus is almost zero. Namely,the intrinsic structure of two fusing nuclei is kept almostunchanged, so is the width of the random force. This sit-uation persists in events which belong to the symmetricGaussian part in Region 2 and in all those in Region 1.A typical event in the asymmetric tail is shown inFig. 2(b). Nucleons are roughly divided into two groupssurrounded by solid lines. However, there appears a smallthird group within the dashed line. Since a few points inthe negative (positive) force region express a set of nu-cleons which escape from the left (right) nucleus, and arebeing absorbed by the right (left) nucleus, a resultantforce made by these nucleons gives a large right(left)-directed component to the random force. These trans-ferred nucleons move in an average potential formed byboth the projectile and the target; they play a role toopen a window .When the two nuclei come much closer, there occurmore events which have more nucleons in the third group.Meanwhile, the other two groups, originating from theprojectile and target, become closer to each other. Conse-quently, the asymmetric tail in the distribution of δF ( R )becomes larger. At the border between Regions 2 and3, it becomes very difficult to distinguish an event inthe center part of the distribution from that in the tailpart and all events are absorbed into a widely spreadingGaussian distribution.From above discussions, it is concluded that the mainmicroscopic origin of the random force, i.e., a two ordersof magnitude enhancement of the random force is gen-erated by individual nucleons in the third group. Thesenucleons also result in the abnormal behavior in the dis-tribution of δF ( R ), i.e., the long tail in Region 2 and amuch larger width in Region 3 compared to Region 1.Next let us extract information for the macroscopicdynamics out of microscopic simulations. Assumingthat the work done by the friction force is completelyconverted into the intrinsic energy E intr ( R ), one getsthe friction coefficient γ ( R ) from the Rayleigh for-mula [6, 16, 19], γ ( R ) ≡ h F fric ( R ) ih P i R , (4)with F fric ( R ) ≡ dE intr ( R ) /dR , E intr ( R ) ≡ E tot ( R ) − E coll ( R ), and E coll ( R ) = P / µ + V ( R ). P denotesthe relative momentum between two CoMs and its meanvalue h P i R at a given R is defined as h P i R ≡ n n X i =1 P i ( t i ) | { t i | R i ( t i )= R } , (5)where P i ( t ) and R i ( t ) are the momentum and coordinateof the i -th event at time t and the following correspon-dence is used: For each event i , a time t i is chosen in (R) F(R) F(R) tot
F(R) F(R) sym
V (R) F ( R ) F ( R ) a nd ( R ) V ( R ) ( M e V ) R (fm)
FIG. 3. (color online) The correlation function h δF ( R ) δF ( R ) i tot [red dots, in (MeV/fm) ] and the frictioncoefficient γ ( R ) (blue squares, in 0 . c /fm). The greyline shows the potential V ( R ). Pink diamonds represent h δF ( R ) δF ( R ) i sym calculated by eliminating events in theasymmetric tail. such a way that the relative distance takes a given value R , i.e., R i ( t i ) = R . A R -dependent correlation functionis defined as h δF ( R ) δF ( R ) i ≡ n n X i =1 δF i ( t i ) δF i ( t i ) | { t i | R i ( t i )= R } . (6)Figure 3 shows the correlation function h δF ( R ) δF ( R ) i and the friction coefficient γ ( R ) which play decisive rolesin the macroscopic description of dissipation phenom-ena. As is seen from Fig. 3, h δF ( R ) δF ( R ) i and γ ( R )have similar shapes and their peaks locate at similar R .The friction coefficient of the fusion process induced bya head-on collision extracted from TDHF calculationsshows similar strong peak structure. As the incident en-ergy E increases, the shape of the curve γ ( R ) ∼ R maychange. When E is high enough, γ ( R ) increases gradu-ally with decreasing R [6, 16, 23].To explore more deeply the dynamical relation betweenthe microscopic motion of individual nucleons and themacroscopic dissipative motion, in Fig. 4 we show thetime correlation function of the random force σ ( R, τ )which is defined as σ ( R, τ ) ≡ n n X i =1 δF i ( t i ) δF i ( t i − τ ) | { t i | R ( t i )= R } . (7)In Fig. 4 one clearly finds the non-Markovian effect.Especially when R = 12 ∼
10 fm, it is important to takeaccount of memory effects generated by the microscopicmotion of nucleons when one tries to properly evaluatesmacroscopic effects of the dissipation. Starting from thegeneralized Langevin equation (1) with memory effects,one gets a generalized fluctuation-dissipation (GFD) re-lation h δF ( t ) δF ( t − t ′ ) i = µT γ ( t − t ′ ) which properly
20 40 60 80 100 120 140 1608101214 R (f m ) (fm/c) (MeV/fm) FIG. 4. (color online) Time correlation function σ ( r, τ ) (7). T symMarkov T totMarkov ( E intr / a ) (cid:215)0.1 T ( M e V ) (a)(b) T sym non- Markov T tot non- Markov ( E intr / a ) R (fm)
FIG. 5. (color online) Effective temperature in the Markovianlimit (a) and the Non-Markovian one (b). The blue line shows p E intr /a with a = A total / takes account of the time correlation of the random force.There are many ways to define the temperature for com-pound nuclei (see, e.g., Ref. [24]). Here we define aneffective temperature for colliding systems by applyingthe GFD relation, T non − Markov ( R ) = 1 µγ ( R ) Z ∞ dτ σ ( R, τ ) . (8)The effective temperature T totnon − Markov as well as theone from the Markovian approximation T totMarkov areshown in Fig. 5. p E intr /a representing the temperatureof a compound nucleus in the Fermi gas model is alsoshown as a reference. Although T totnon − Markov and T totMarkov differ by one order of magnitude, they both show a peakaround the range where the asymmetric tail appears inthe distribution of δF ( R ). These peaks are related to thefact that the relative motion for events in the asymmet-ric tail part of the δF ( R ) distribution is strongly affectedby a few transferred nucleons between two fusing nuclei,i.e., by those in the third group of Fig. 2(b). The macro-scopic dynamics of the relative motion described by theone-dimensional Langevin equation (1) is not appropri- ate in Region 2. In other words, the appearance of thenon-Gaussian distributed random force indicates a neces-sity of introducing a new macroscopic DoF. Whether ornot this new DoF may be related to the formation of aneck is an open question [1, 25].After eliminating the events in the asymmetric tail inthe distribution of δF ( R ), one gets effective temperatures T symnon − Markov and T symMarkov which are depicted in Fig. 5.The correlation function h δF ( R ) δF ( R ) i after eliminatingthe asymmetric tail is also shown in Fig. 3. T symnon − Markov shows a consistent feature with p E intr /a in Region 3.While T symMarkov is by an order of magnitude smaller than T symnon − Markov . That is, the amount of energy dissipatedfrom the relative motion into the intrinsic DoFs couldbe more properly described by the generalized Langevinequation with memory effects.When the incident energy E is far above the Coulombbarrier, the non-Gaussian fluctuation and the non-Markovian effect become less pronounced [23]. It will beinteresting to study the dependence of the non-Gaussianfluctuation and the non-Markovian effect on E as wellas the impact parameter and the reaction system. Thespin-orbit coupling is important to properly reproducethe dissipation in heavy-ion fusion reactions [26]; e.g.,the so-called “fusion-window” problem was solved in thefirst quantitative TDHF calculations with the inclusionof the spin-orbit interaction [26]. One may expect moredissipations if the spin-orbit coupling effects are includedin the ImQMD simulations.In summary, we have discussed the generalizedLangevin dynamics with memory effects by using boththe macroscopic and microscopic information extractedfrom ImQMD simulations for the fusion process aroundthe Coulomb barrier. It is found that the dissipationdynamics of the relative motion between two fusing nu-clei is associated with non-Gaussian distributions of therandom force. In addition to the macroscopic informa-tion like the friction coefficient and the potential for therelative motion, the microscopic information of the ran-dom force as well as of its time correlation function and aproper treatment of the non-Markovian (memory) effectin the Langevin dynamics are decisive for the dynamicsof emergence in the nuclear dissipative fusion motion.We thank G. Adamian, P. Danielewicz, Q. F. Li, B.N. Lu, R. Shi, S. J. Wang, Y. T. Wang, Z. H. Zhang,E. G. Zhao, K. Zhao, and Y. Z. Zhuo for helpful discus-sions. F.S. appreciates the support by Chinese Academyof Sciences (CAS) Visiting Professorship for Senior Inter-national Scientists (Grant No. 2011T1J27). This workhas been partly supported by MOST of China (973 Pro-gram with Grant No. 2013CB834400), NSF of China(Grants No. 11005155, No. 11075215, No. 11121403,No. 11120101005, No. 11275052, and No. 11275248),and Knowledge Innovation Project of CAS (Grant No.KJCX2-EW-N01). The results described in this paperare obtained on the ScGrid of Supercomputing Center,Computer Network Information Center of CAS. ∗ [email protected][1] C. Shen, G. Kosenko, and Y. Abe, Phys. Rev. C ,061602(R) (2002); V. I. Zagrebaev, A. V. Karpov, andW. Greiner, Phys. Rev. C , 014608 (2012); Y. Aritomo,K. Hagino, K. Nishio, and S. Chiba, Phys. Rev. C ,044614 (2012); K. Siwek-Wilczynska, T. Cap, M. Kowal,A. Sobiczewski, and J. Wilczynski, Phys. Rev. C ,014611 (2012); Z.-H. Liu and J.-D. Bao, Phys. Rev. C , 034616 (2013).[2] G. G. Adamian, N. V. Antonenko, W. Scheid, andV. V. Volkov, Nucl. Phys. A , 409 (1998); J.-Q.Li, Z.-Q. Feng, Z.-G. Gan, X.-H. Zhou, H.-F. Zhang,and W. Scheid, Nucl. Phys. A , 353c (2010); Z.-G. Gan, X.-H. Zhou, M.-H. Huang, Z.-Q. Feng, andJ.-Q. Li, Sci. China-Phys. Mech. Astron.
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